Representation Space

Vision-language navigation requires aligning language with visual observations while maintaining spatial understanding over time. Geometry foundation models (GFMs) expose intermediate representations throughout their hierarchy, but how navigation policies should use these features and retain historical geometric evidence remains unresolved. We introduce \method{}, a streaming VLN framework that addresses these questions across \textbf{representation depth} and \textbf{navigation time}. Hierarchical GFM--VLM fusion couples earlier, intermediate, and later GFM representations to successive policy stages instead of repeatedly injecting a terminal feature. Navigation-aware GFM memory retains historical VGGT global-attention KV states according to instruction relevance, geometric confidence, and transition novelty under a bounded per-layer budget. Retained states provide geometric context for subsequent observations before fusion with the policy. Across R2R-CE and RxR-CE, \method{} achieves strong performance using a single RGB stream without additional navigation-specific external data. Controlled ablations show that multi-depth coupling substantially outperforms repeated terminal-feature injection at matched fusion locations. Bounded navigation-aware retention preserves navigation performance while considerably reducing GFM-KV memory relative to larger-memory temporal retention. These findings support jointly examining the geometric representations exposed to the policy and the historical evidence retained for future inference. Code will be released upon acceptance at https://humanoid-research.github.io/adageovln/.

How neural representations preserve the structure of input changes connects representation analysis with internal intervention. We study operable representational content through compatible actions of reference transformations on neural features. We characterise when a transformation descends through an encoder, and give a linear setting in which the defect is governed by the transformation's demand for discarded information, measured in the metric the representation induces. On a rectifier the failure to realise a transformation has two distinguishable sources --- what the source region has already made unrecoverable, and what it costs to satisfy every region the transformation visits with one operator --- and for a \textit{measured} harmonic carrier the same question has a closed answer: a linear realisation exists exactly when the retained harmonic blocks are invariant under the action. Using colour as the in-depth instance, we find that hue orbits in frozen visual features concentrate 84--88% of their energy in the first two harmonics with rotation planes shared across shapes, that this organisation is substantially inherited from input and architecture and is reshaped by training and depth, and that the measured structure supports prediction, transport from new starting states, and composition --- with global and local realisations differing sharply in which they achieve. Guided by the measurements, we construct a compact interface whose rotation action is fixed by the structure and never fitted: it reads hue zero-shot at 3.4$^\circ$ median error on unseen shapes. Theory, structural measurement, and construction together establish transformation laws as a concrete object connecting the understanding of neural representations to their design.

We characterize when compressed vector representations admit exact linear or affine readouts of a finite lexicon's truth conditions: one fixed map per predicate, sending each entity vector to the corresponding truth vector. A necessary and sufficient row-space condition determines existence; the augmented truth matrix has rank r, giving minimum dimension r in the linear case, and r-1 in the affine. Exact readouts return values in a shared truth basis on which Boolean connectives act unchanged; separability alone requires an intervening threshold. For binary relations, exact bilinear readout of identity or strict total order requires linearly independent entity vectors. Experiments with GloVe and word2vec distinguish exact affine recovery, linear separability, and held-out prediction: most predicates are strictly separable, but none admits an exact affine readout from the pretrained embeddings. Supervised transductive training attains exact affine recovery to numerical precision at every tested dimension meeting the bound. At the embeddings' original dimension, geometries constrained to exact linear recovery retain 98-99 percent of the pretrained variance on the feature norms, and 80-83 percent on the WordNet lexicon.

Few-shot audio classifiers may rely on foreground-background co-occurrences and fail when those correlations shift. On SpurAudio, the resulting representation shift is concentrated and class dependent: for ResNet12, the top 10 percent of channels explain 82.80 percent of the null-corrected shift contribution. We propose SAMPLESELECT, which predicts a fixed-budget feature mask independently for each input while keeping the encoder and source classifier frozen. Training uses differentiable Gumbel Top-k selection with foreground classification and cross-background contrastive losses; inference uses deterministic Top-k masks and support-only linear adaptation. Across ResNet12 and Conv64 in 5-way 1-shot and 5-shot evaluation, SAMPLESELECT gives the best OOD accuracy among the compared methods and improves the matched full-representation control by 4.90-8.38 percentage points. Ablations and representation analyses further support the learned selection mechanism. Code is available at https://github.com/Cross-Innovation-Lab/SAMPLESELECT/

A language model's representation geometry is not predetermined; it evolves as the model runs. A faithful account of that geometry must capture that dynamic process, and so cannot be based solely on model-independent statistics such as co-occurrence. Here we introduce a mean-field analysis of attention. The average attention from one token to another defines a kernel that carries representations layer to layer and can be iterated through the network to model how the geometry is transformed. We condition this average two ways. Conditioned on a whole corpus, the kernel predicts the average-case evolution of representation geometry. Conditioned instead on a single context, it predicts the expected geometry for that context. A head's departure from that prediction, its \emph{mean-field deviation}, isolates the context-specific computation that the mean field misses. Under the corpus-conditional reading, the kernel yields an open-loop model: from the input embeddings and the frozen weights alone, we can iterate the kernel and the model's own MLPs over token representations, never consulting a measured deviation at any layer. The resulting prediction is highly accurate. In early training the model and its corpus mean field are indistinguishable. Replace every attention head with its mean field, and the substitution leaves the loss on real text unchanged. Around the onset of induction, the two diverge, and the gap widens as representations become contextualized. Under the context-conditional reading, deviation from the mean field is a task-agnostic measure of context-specific computation. The residual decomposes additively into unusual attention routing and contextualization of the transported values. Across controlled induction and few-shot settings, greater deviation tracks greater reliance on in-context information.

Transformer representations evolve through learned additive transformations that either preserve their current direction or redirect it. We study this evolution as a functional geometry, decomposing learned updates into parallel and perpendicular components. Across pretrained models, we find substantial parallel components beyond the residual identity path. We then apply the decomposition in two spaces: to attention and MLP updates relative to the hidden state, and to attention value aggregation relative to the current token's value. Targeted edits reveal a strongly space-dependent asymmetry: exclude-self value-space parallel manipulation is markedly more robust than residual-space and perpendicular counterparts, preserving the direct self message while scaling only the non-self aggregate. The same decomposition gives a component-resolved description of compression-induced update error: perpendicular error separates compression methods more clearly than parallel error. Extensive experiments further demonstrate that full-aggregate parallel suppression during from-scratch pretraining lowers validation-loss trajectories and improves downstream averages, with the value-space variant strongest. Together, these results connect representation geometry to editing robustness, compression diagnosis, and training-time intervention. Code is available in the \href{https://github.com/Shwai-He/Transformer-Geometry}{project repository}.

Mixed-curvature representation learning seeks to capture rich geometric structures that cannot be adequately modeled by a single curvature regime. Existing approaches largely rely on product manifolds, which require manually specifying how different curvature spaces are combined and separate their curvature contributions across factors. We introduce the $\mathbb{SL}(n)$ space, a representation geometry defined by the simple $\det(A)=1$ constraint and a left invariant Schatten-$p$ Finsler structure. Despite this minimal construction, $\mathbb{SL}(n)$ exhibits pointwise negative, zero, and positive flag curvature around a common flagpole, while its mixed-curvature and curvature-coupling capacities are asymptotically maximal relative to the intrinsic geometric upper bound. Beyond geometry, its noncommutative group structure provides inherent order sensitivity, and its non-nilpotent Lie algebra admits nonzero nested Lie brackets at arbitrary depth, enabling deep order-aware composition. Empirically, $\mathbb{SL}(n)$ consistently outperforms a broad range of representation manifold baselines across graph benchmarks at different scales. It reduces average distortion over the strongest baselines by $44.3\%$ on KEGG and $40.5\%$ on HumanCyc, and improves Hits@20 by $42.8\%$ on OGBL-PPA. Experiments on Flickr30k-Order further support its ability to capture higher order dependencies from ordered composition. Together, these results show how a seemingly simple structural constraint can yield unexpectedly rich geometry, capacity, and composition within a unified representation space.

