Representation Space

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.

End-to-end neural audio models achieve high-fidelity compression and generation. We might read that performance as evidence they directly represent interpretable features such as pitch and timbre, but a model can produce plausible outputs without doing so. A model may encode these features in any reachable basis, but regardless of which, the features are well described as compositions of time-frequency-localized primitives. Whether state-of-the-art encoders preserve access to these primitives, and thus to compositions of them, remains unclear. Through theoretical analysis and controlled experiments, we show that several state-of-the-art strided convolutional encoders impose two structural bottlenecks, both predictable from architecture and signal structure, on access to these primitives: (1) they collapse primitives into alias equivalence classes, establishing a bound on representational capacity, and (2) they limit the frequency resolution available to learned filters, restricting separability. For well structured data, we find collapse rates of 31-35% and filter bandwidths 10-35x above the theoretical resolution bound, confirming that both bottlenecks arise under realistic signal conditions. We then introduce Gabor Latent Refactorization (GLRF), a lightweight post-hoc intervention that re-expresses encoder latents in a frequency-localized basis, reducing filter bandwidths from 10-35x to 1.5-3x of the theoretical resolution bound while preserving reconstruction fidelity and improving control over attributes like pitch. These results show that the encoders in question predictably degrade access to frequency-localized primitives, entangling the features that depend on them, and that a lightweight, retraining-free intervention can recover much of that access, improving steerability and interpretability.

Large Language Models (LLMs) frequently exhibit sycophancy, where they agree with a user's statement even when incorrect. While sycophancy is often treated as a single defined behavior, it can manifest in substantially distinct ways and circumstances, raising the question of whether this multi-faceted nature is reflected in its internal mechanisms. To address this gap, we dissociate the representations of sycophancy into factual and opinion subtypes -- motivated by the distinction between verifiable claims and subjective beliefs. We train linear probes and construct steering vectors on activations of one subtype and evaluate their transfer to the other subtype to measure to what extent they share representations. We find evidence that different LLMs represent these subtypes differently, with either more unified or more distinct and causally interfering representations. This method of dissociation offers a promising framework for studying the representational structure of complex model behaviors.

We characterize observability in representation learning through Platonic Projection Structures (PPS), an operator-theoretic framework for analyzing representation accessibility under partial observation. Rather than treating observable outputs as direct reflections of latent representations, PPS models observation through a self-adjoint positive semidefinite operator acting on a latent representation space. A system is represented as a triple $(H, Π, O)$, where $H$ is a latent representation space, $Π\succeq 0$ is an observation operator, and $O(v)=\langle v,Πv\rangle$ defines an induced scalar observable. Observability is characterized by the quotient geometry $H/\ker(Π)$, representing equivalence classes of latent states indistinguishable under observation. We show that quantum measurement and representation inference under linear observation models share this operator-theoretic structure while differing in the algebraic properties of their observation operators; the correspondence is structural rather than physical. Representation transfer and knowledge distillation can likewise be interpreted as approximate preservation of observable geometry through $ΦΠ_T \approx Π_S Φ$. PPS also reveals a structural limitation of output-based interpretability: latent components in $\ker(Π)$ are inaccessible from induced observables, imposing intrinsic constraints on attribution and explanation methods. Controlled empirical validations demonstrate kernel-invariant observability, projection-induced attribution gaps, and rank-controlled observable geometry in latent representation spaces. PPS thus provides an explicit characterization of observability through operator-induced quotient geometry and a unified perspective on representation accessibility, interpretability, and projection-mediated inference.

Grokking -- generalization long after training-set interpolation -- has been accelerated by structure-agnostic interventions (gradient filtering, weight-norm clamping, geometric penalties). Whether the delay specifically measures the time to form task-structured representations has remained observational. We test it causally by injecting representational priors of varying content into a one-layer transformer learning modular addition, via a supervised-contrastive loss whose positives encode (i) the task's true structure ($(a+b) \bmod p$), (ii) a coherent-but-wrong sibling ($(a-b) \bmod p$), or (iii) a random partition -- all with identical loss form, strength, class sizes, and geometry. Whether generalization occurs follows a clean gradation: true 22/30 runs, sibling (same periodic features, wrong combination) 14/15, random (only memorizable) 0/20 (Fisher $p=1.3\times10^{-7}$). A weight-norm-matched control replaying the norm trajectory onto plain cross-entropy generalizes 0/15, ruling out the norm as mediator. Probes show structure formation precedes and predicts generalization in all runs. Only the true structure also accelerates grokking (up to $2.75\times$), but this is dose-dependent and bimodal. We then confirm the mechanism by prediction: because the acceleration is gated by a weight-norm side-effect, clamping the norm during training yields a reliable, standalone accelerator with a median $8.6\times$ speedup (up to $22\times$ on the fastest seeds, under 1000 epochs), growing monotonically as the norm is held lower; the residual stalls also vanish, though significant only pooled over the two mitigations run at both strengths ($0/40$ vs $6/20$, $p=7.7\times10^{-4}$), not per method. The grokking delay is, causally, the time to form the right representational structure -- decided at the level of features, not labels.

We propose NetinfoGC, a framework for graph classification that extends the Network Usable Information (NUI) paradigm to graph-level learning. Unlike conventional graph neural network approaches that rely on end-to-end training of black-box embeddings, NetinfoGC constructs a family of permutation-invariant graph representations derived from propagation-based mechanisms and classical structural descriptors, including graph centrality measures. To evaluate representation quality, we introduce a training-free NUI estimation procedure based on clustering consistency with ground-truth labels, providing a proxy for task-relevant information without supervised learning. We further exploit the same representations using sparse-group LASSO regularization, enabling automatic selection of informative structural descriptors while suppressing redundant ones. Experiments on benchmark datasets show that classical centrality measures are highly competitive with learned propagation-based representations, and in several cases yield superior performance. Moreover, we observe a strong correlation between estimated NUI and downstream classification accuracy, validating NUI as an effective measure of representation utility. Overall, NetinfoGC provides a unified and interpretable framework for evaluating and exploiting graph representations without requiring end-to-end neural training.

The classical Kolmogorov--Arnold representation theorem states that any continuous multivariate function can be exactly decomposed into a finite composition of univariate continuous functions and addition operations. This foundational result has recently inspired the development of Kolmogorov--Arnold Networks (KANs) in classical machine learning, as well as their extensions into the quantum domain (QKANs). In this paper, we establish two quantum analogues of the Kolmogorov--Arnold representation theorem for continuous unitary-valued maps of several variables within an open $1$-neighbourhood of the identity matrix $O_1(\mathbf{I}) \subset \mathcal{U}(n)$. First, we prove a representation theorem that yields an exact additive decomposition inside the matrix exponent of anti-Hermitian-valued maps. Second, due to the non-commutative nature of quantum operators, we derive a factorised version expressing the target unitary map as a finite sequential product of univariate matrix exponentials. Finally, we provide