Low-Rank Adaptation (LoRA) is an effective approach for adapting large pretrained models by learning low-rank weight updates. In practice, the LoRA rank is used to control an adapter's parameter budget and representational capacity. We show that this view is incomplete: while the nominal rank determines the representational capacity, the optimizer shapes how much of that capacity is used in the induced weight-space updates. In a case study of GPT-2 adaptation with LoRA, we observe a strong rank-dependent optimizer effect. Despite using the same nominal rank, AdamW often produces per-step updates with concentrated singular spectra and low effective rank, whereas Muon uses a richer set of directions and benefits more consistently from increasing LoRA rank. These observations motivate ISO-LoRA, an optimizer that couples the LoRA factor updates through spectral descent on the induced tangent perturbation in weight space. ISO-LoRA promotes updates that distribute energy more evenly across singular directions, improving rank utilization while preserving compatibility with the LoRA parameterization. We complement this design with theoretical guarantees showing that ISO-LoRA can achieve higher effective rank than standard factor-wise optimizers through a one-step analysis under a stylized spiked-gradient model. We validate this design on language-model adaptation across 0.1B-7B-parameter models, where ISO-LoRA improves effective rank and downstream performance, with the strongest gains at moderate-to-large LoRA ranks. Our results highlight rank utilization as a key factor in LoRA optimization and suggest that optimizer design offers an important path toward stronger parameter-efficient adaptation.

Matrix-valued memories make rank the natural budget of a learned representation: the number of independent directions a state spans bounds what it can bind, compose, and track. We report causal evidence, on a group-composition testbed trained under a hard single-state bottleneck with a fixed decoder that cannot launder rank, that gradient descent recruits precisely the rank the task's algebra demands. A companion paper [Larson, 2026a] establishes the analogous recruitment and causal necessity pattern on a $K$-pair associative-binding testbed, where exact recovery provably requires state rank at least $K$; this paper inherits that instrument and extends the rank law from a scalar capacity bound to a representation-theoretic one. We train toward chosen minimal faithful reference representations embedded in larger matrices. On group-composition state tracking over five finite groups spanning the solvable/non-solvable divide, the recruited rank equals the group's minimal faithful real representation dimension $d_{\min}$ (Spearman $\rho = 0.9747$, the design's tie-capped maximum), the dimension-matched solvable/non-solvable pair $S_4$/$A_5$ is statistically equivalent under a pre-registered test, and a pre-registered force-rank test separates a guaranteed similarity ceiling from empirical recovery at the target dimension: one rank below $d_{\min}$, cosine similarity is capped by the target's tied unit spectrum at $\sqrt{(d_{\min}{-}1)/d_{\min}} \le 0.894$, below the $0.9$ threshold in every group by construction, with observed cells at 86-95% (mean 91%) of that ceiling; at $d_{\min}$, not guaranteed a priori, recovery clears the pre-registered anchor-relative bar at four seeds per group in all five groups. Within this testbed, measured effective rank tracks representation dimension; the matched-dimension $S_4$/$A_5$ comparison establishes equivalence within the pre-registered tolerance.

Latent embeddings have become a central data abstraction in modern machine learning, especially in biomedicine, where foundation models are increasingly used to encode multimodal data like clinical text, medical images, omics, and physiological signals. However, the utility and value of these representations depends on understanding their quality, structure, and the information they encode. Existing analysis workflows for evaluating representations remain fragmented across custom scripts, isolated metrics, and most importantly lack multimodal analysis, limiting accessibility and reproducibility. We present LatentVerse, a representation analysis resource that combines a web-based visual analytics platform for accessible, report-driven exploration with a command-line interface for scalable technical workflows. LatentVerse unifies diagnostics for various representation quality metrics and extends to multimodal settings by decomposing embeddings into shared and modality-specific components. We evaluate LatentVerse through controlled unimodal and multimodal simulations, discovery-oriented analyses on real biomedical embeddings, and a user study across diverse use cases. By supporting thorough and interpretable evaluation of latent spaces, LatentVerse makes foundation model representations more understandable in biomedical and data science applications.

Existing bias auditing methods typically rely on model outputs, requiring costly benchmarks or judge models and potentially missing internal shifts that never appear in generated text. We propose a reference-based method that audits bias in hidden-state representations across related model variants, for example before and after fine-tuning. Because fine-tuning reshapes representation geometry, absolute hidden states are not directly comparable, so we encode each sentence by its similarities to a fixed set of anchor sentences, yielding relative representations in a shared comparison space. There we measure how target groups shift in their association with positive and negative attributes, a quantity we call the Representational Bias Shift $ΔB$. Across three model families and the WildGuardMix, DecodingTrust and ToxiGen benchmarks, $ΔB$ correlates with output-level bias change in 15 of the 18 settings we test, reaching $|r| = 0.84$ ($p < 0.001$) under full fine-tuning and becoming more model-dependent under parameter-efficient adaptation. Thresholding $ΔB$ detects checkpoints whose bias increased with ROC AUC between $0.65$ and $0.99$, and on WildGuardMix and DecodingTrust it separates them better than a SEAT-based baseline for all three families. $ΔB$ is also stable under changes to the anchor set, attribute sets and target templates. Our method requires no task-specific evaluation data and audits a model in about three minutes, using $3$-$50\times$ less compute than the output-level benchmarks considered here. We view it as complementary to output-based auditing rather than a replacement for it.

Vision transformers (ViTs) have achieved remarkable generalization across visual domains, yet little is known about how they internally represent the structure of the world. To address this gap, we use Cross-Layer Transcoders (CLTs) to read concept circuits from ViTs: directed graphs whose nodes correspond to sparse, interpretable concepts and edges capture concept interactions across layers. Our method yields two complementary views of model behavior. The global concept circuit is input-invariant and can be recovered directly from learned cross-layer weights, exposing the reusable "world knowledge" encoded in the model. The instance concept circuit is input-dependent and identifies the concepts and pathways actually used for a specific prediction, enabling faithful example-level explanations. We demonstrate the utility of concept circuits in three ways: (1) Automatic spurious correlation discovery: leveraging the statistics of our global concept circuits to identify shortcut dependencies within the model. (2) Spurious correlation removal: intervening on the instance concept circuit to steer the model towards correct predictions. Empirical results show that our method outperforms existing counterparts by 11.0% on the Waterbird dataset. (3) Model comparison: contrasting the global concept circuits of different foundation models (e.g., CLIP vs. DINO) to reveal how supervision paradigms shape representational structure. Our code is available at https://github.com/deep-real/VisionCLT

Recent state-space models (SSMs) such as Mamba achieve language modeling performance comparable to transformers despite relying on fundamentally different architectures. This raises an important question: how do these structural differences influence the geometry and functional nature of their internal representations? We study this question through a multi-scale analysis of representations in transformers, SSMs, and hybrid architecture. First, we find that SSMs distribute their representational information evenly across all dimensions, whereas transformer representations are heavily dominated by a single principal direction. By evaluating hybrid architectures, we observe that the representation space becomes increasingly skewed toward a single dominant direction after each attention layer. Next, we explore how the different geometric spread of representations impacts representational capacity through compressibility. Surprisingly, we find that despite their contrasting geometric structures, both architectures exhibit tightly matched effective capacities. We further investigate whether this skewed geometry affects how concepts are encoded. Using rank-constrained probes, we demonstrate that both architectures encode concepts in subspaces of surprisingly similar dimensionality. Furthermore, we demonstrate that the transformers' dominant principal direction does not inherently encode more conceptual information. Finally, we zoom in and examine the alignment between manifolds, either by analyzing representations of specific topics or by looking at the nearest neighborhoods of tokens, and find that they are highly aligned. Ultimately, our analysis suggests that while transformers and SSMs induce different usage of latent space, they display a striking functional convergence at the level of local semantic manifolds.

Comparing intelligent systems under deployment constraints requires more than predictiveaccuracy.This paper develops Compression-Prediction-Resource Intelligence Efficiency (CPR-IE) as a protocol-relative ordering by representational economy, predictive quality, and resourceburden. The analysis separates two questions-how raw resource consumption is represented, andhow the resulting attributes are aggregated. Proportional-increment composition uniquely yieldslogarithmic cumulative burden, and context-independent ratio response yields power responsesto compression, prediction, and burden; with reference normalization the representation is I(C,P,T).We prove Pareto consistency, unit invariance, boundary behavior, trade-off identities, ranking-stability regions, and cross-task aggregation. A translog parent model makes interaction restrictions explicit, and further results establish cardinal and ordinal identification, sub-Gaussianfinite-sample ranking guarantees, robust selection under exponent uncertainty, and deterministicregret bounds. Minimum description length, algorithmic complexity, proper scoring rules, varia-tional inference, and Landauer's principle motivate measurement choices but do not entail theformula. CPR-IE is a constructed efficiency representation, not a universal law or a definition ofintelligence itself.

Synthetic data augmentation has become a common strategy for addressing class imbalance in NLP, but most approaches focus on the quantity and diversity of generated examples rather than their geometric relationship to real training data. We investigate this question in the context of discourse pragmatic function classification, a task where data sparsity is a structural feature rather than a collection artefact. Using 410 manually annotated instances of the English word look drawn from the British National Corpus, spanning four functions: Attention Signal, Directive, Discourse Marker, and Interjection. We generate synthetic training examples with Llama 3.1 and partition them by their cosine distance from real training data in RoBERTa embedding space. We compare six training conditions that differ in the placement of synthetic examples relative to the empirical decision boundary, while holding augmentation quantity constant across conditions. All augmented conditions improve macro F and accuracy over the real only baseline, but core proximal examples (NEAR) yield the largest gains in macro F (0.113), while a distance balanced mix achieves the highest accuracy (0.748). No condition improves AUC, indicating that augmentation shifts the decision boundary rather than improving the model's underlying probability estimates. These findings suggest that where synthetic examples land in representation space matters as much as how many are generated, with implications for low resource pragmatic classification more broadly.

This paper examines the relationship between the parameters of autoencoder models and the statistical properties of the data on which they are trained. Autoencoders are defined as models with an encoder-decoder architecture, trained to reconstruct input data through a compressed latent representation. It is proposed that the model parameters can be viewed as a dense vector representation of the corresponding sample. To test this hypothesis, a theoretical and experimental study is conducted in which a vector representation is formed based on the spectral characteristics of the autoencoder parameter matrices. Theoretical analysis shows that the singular values of the model parameter matrices are related to the eigenvalues of the covariance matrix of the training data, ensuring the transfer of information between the data space and the parameter space. Experimental results on the CIFAR-10 and FashionMNIST datasets confirm that the resulting vector representations allow for a high degree of accuracy in distinguishing between models trained on different data subsets, without resorting to complex vector generation algorithms or using the original samples. These results suggest that the parameters of trained autoencoders can be viewed as sample representations.

Smoothness of a function on the real line is reflected in the decay of its Fourier transform, which suggests that smoothness of a function in $L^2(G)$ for a group $G$ should mean concentration of the Fourier coefficients at low frequency. Such a reading presupposes an ordering of the irreducible representations of $G$, but for non-abelian $G$, no ordering is canonical. Given a symmetric generating set $S$, the Laplacian of the associated Cayley graph is block diagonal over the dual, and we order the irreps by the mean of the eigenvalues in each block. This produces an ordering function $ω:\widehat{G}\to\mathbb{R}$ that depends only on the pair $(G,S)$. This function is bounded between zero and two, vanishing only at the trivial representation and achieving the upper bound exactly when the Cayley graph is bipartite. We then ask how much freedom the construction has. Within the class of operators satisfying natural axioms, the induced orderings are exactly the real functions on the dual vanishing at the trivial representation and agreeing on conjugate pairs, and the orderings coming from inversion orbits of conjugacy classes form a basis for them. We cut the freedom down further by requiring two additional inputs: nonnegativity of the class weights and a declaration of which group elements count as uniform incremental changes, which pins the operator to the Cayley-Laplacian up to positive scale. We observe that the construction persists for compact groups even though the Cayley graph does not, and we extend the theory to finite sets carrying a transitive group action, where the acting group selects which frequencies exist and the generating set orders them. The answer to the title question is therefore that smoothness is a property of a function together with a choice of group and generating set, not of the function alone.

Recent work identifies a mid-depth band of verbalisable, causally potent representations in a standard feedforward transformer --- a functional analogue of a global workspace. Whether the same workspace functionality emerges when depth is implemented through recurrence rather than a stack of distinct layers remains unknown. Looped and depth-recurrent transformers provide a direct test of this question because they reuse the same weights across depth. We extend the Jacobian lens to iterated architectures using a virtual-unrolling adapter. We apply the full workspace suite --- lens fitting, readout, and eleven causal experiment families --- to Ouro-2.6B (48 layers looped 4 times, deeply supervised) and Huginn-0125 (a 4-layer core recurred 16 times, trained for latent reasoning), using Qwen3.6-27B (64 untied layers) as the standard baseline. We find that a workspace forms in the iterated part of each architecture, but that recurrence changes how it can be accessed. Ouro reconstructs workspace content in every loop, and linear transport cannot carry that content across loop boundaries; writes and ablations must therefore span every remaining loop. Huginn carries content forward across all sixteen recurrences, while reads, writes, and ablations act only within a sliding window of roughly two recurrences. Whether newly injected content can be verbalised tracks explicit per-iteration supervision; whether existing content can be steered does not.

Clinical LLM assistants must reason over multi-visit patient trajectories, yet whether the compact history representations used to scale them---retrieval, structured timelines, LLM summaries, agentic memory---preserve the longitudinal signal clinical reasoning needs has not been measured. We introduce ClinTraceBench: 385 MIMIC-IV-derived verified dialogues with event-ID provenance, a nine-task taxonomy (T1--T9), and L0--L4 deterministic + L5 human-audit validation (98.92% agreement). We evaluate eight history representation strategies---a no-context floor, \textit{last-visit-only}, \textit{full-context}, BGE-M3 \textit{dense-retrieval}, two compression schemes, and two agentic-memory systems (\textit{Mem0}, \textit{A-Mem})---across four backbones (DeepSeek-V3, GPT-4o-mini, Haiku4.5, Sonnet4.6) on 6{,}271 questions: 32 cells, 200{,}672 predictions. Four findings: (SP4) a controlled T3 injection probe isolates compression-induced \textit{relation} loss---with the attribution sentence present \textit{before} construction, \textit{Mem0}, \textit{A-Mem} and \textit{llm-summary} still recover only 0--5.3% of the injected positives; (SP1) compressed strategies pay an aggregation tax on multi-visit trends and cross-patient comparisons; (SP2) the blind-to-full gap spans $+29.8$~pp (GPT-4o-mini) to $+62.7$~pp (Haiku); (SP3) abstention scales non-monotonically with context length. On the Pareto frontier Haiku dominates Sonnet under \textit{full-context} ($25.76 vs.\ $106.21), inverting the ``biggest backbone wins'' heuristic.

Neural networks are increasingly employed to identify both well-defined and ambiguous concepts, yet output-level metrics reveal little about how those concepts are represented internally. Our study asks if these networks exhibit \textit{conceptual separation}: if examples of the same concept form coherent representations, and whether related concepts lie closer together in the representation space. We examine this conceptual organisation in Convolutional Neural Networks (CNNs) and Large Language Models (LLMs) through geometric and distributional analysis of their internal activations. In CNNs, familiar ImageNet concepts form coherent and semantically ordered representations, while this coherence weakens for unseen concepts and suffers within-class domain shift. In LLMs, clearly distinct domains remain well separated, related subdomains move closer together, and the distinction between ambiguous topics collapses at both the mean and covariance level. These results suggest that conceptual separation can reveal structure that output accuracy alone cannot, and may serve as a useful diagnostic of how robustly a model represents the concepts it is asked to identify. Code and data available on \href{https://github.com/JaeeRoshniCapstoneProject/Are-You-Thinking-What-I-m-Thinking-Examining-Conceptual-Separation-in-Neural-Architectures}{GitHub}.

Language models learn about grammatical number primarily from co-occurrence, and show frequency effects as a result---sometimes taken to indicate that they do not learn abstract ``rules'', and are instead dependent on specific lexical items. Testing generalization with text stimuli alone cannot settle this debate, since distributional cues (is/are, this/these) easily give number away. We instead use cross-modal generalization as a tool to investigate abstractions in LMs that can also accept visual inputs (VLMs), restricting the evidence that diagnoses number to an extra-linguistic modality. We teach VLMs pairs of new nouns by adding new embeddings and only updating them during learning, comparing conditions where number is diagnosed by visual cues alone against ones where it is disambiguated by text. Across behavior, representational dynamics, and causal mechanisms, we find non-trivial evidence for cross-modal generalization across both exposure conditions, and that linguistic vs. extra-linguistic cue conditions are treated in similar ways in the internal mechanisms of the model. This suggests that statistical learners like VLMs can generalize beyond surface-level co-occurrence and show genuine abstraction-compatible behavior.

Partially observable locomotion requires a policy to act when task-relevant properties of the robot--environment state are not fully specified by instantaneous observations. Existing approaches often address this challenge by explicitly estimating missing physical variables or processing extended observation histories through structured architectures. We take a different view: partial observability is fundamentally an information-retention problem. The decisive question is not how task-relevant information enters the network, but whether the policy's internal state retains it. Guided by this perspective, we propose SleepWalking for Robot Locomotion (SWAQ), a one-stage end-to-end framework that uses next-step privileged physical reconstruction to shape what a recurrent history representation retains during policy learning, while the deployed actor uses only a direct history-to-action pathway. Under aligned training settings, SWAQ achieves a 15.0% higher peak mean terrain level than DWAQ, the strongest non-exteroceptive baseline, while using 44.4% fewer inference MACs per control step. Layerwise probes further show that information associated with the reconstructed physical variables remains linearly decodable through the policy head up to the layer preceding the action output. Complementary theoretical analysis relates privileged-variable recoverability to the achievable-return gap between history-based and privileged-information policy classes. These results suggest that semantic objectives can structure learning without requiring a corresponding architectural decomposition of the deployed controller.

The extent to which large language models for code rely on memorization over genuine understanding remains highly debated. While current literature frequently reports widespread memorization, evaluating the underlying probing techniques across dense architectures reveals a severe breakdown in their utility at scale. Traditional encoder-style probes using perturbations such as synonym fuzzing or dead-code insertion struggle to expose memorization in scaled models, even on known-contaminated benchmarks, and decoder-style probes that rely on log probabilities show similar performance degradation. The specific mode of failure for these probes, particularly why such techniques disrupt smaller models but fail to impact larger ones, motivates us to untangle representation load from memorization rather than treating them as a single phenomenon. By applying invertible mathematical transforms to numeric problems, we isolate these two factors and reveal that scaled encoders successfully absorb substantial representation load while still converging on the correct family of solutions. In practical software engineering, this ability to adapt to varying surface forms is what truly matters for usability and generalizability in LLM and agentic applications. Whether a specific solution was seen during training becomes a much less pressing question because although memorization inflates scores on contaminated benchmarks, factoring out representation load makes it debatable how much we should truly care if a functional answer was originally memorized. Future evaluations must therefore be built around separating these phenomena rather than relying on methodologies that quietly entangle them.

Learning world models from offline trajectories enables agents to accomplish different tasks through planning. Object-centric (OC) representations, which decompose a scene into a set of slots that bind to its objects, have been proposed as an inductive bias for world models that are more sample-efficient and generalize better. Yet prior object-centric world models (OCWMs) take the slot encoder as given and evaluate only in-distribution, leaving open whether the object-centric bias actually delivers for planning and what within the OCWM drives it. We conduct a controlled study of OCWMs for visual model-predictive control along two axes: object-centric representation quality and generalization under distribution shift relative to scene-centric models. We find that (i) planning success correlates positively with unsupervised slot-quality metrics (FG-ARI, mBO), though the gains saturate at high slot quality; (ii) with well-bound slots, the auxiliary proprioception inputs and masking inductive bias that prior methods relied on become unnecessary; and (iii) under unseen distribution shifts, the OCWM with well-bound slots is more robust overall than the end-to-end trained scene-centric LeWM, while DINO-WM, built on similar frozen pretrained features, remains competitive -- pointing to pretrained features as a key contributor to robustness.

Gated attention is an effective approach to mitigate attention sinks and enhance the representational capacity of attention. To further extend its effectiveness-efficiency Pareto frontier, we propose a Hybrid Gated Attention (HyGA) framework that contains three types of gating strategies. Specifically, these gates leverage diverse information from multiple stages of attention, and collaboratively build element-wise/head-wise gating from multiple perspectives, capturing intra-head and cross-head information interactions. Through our hybrid gating components, HyGA could provide multi-source modulation signals, enabling more comprehensive control over information flow and improving the representational capacity of attention. We also introduce low-rank matrix decomposition and learnable attention sink to further enhance training efficiency and stability. In experiments, we evaluate HyGA on widely-used benchmarks based on different backbones. The experimental results show that our HyGA comprehensively improves both training loss and various downstream performances compared with Gated attention. HyGA has also been verified to achieve the best performance at different computation costs, with comprehensive model analyses for better understanding. The proposed HyGA sheds light on a more effective, efficient, and stable attention mechanism.

Full justified representation (FJR) is among the strongest known satisfiable proportionality axioms for approval-based committee elections. Recent work has shown that an FJR committee can be found in polynomial time, but verifying whether a given committee satisfies FJR remains coNP-complete. We introduce FJR+, a strict strengthening of FJR and EJR+ that can be verified and satisfied in polynomial time. We then analyze the Residual-Budget Greedy (RBG) algorithm and prove that it selects a partial committee such that every size-$k$ completion satisfies FJR+. This freedom allows us to use sequential Phragmén to obtain a priceable completion. The resulting rule always satisfies FJR+ and the sub-core, and it is priceable whenever at least $k$ candidates receive an approval. We also obtain a Droop-quota version of FJR+. Finally, we extend FJR+ to approval-based participatory budgeting with arbitrary project costs. A project-specific version of RBG computes this property in polynomial time and can be continued to a priceable outcome satisfying a cost-based version of the sub-core.

AlphaFold3-class structure predictors pair a representational trunk, which processes sequence and context, with a diffusion module, which generates atomic coordinates. How biological information changes as it crosses this architectural boundary remains poorly understood. We analyze per-residue activations from the Pairformer trunk and diffusion module of Boltz-1 using linear probes, sparse autoencoders (SAEs), and causal interventions. From the trunk, both geometry (secondary structure, disorder) and sequence chemistry (amino-acid identity, signal peptides, disulfide-bond annotations) are linearly decodable. In the diffusion module, the two diverge. Secondary structure transfers essentially unchanged, whereas sequence chemistry is strongly attenuated. We then test whether decodable directions can steer the model, intervening on the final trunk single representation that conditions the diffusion module. Helix and coil directions change predicted structure dose-dependently against matched-norm random controls, but a beta-strand direction that is highly predictive (F1 =0.82) produces no measurable increase in strand content: linear decodability does not imply causal influence at the site we tested. The same probes also score markedly lower against sparse SwissProt annotations than against dense DSSP labels, because unannotated residues that the model gets right are charged as false positives; such scores are therefore lower bounds. Finally, supervised probes outscore single SAE features wherever a label already exists. We release the trained trunk and diffusion SAEs, Boltz-1 per-residue activations, and the analysis code.

In this work, we study the information bottleneck under perfect privacy, with particular emphasis on the active-rate regime, where the representation-rate constraint is binding and directly limits the achievable utility. The goal is to construct a representation that preserves utility-relevant information while remaining statistically independent of a sensitive variable. This exact independence requirement introduces an additional constraint beyond the classical rate-relevance tradeoff and must be explicitly incorporated into the optimization. To this end, we develop an alternating direction method of multipliers (ADMM)-based method tailored to the resulting problem structure. Under suitable regularity conditions, we establish global convergence of the generated sequence, characterize its convergence rate through the Kurdyka-Lojasiewicz exponent, and extend the analysis to inexact block updates.

Distance-based reliability estimation assumes that a representation's geometry reflects its trustworthiness, yet this assumption is rarely tested under training interventions that reshape geometry directly. We audit this assumption under domain-adversarial representation learning using a disentanglement dose-response ladder. Three checkpoint families share the same architecture and a 16-dimensional representation, differing only in orthogonality strength (lambda = 0, 1, 5). Representation geometry changed substantially with disentanglement strength: the condition number shifted by two orders of magnitude (Kendall tau = 0.84, exact p = 2.8e-5). This change was not accompanied by improved reliability estimation: Mahalanobis-distance AUROC (ISIC-test vs. PAD-UFES) remained flat and below chance (about 0.40) at every level, with no significant association with any of five geometry metrics tested. The same failure was observed for cosine-to-centroid and pooled k-nearest-neighbor scorers, plus three non-distance-based scorers: an energy-based confidence score, Virtual-Logit Matching, and a kernel density estimator. Seven of eight scorers converged on the same result; the energy-based score showed an isolated upward trend that we report but do not treat as evidence against the overall pattern. A supervised probe with no access to the training objective recovered domain membership from the identical embeddings at 0.72-0.81 AUROC across every level, showing that the relevant information was not absent from the representation. These findings indicate that classification performance alone can overlook whether information in a learned representation is organized in a form that downstream reliability estimators can use. Information can remain decodable while becoming largely inaccessible to non-probing reliability estimators.

Hyperbolic geometry has recently emerged as a powerful representation space for multimodal learning, as it naturally captures hierarchical semantic structure across modalities. Despite this progress, how such representations behave under continual learning poses fundamentally different challenges that remain underexplored. This work provides a geometric perspective on this problem and establishes a theoretical foundation for representation preservation in hyperbolic space, showing that preventing forgetting requires cross-modal invariance under a shared hyperbolic isometry. We further show that forgetting in hyperbolic continual learning involves both semantic relation drift and hierarchy-related distortion, motivating preservation of both cross-modal relational structure and hierarchical geometry. Guided by these insights, a principled continual learning framework is derived that preserves essential geometric structure while allowing effective adaptation to new tasks. Experiments on continual multimodal benchmarks corroborate the effectiveness of the proposed approach.

Generative AI models are primarily designed to imitate the data distribution, an objective that neither corrects diversity lost by a learned generator nor defines how generation should extend beyond the diversity of the data itself. We introduce Imaginative Generative AI (IGA), a framework that makes diversity part of the target-distribution design problem: among distributions close to a reference, IGA selects one whose spectral diversity reaches a prescribed level. Diversity is measured by the von Neumann entropy of the generated distribution's kernel covariance operator in a fixed representation space, providing a reference-free representation-guided measure of how broadly probability mass occupies embedding directions. The spectral entropy of the population data distribution defines an Entropy Wall. Below the wall, IGA performs diversity repair, recovering variation that a learned generator has lost while remaining within the diversity level of the data. Beyond the wall, the data distribution itself becomes infeasible, and IGA deliberately departs from it to produce distributions with greater representation-relative spectral diversity, an operational notion of imaginative generation. These regimes form a single regularization path from imitation to imagination and define an i.i.d. target distribution at each prescribed diversity level. We develop the theory of this entropy-constrained projection and show that, under a KL anchor to a pretrained generator, the optimum satisfies a self-consistent exponential-tilt relation. This characterization leads to IGA Guidance, a retraining-free inference-time method for score-based and diffusion models, including DDPM and DDIM samplers. Experiments on synthetic and vision benchmarks demonstrate diversity repair below the Entropy Wall and controlled spectral extrapolation beyond it.

Focus particles such as "even" and "only" are central to formal semantic theories that posit structured representations over sets of alternatives. "Even" highlights unexpected or extreme alternatives, while "only" enforces exclusivity. If such scalar representations are robust and generalizable, they should give rise to consistent judgments across contexts and systems. In this work, we test whether humans and large language models (LLMs) construct stable scalar representations from sentences containing these particles. Using a dataset of approximately 100 items, participants and models were asked to make scalar judgments. Preliminary results suggest that similar outputs across humans and LLMs may arise from different underlying mechanisms.

Recent advancements in deep reinforcement learning have increasingly favored simplified, highly parallelized paradigms. Notably, the Parallelized Q-Network (PQN) algorithm enables off-policy value learning without relying on experience replay buffers or target networks. However, the representational capacity and computational efficiency of visual encoders operating in these buffer-free settings remain comparatively underexplored. In this work, we systematically investigate the architectural design space of Convolutional Neural Networks within PQN. We evaluate eight distinct CNN topologies while explicitly characterizing their parameter and computational requirements. We further study the effect of multiplicative representation learning and advanced value estimation by integrating the Hadamax encoding paradigm with categorical, ensemble, and dueling value heads. Extensive experiments on Atari-57 show that our final composite architecture, Aftab, achieves an Interquartile Mean (IQM) Human-Normalized Score of 6.592, compared with 2.715 for the standard PQN baseline, together with a 0.86 Probability of Improvement over PQN. We additionally evaluate Aftab on Procgen-Hard to assess performance under procedurally varying visual environments. Aftab achieves a normalized learning-curve Area Under the Curve (nAUC) of 0.541 compared with 0.216 for PQN. Overall, the results demonstrate that carefully designed encoder topology, multiplicative feature interactions, and advanced value-estimation heads can substantially improve performance within a parallelized, replay-free Q-learning framework while preserving its memory-efficient training paradigm. The complete Aftab framework, including model definitions, training configurations, reproducibility settings, and raw experimental logs, is open-sourced at https://github.com/tahashieenavaz/aftab

Autonomous agents now carry out entire data analyses, selecting cohorts, joining tables, and fitting models with little step-by-step supervision. When such an analysis turns out to be wrong, someone must determine which operation caused it. A recent approach does this without any labelled mistakes, learning instead from analyses known to be sound and flagging operations that depart from what that model predicts; how reliable such audits are has not been studied. This paper supplies that analysis. The choice of score determines whether an error can be localized at all. If each operation is scored by how surprising it is given the operation immediately preceding it, then operations that merely inherit an earlier error are indistinguishable from correct ones, so one mistake produces one flag; scores computed against a longer reconstruction of the intended analysis instead spread a single mistake across many operations. We quantify how far they spread, and how to choose the comparison length when an error accumulates gradually rather than at once. We then give procedures that control the proportion of falsely flagged operations within a single audited analysis, requiring only that sound analyses be exchangeable rather than that the fitted model be correct, and we quantify how much the guarantees weaken when the model is imperfect or when the analysis was selected for review in a way that depends on its content. Finally we establish a limit on what any such audit can report: errors below a certain magnitude cannot be attributed at all, being indistinguishable from ordinary variation among sound analyses. This limit falls so slowly as more sound analyses are collected that at the representation sizes now in use a hundredfold increase reduces it by under two percent, so the dimension of the representation rather than the volume of training data is the binding constraint.

Representation theorems in decision theory establish that behavior satisfies certain axioms if and only if it can be rationalized by a well-defined objective. I argue that this ``if and only if'' structure provides a potentially useful foundation for label-free evaluation and regularization of LLMs and other AI systems. Axiom compliance can be checked from the model's own responses to synthetic choice problems, with no external labels or human feedback, and the penalties are readily computable. Because the axioms are necessary and sufficient, the resulting checks exhaust the implications of the relevant rationality standard for the elicited data: a model that passes cannot be rejected on rationality grounds by any further test of the same data. I discuss three instantiations: probabilistic coherence via a theorem of de Finetti, preference rationality via Afriat's theorem, and subjective expected utility via a theorem of Echenique and Saito (2015), each yielding a continuous penalty that is zero whenever behavior can be rationalized. Since coherence does not restrict which objective rationalizes behavior, these penalties complement rather than replace other evaluation and training signals.

Multi-head attention layers produce vector representations that support multiple downstream tasks. We establish bounds on the number of heads required in two simple and concrete multi-task scenarios. In the first scenario, a vector representation is sought so that linear predictors can compute both the smallest and largest numbers in a given list. In this case, it is known two attention heads with small embedding dimension and bit precision level suffice. We prove that a single attention head requires exponentially higher embedding dimension or precision level. In the second scenario, a vector representation is sought so that a polynomial threshold function can compute the XOR of a given string of $n$ bits. This scenario is analogous to the first one for $n=2$, since XOR is readily computed by a linear function using a vector representation that encodes both the AND and the OR of the two bits. We observe that $n$-bit XOR requires the product of the number of heads and the polynomial degree to be at least $n$, and we construct multi-head attention layers that match this lower bound. These results generalize to arbitrary (symmetric) Boolean functions, where the bound is given in terms of the threshold degree.

Recent advances in Diffusion Transformers (DiTs) have enabled remarkable progress in visual synthesis, benefiting from their superior scalability. To facilitate DiTs' capability of capturing meaningful internal representations, recent works such as REPA incorporate external pretrained encoders for representation alignment. However, the underlying mechanisms governing representation learning within DiTs remain poorly understood in the community. To this end, this paper first presents a systematic analysis of the representation dynamics of DiTs via quantifying the diversity of block-wise representations. Specifically, we introduce a novel metric, termed the Weighted Diversity Score (WDS), to measure the representational discrepancies across different blocks. Through extensive investigations on the evolution and influence of internal representations under various settings, we reveal that representation diversity across blocks is a critical factor for effective representation learning in DiTs. More importantly, WDS exhibits a strong correlation with synthesis quality across diverse settings, model scales, and training stages (Pearson's $r=-0.869$ with $\log(\text{FID})$), suggesting its potential as an indicator to reflect model performance and a principled guide for model optimization. Based on this key finding, we propose DiverseDiT++, a novel framework that explicitly promotes diverse representation learning. Concretely, our method incorporates long residual connections to diversify input representations across blocks and a representation diversity loss to encourage blocks to learn distinct features. Extensive experiments on ImageNet $256\times256$ and $512\times512$ demonstrate that our DiverseDiT++ yields consistent performance gains and convergence acceleration when applied to different backbones with various sizes,...

All Transformer-based large language models compute attention via the Euclidean inner product, an architectural choice that Dong et al. (2021) proved causes representational rank to decay doubly exponentially with depth in pure self-attention stacks. We develop a theoretical framework that targets this structural limitation at the mathematical level by replacing the flat Euclidean metric with learned per-token Riemannian metrics. Our contributions are threefold. (1) We prove that Riemannian attention scores with heterogeneous per-token metrics are non-Gram---they cannot be factorized as QK^T with factorization dimension O(d). We are explicit that this is a structural observation, not a proof of rank preservation. (2) We establish that low-rank metric factors render all geometric operations tractable: geodesic distance in O(dr) per token and metric inversion in O(dr^2) via the Woodbury identity---both far below the O(d^3) cost of a general matrix---making Riemannian attention feasible at billion-parameter scale with negligible overhead. (3) We present the Fiber Bundle Transformer, a complete architecture specification in which each token position carries its own Riemannian metric, attention is geodesic distance computation, feed-forward updates use metric-preconditioned steps, and the connection carries explicit curvature and torsion proxies. We derive formal predictions about correctly implemented geometric architectures and identify the central open problem: proving or disproving that heterogeneous Riemannian metrics prevent the rank collapse that row-stochastic attention matrices otherwise cause. This paper presents theoretical analysis and architectural design; empirical validation is the subject of future work.

Next-destination prediction in continuous-time dynamic graphs (CTDGs) commonly ranks an observed interaction against sampled negative destinations. The resulting score is conditional on both the negative distribution and the number of candidates chosen by the researcher. We show that a non-uniform negative distribution changes the Bayes-optimal ranking, while even a finite candidate set drawn uniformly can destabilize model rankings and measured module effects. Time-varying source-destination history membership and model operations that use this information directly transmit the sampler's influence to the evaluation score. We examine this mechanism using a factorial evaluation of repeated and new positives against seen and unseen negatives, a minimal scorer based solely on pair-history membership, and controlled representation interventions. Across six models on LastFM, MOOC, Reddit, and Wikipedia, at least one model pair changes relative order between the expected Uniform-20 metric and the full catalog on three of the four datasets. The measured effect of the same module also changes in magnitude and direction with the candidate-set size and training objective. These results establish that model-superiority and ablation conclusions from sampled-negative benchmarks are conditional on the stated candidate configuration. All-entity ranking evaluates every destination in a fixed catalog, eliminating negative-selection freedom and sampling variation while retaining the original CTDG scorer. We therefore recommend all-entity ranking as the primary evidence for architecture comparisons on CTDG benchmarks with an enumerable, fixed destination catalog.

Understanding and exploiting the training dynamics of overparameterized deep neural networks remains a central challenge in modern machine learning. Recent evidence on Neural Collapse (NC) shows that class representations and classifiers exhibit highly structured geometry, while the Tunnel Effect suggests that only a subset of layers is essential for feature extraction. We combine these two perspectives and propose an NC-inspired training framework for simplifying deep networks during training. Our method monitors representation dynamics through the Inverse Fisher Criterion, a stable and efficient proxy for the variability collapse behavior, to identify both the split point between feature extraction and classification and the training stage at which simplification becomes viable. We then replace the trailing layers with a lightweight classification head and continue training the reduced model. Experiments on image-classification benchmarks across MLP, VGG, and ResNet architectures show that the proposed method achieves substantial parameter reductions while maintaining accuracy comparable to that of the full model. Code to reproduce the experiments can be found at: https://github.com/LorenzoSciandra/NNS.

Learning-to-optimize (L2O) methods accelerate repeated optimization by training models to predict solutions, warm starts, branching decisions, or other forms of solver guidance. A critical yet largely overlooked component of these pipelines is the feature function that maps problem instances to inputs for machine learning models. Existing L2O methods typically rely on hand-crafted features, making representation design manual and largely fixed across domains. We introduce FunL2O, the first unified framework for automating feature design through LLM-driven program evolution for L2O. In a FunSearch-style loop, an LLM proposes executable feature functions, while a fixed evaluation process retrains the original L2O model and measures downstream optimization performance. We evaluate FunL2O on linear and quadratic programming tasks involving solution prediction and warm-starting, as well as on mixed-integer optimization tasks using GNN-guided backdoor branching and Predict-and-Search. Across continuous and discrete optimization tasks and four LLMs, the evolved features consistently outperform hand-crafted representations. These results establish LLM-driven feature evolution as a general and effective approach to automating representation design in L2O.

Latent communication in large language model (LLM)-based multi-agent systems (MAS) transmits continuous internal representations instead of text, but greater representational capacity does not establish that the receiver uses task-relevant information. End-task performance alone also cannot reveal whether an observed effect depends on message presence, content generated for the evaluated example, or information supplied by a separate agent. We introduce a causal audit that applies controlled message replacements at the boundary where the sender-produced representation enters the receiver. Four message settings support five measurements of encoded sender information, receiver sensitivity to message presence and identity, the task value of example-specific content, and the additional value supplied by a separate agent. We apply the audit to latent relay with Qwen3-4B and Qwen3-8B on GSM8K, ARC-C, and MATH-500. On GSM8K, the Qwen3-4B overall performance effect of -1.00 percentage point decomposes into a -6.17-point effect retained by an other-example message and a +5.17-point effect attributable to example-specific content; both component directions reverse at 8B. On MATH-500, the Qwen3-4B gain of 15.00 points comprises 8.33 points retained by an other-example message and 6.67 points attributable to example-specific content, while the 8B gain is dominated by the former component. Self-substitution comparisons further show that example-specific content and other-agent value are distinct. These results show that aggregate accuracy does not identify how a latent message affects the receiver and motivate controlled message comparisons as a standard evaluation for latent communication.

When it comes to generating vector representations of words, current language models are achieving high-quality results. However, what is not known is the extent to which knowledge about semantic relations is represented in the geometry of the semantic spaces created in this way. In order to answer this question, we study the relation geometry of such semantic spaces from three perspectives. We first examine whether words standing in a particular relation to a target word~(called relata) occupy the same region in semantic space, and whether the regions corresponding to different relations are distinct from each other. We then verify to what extent semantic spaces reflect certain well-known properties of relations, such as symmetry, asymmetry, and transitivity. Finally, we consider which information about the target words and relata is more important for relation geometry: their surface forms, or their contexts. We conduct experiments on six semantic relations using causal, masked, and diffusion language models. The results show that relata in asymmetric relations relatively clearly occupy a distinct region in semantic space. Asymmetric relations' properties are only moderately well encoded in the semantic space, yet better than those of symmetric ones. Furthermore, when considering the question which information source has the strongest impact on results amongst the models we evaluated, we find that lexical information tends to be more important for the causal language model, whereas contextual information is more important for the masked and diffusion language models. Our results empirically show that relation geometry is not equally well-represented for all relations in semantic space, suggesting that there is a difference in how well semantic relations might be learned from distributional information alone.

LLMs are widely regarded as alien intelligences, systems whose cognitive operations are fundamentally unlike our own. Apparent similarities to human cognition are therefore often seen as the result of anthropomorphic projection. We argue that this framing is mistaken. LLMs clearly differ from humans in important respects, including their physical substrate, learning history, and the environments with which they interact. These differences make it all the more striking that contemporary LLM-based systems converge with human cognition on a number of principles of cognitive organization with longstanding support in cognitive science. We identify structural correspondences across five dimensions: inferential organization, computational architecture, representational structure, prediction-driven learning, and reinforcement-learning-like mechanisms supporting goal-directed action. These correspondences support a broader model of intelligent cognition in which core principles long used to explain human intelligence also characterize contemporary LLM-based systems.

Pretrained molecular encoders are commonly evaluated through downstream prediction, but predictive accuracy alone does not establish that a learned representation captures reproducible scientific structure, adds information beyond strong conventional baselines, or transfers out of distribution. We present a reliability-aware audit of generic molecular representations for human olfaction across four distinct claims: global perceptual geometry, incremental predictive value beyond chemistry, cross-dataset replication, and mixture transfer to unseen components. Using the Keller-Vosshall and Bierling single-molecule rating datasets and the Ma binary-mixture dataset, we compare MoLFormer and ChemBERTa against RDKit descriptors and Morgan fingerprints under identity-controlled and matched evaluations. Human three-attribute rating geometry, based on intensity, pleasantness, and familiarity, is reproducible across participant splits (median RSA 0.743 and 0.855), whereas model-human alignment is substantially weaker (RSA 0.019-0.158). Learned embeddings do not consistently outperform conventional representations in global alignment, and MoLFormer provides no clear incremental predictive value beyond a combined RDKit-Morgan baseline in either single-molecule dataset. Human geometry shows positive but incomplete agreement across 63 shared molecules (RSA 0.331; 95% bootstrap interval [0.204, 0.507]). Under one strict unseen-component mixture split, incremental effects are outcome- and representation-dependent, with all intervals crossing zero. These results establish empirical boundaries for the evaluated generic molecular encoders and motivate a broader evaluation principle: representation quality in scientific domains should be assessed separately for target reliability, structural alignment, incremental information, replication, and out-of-distribution transfer.

Large language models (LLMs) have been applied to sequential recommendation by formulating it as a natural language task. Previous work has improved personalization by incorporating collaborative and sequential signals through input conditioning or LLM fine-tuning. However, existing approaches often rely on one or more of the following: LLM fine-tuning, additional architectural modules, representation distillation, or item-level conditioning over long interaction histories, increasing training complexity and deployment cost. We propose REPREC, a lightweight framework that reformulates LLM-based sequential recommendation through lightweight user representation alignment. REPREC maps a fixed-size user embedding from a frozen sequential encoder into a small set of learned soft tokens through a lightweight MLP injector that conditions a frozen LLM, leaving both pretrained backbones unchanged while training only the injector. We conducted exhaustive experiments on multiple benchmark datasets and demonstrate that REPREC consistently outperforms LoRA while remaining compatible with different pretrained sequential encoders and LLM backbones, enabling a modular and production-friendly recommendation pipeline without modifying either pretrained component. The gains are particularly pronounced for casual and core users across all datasets, highlighting REPREC's effectiveness in low-data regimes. Finally, when trained on short prompt histories and evaluated with longer contexts, REPREC maintains 85-100% of LoRA's performance while reducing per-epoch training time by an average of 1.51X, demonstrating an effective balance between recommendation quality and computational efficiency for production deployment. The code is available at https://github.com/phdbotcode/REPREC

Weakly supervised hierarchical models exhibit a persistent asymmetry: coarse lesion-type features are preserved under reconstruction while fine-grained malignancy cues degrade---a pattern with direct consequences for the clinical reliability of breast cancer screening pipelines. We introduce gradient-based orthogonal latent decomposition for hierarchical Variational Autoencoders~(H-VAEs) to mechanistically explain this asymmetry. The latent space is partitioned into a task-aligned component~($z_1$), shaped by coarse supervisory gradients, and an orthogonal residual~($z_{\text{res}}$) capturing remaining representational capacity. On3,550 mammographic Regions of Interest(ROIs) from CBIS-DDSM, only~$\sim$4.4% of latent magnitude aligns with supervisory gradients, leaving~$\sim$95.6% in the orthogonal residual upon which fine-grained pathology prediction primarily depends. The model achieves Stage-1AUC0.866 and Stage 2AUC0.552, with a reconstruction stability gap of $Δ_{\text{diag}}=5\%$ ($p=0.005$) and a classification gap of $Δ_{\text{AUC}}=0.314$ ($p{<}0.001$). Latent ablation confirms that features for both tasks reside heavily in~$z_{\text{res}}$, structurally explaining why reconstruction degrades pathology stability disproportionately. Comparisons with Multi-Instance Learning~(MIL) and Multi-Task Learning~(MTL) confirm generalization across architectures and modalities. These findings reveal that in high-dimensional spaces, a single coarse supervisory signal isolates only a sparse 1D latent direction, forcing critical fine-grained features into the vulnerable residual subspace.

Surprisal Theory is often characterized as a computational-level explanation per (Marr, 1982). We argue in this work that, even though a computational level narrative has been used to support "representation-agnostic research" within computational psycholinguistics, the movement toward black box systems embodied by large language models (LLMs) does not exempt modelers using the surprisal metric from the representational decisions required by computational-level characterizations. In fact, we argue that the uncritical use of LLM-surprisal obfuscates the representational and algorithmic-level commitments of different models. In three analyses, we show that the choice of algorithm and model architecture play significant roles in the computation of language model probabilities. We advise that researchers who wish to test Surprisal Theory re-evaluate the practice of treating large language model probabilities as interchangeable

Highly nonlinear chaotic dynamical systems remain difficult to model due to fundamental trade-offs between complexity, expressivity, and data efficiency. Modern machine learning methods achieve strong predictive performance but often rely on a-priori system knowledge or curated data with limited interpretability. Koopman operator theory offers a promising direction via linear representation in an infinite-dimensional observable space. However, many data-driven Koopman methods seek globally valid operators for which useful finite-dimensional spectral embeddings remain difficult to identify under these constraints. To overcome associated limitations, we introduce Fuzzy Spectral Region Decomposition (fSRD), a fully automated learning framework for estimating finite Koopman representation via multiple operators. The proposed method realizes a data-adaptive framework for assembling locally invariant embeddings, termed Invariant Decomposition. fSRD achieves highly accurate linear reconstructions of nonlinear systems while learning finite-dimensional representations of their induced evolution operators, bridging interpretable operator-theoretic models with expressive data-driven sequence learning. These embeddings are adaptively constructed via a global fuzzy tree model, drawing inspiration from fuzzy neural architectures to learn the induced dynamics while prioritizing parsimonious solutions. Empirical results across canonical chaotic systems (e.g., Lorenz and Duffing) and high-dimensional real-world data demonstrate strong predictive accuracy, interpretability, and robust expressivity across data-rich and data-limited regimes, highlighting the method's generality.

Representation is a central concept in modern machine learning, where it usually refers to internal encodings that support learning and generalization. As models scale and their capabilities become increasingly human-level, this representational language sometimes shifts from an engineering context into the more philosophically loaded domain of mental representation. We argue that this is the case for recent claims about the convergence of representational properties across different AI models. In particular, we assess the arguments developed in The Platonic Representation Hypothesis, according to which this convergence is driven by a unified structure of reality. We examine this claim by introducing arguments and ideas from debates about mental representation in the philosophy of mind. We argue that these philosophical resources can clarify what is at stake in such claims, explain why alignment evidence alone is insufficient for strong metaphysical conclusions, and suggest directions for future research.

Cooperative multi-agent RL systems routinely use team-averaged rewards, a feedback-attribution choice that gives each agent the team outcome regardless of its individual contribution. We ask whether this leaves a measurable signature, geometric or behavioral, on learned representations. We propose EffRank/$n$ (effective rank normalized by agent count) and $D_\text{act}$ (mean pairwise KL divergence between agents' action distributions) as low-overhead diagnostics for reward-attribution effects, then test them on competent MAPPO agents in SMACv2 \texttt{protoss_5_vs_5}, where unit type is encoded in the observation. In an observation $\times$ reward-attribution comparison (unit type observed vs.\ masked; individual damage-contribution reward vs.\ shared team reward), geometry follows observation rather than reward. With unit type observed, shared and individual rewards have similar EffRank/$n$ ($0.31{\pm}0.03$ vs.\ $0.29{\pm}0.02$) and probe accuracy ($0.75{\pm}0.05$ vs.\ $0.73{\pm}0.05$, both $\gg 1/3$ chance), while $D_\text{act}$ leans higher under individual rewards ($1.23{\pm}0.06$ vs.\ $1.07{\pm}0.20$). Masking unit type cuts the above-chance probe signal by more than half, to $0.49$ in both reward arms. In short: individually rewarded agents are competent and separable by role, but on SMACv2 the observation explains the geometry and reward attribution shows up mainly in behavior. Thus geometric diagnostics must control for observed role information and test persistent roles that are not directly observed. EffRank/$n$ and $D_\text{act}$ add $<$5% overhead.

Introduction: Biological systems face anatomical and metabolic constraints, including costly synaptic maintenance and limited connectivity. These constraints favor neural codes that compress behaviorally relevant information into low-redundancy patterns. We test whether an excitatory competitive Hebbian rule can support synaptic resource allocation under such constraints and whether the resulting representations occupy a more favorable cost-performance regime than reference learning rules. Methods: Representational cost is quantified using mutual-information-based measures derived from the Variational Information Bottleneck. Experiments use fixed audiovisual embeddings from three audiovisual benchmarks (AVE, Kinetics-Sounds, VGGSound100) to isolate downstream associative plasticity. Hebbian learning is compared with Dense Difference Target Propagation (DDTP) and backpropagation (BP) under matched sparsity and architectural constraints. Results: Hebbian learning achieves lower task-information cost (CTI) than sparse BP and DDTP in the main compressed comparisons, while reaching CTI values comparable to shallow BP with nonnegative weights. Rather than uniformly improving classification performance, Hebbian learning shifts the trade-off between task-relevant information and representational cost, yielding lower CTI at comparable functional performance in several settings. Discussion: The results indicate a cost-performance trade-off rather than uniform accuracy gains. For a given level of task-relevant information, Hebbian representations retain less input information while preserving functional performance, although accuracy is slightly reduced on some datasets. These findings support interpreting Hebbian learning as a mechanism for synaptic resource allocation rather than as a general strategy for maximizing audiovisual classification accuracy.

Multimodal Contrastive Learning (CL) has shown significant performance in aligning representations across various data modalities and improving downstream tasks, especially in healthcare. It works by minimizing the distance between matched (positive) data modalities, while maximizing the distance between mismatched (negative) samples. Traditional CL frameworks typically assume instance-based correspondence within data batches, treating all non-paired samples as negatives. However, this assumption often fails in medical settings, where samples may share high-level semantic attributes, leading to false negatives that degrade representation quality. In this paper, we propose Multimodal Semantic-Aware Contrastive Learning (MseaCL), a CL framework trained on a pediatric cohort of 3D brain magnetic resonance imaging (MRI) scans and radiology reports. The goal of this framework is to mitigate the impact of semantically similar false negative samples by incorporating semantic similarity between radiology reports, as a guiding signal during the learning process. Our results indicate that applying this framework as a pretraining stage can achieve notable improvements in downstream tasks, e.g., at least a 22.6% increase in the area under the receiver operating characteristic curve (AUC) of pediatric brain tumor molecular classification, demonstrating its potential for more robust and semantically aligned multimodal representations in clinical applications.

Muon-style optimizers apply a polar map to matrix momentum, but their updates also depend on the representation of each parameter block before orthogonalization. We study this representation choice as a form of optimizer geometry and introduce {\method}, a family of Muon-style optimizers that shares the same momentum rule and Newton--Schulz backend across native, nearest-square, skinny, and vector representations. Each Frobenius-isometric representation induces a distinct polar steepest-descent geometry, in which the shorter matrix dimension determines the number of supported singular channels, the pullback scaling, and the constants in stochastic nonconvex convergence bounds. In a teacher--student model, curvature collapse and an isotropic Marchenko--Pastur spectral profile connect early-stage dissipation to the represented nuclear-to-squared-Frobenius norm ratio. Pretraining experiments on LLaMA2-130M and LLaMA2-600M, together with fixed-momentum diagnostics, show that balanced non-native representations can match the performance of the native representation, whereas reducing the shorter dimension weakens the scaling and singular-channel support, leading to behavior that increasingly resembles normalized momentum.

Contrastive Language-Image Pretraining learns a shared representation space through large-scale contrastive learning. However, existing methods that enforce global consistency regularization overlook a key challenge: the inherent information asymmetry between images and text: captions typically describe only one specific aspect of an image, thus images with similar visual content can be paired with completely divergent textual content and semantic information. Consequently, global regularizers inadvertently impose constraints between visually similar images whose captions describe divergent aspects, introducing semantic distortion into the representation space. We propose AspectCLIP, a framework that reformulates consistency regularization to respect this one-to-many structure. AspectCLIP first partitions training samples into attribute clusters based on textual similarity to identify aspect-coherent groups, then applies full cyclic consistency within each cluster while restricting cross-cluster regularization to prototype-level comparisons. This aspect-guided regularization enforces strict geometric alignment only when images and texts describe a consistent facet, while allowing flexibility across divergent aspects. Extensive experiments on downstream tasks demonstrate that AspectCLIP consistently outperforms traditional methods and achieves a more structured representation space.

The $\mathcal{O}(N^2)$ complexity of attention over $N$ tokens remains a computational bottleneck in transformer models. Vector-Quantized (VQ) attention reduces this to $\mathcal{O}(MN)$ by representing keys with $M$ codewords, but applies uniform codebook capacity regardless of where attention mass concentrates: high-attention regions of key space may be coarsely approximated while low-attention regions waste representational capacity. We propose Adaptive Vector-Quantized (AVQ) Attention, which adaptively allocates codebook capacity based on attention importance. Starting from a small set of codewords, our method identifies the most important codes during the forward pass and refines them with pre-learned child codewords, achieving fine-grained quantization where it matters most while maintaining coarse quantization elsewhere. We develop an implementation using custom Triton kernels that enables the full adaptive refinement process, including importance scoring, child codeword insertion, and parent contribution replacement, to be carried out within the tiled computation paradigm of Flash Attention with minimal overhead. Our approach maintains $\mathcal{O}(MN)$ complexity while achieving improved accuracy-efficiency trade-offs compared to fixed-codebook VQ-attention.

Inverse design is an emerging data-driven paradigm for efficiently navigating vast chemical spaces to discover new materials with targeted properties, and in the context of heterogeneous catalysis, surface generative models have recently advanced this goal by directly generating catalyst surface-adsorbate structures. However, these models typically operate at the slab level and do not provide the corresponding parent bulk structure, making it difficult to assess bulk-dependent properties such as formation energy, surface energy, crystallographic symmetry, and synthesizability. Here, we address this missing slab-to-bulk connection as a retrieval problem and introduce CatRetriever, a contrastive representation learning model that aligns slab and bulk crystal representations in a shared latent space. From a slab query, CatRetriever accurately retrieves plausible parent bulk candidates with R@1 > 91% and R@3 > 98% on both the in-distribution and holdout evaluation sets. We further extend the CatRetriever framework into an adsorption energy targeted bulk discovery pipeline that combines bulk retrieval, generative search space expansion, and adsorption energy distribution analysis. This workflow evaluates candidates by both structural compatibility with the query slab and their ability to access the target adsorption energy range across diverse surface environments. CatRetriever therefore provides a scalable route for connecting catalyst generative models with physically plausible and adsorption energy compatible bulk catalyst discovery.

Grokking is a phenomenon in which neural networks initially memorize training data and only later exhibit strong generalization after prolonged optimization. Despite extensive recent study, the factors influencing the emergence and timing of grokking remain incompletely understood. We investigate the relationship between representation geometry and delayed generalization. We find that dimensionality collapse consistently precedes the onset of grokking in all evaluated settings. Motivated by these observations, we introduce Geometric Dimensionality Regularization (GeomDR), a simple spectral regularizer that modifies the effective dimensionality of hidden representations during training. Across modular addition, modular division, and permutation composition tasks, GeomDR consistently alters grokking dynamics and can substantially accelerate the onset of generalization depending on the intervention schedule and target dimensionality. In several settings, grokking is accelerated by up to 52 times relative to standard AdamW training. Similar qualitative effects are observed in both multilayer perceptrons and transformers. Together, these results suggest that representation geometry can serve as an effective control signal for grokking and provide evidence that geometric interventions offer a practical approach for studying and influencing delayed generalization in neural networks.

Under the free energy principle, a predictive system does not observe reality directly; it maintains a generative model of the world and experiences that model's best current hypothesis. Can a synthetic environment be made consistent enough that a predictive system's own inference machinery adopts it as this default hypothesis, permanently displacing the environment that first shaped it? We call this state ontological inversion. Because inducing and monitoring such a transition in a nervous system is neither ethical nor technically feasible, we study the underlying computational problem through a controlled proxy: a convolutional variational autoencoder paired with a recurrent latent predictor, whose evidence lower bound objective is mathematically identical, up to sign, to variational free energy itself. The network is trained first on a baseline visual domain, then on a mixed stream in which a swept rehearsal ratio r controls how much baseline content persists during transition to a target domain. Representational capacity, what the latent space can discriminate, is tracked separately from default behavior, what the system generates when left unconstrained. Across a full sweep of 90 runs, the two diverge sharply: representational accuracy stays near ceiling, 0.97 to 0.998, regardless of r, while default behavior spans nearly the system's entire range depending on r alone, a decoupling of learning from acceptance. More strikingly, at intermediate r the system's default output rises toward the target domain, then partially reverts toward the baseline while training continues unchanged, a structural failure we term cognitive relapse. Resistance to reality-adoption is not reducible to learning speed; it is a structural property with its own distinct failure modes, established here as a computational existence proof and nothing further.

Modern AI systems are increasingly being evaluated for their ability to reason, code, prove theorems, use tools, and long-horizon research tasks. These are powerful capabilities, but they share a structural limitation: the representational frame within which the model operates, including its conceptual vocabulary, the space of admissible solutions it can search, and the criteria by which success is evaluated, is typically fixed and supplied in advance. This paper argues that building stronger intelligent systems capable of open-ended innovation requires additional classes of operations: the creation, stabilization, and reuse of new representational primitives, which alter the space being searched rather than simply searching within it. We characterize the distance between current AI systems and genuinely open-ended intelligence through two gaps. The first is the vocabulary gap, the difficulty of inventing and stabilizing new representational primitives rather than merely recombining existing ones. The second is the verifier gap, the difficulty of judging the value of a new primitive when its full payoff may be visible only after future reuse. We interpret both gaps through a unified framework of intelligence as cognitive discrepancy reduction. By viewing intelligent behaviors as a sequence of cognitive transformations, we distinguish intra-space transformations which operate within a fixed representational frame, from generative transformations which may modify the frame itself. On this basis, we propose a ladder of innovation autonomy and outline several directions for advancing open-ended AI, including objectives that reward useful representational change, persistent memory architectures for invented primitives, and adaptive verification mechanisms capable of evolving alongside the representations they evaluate.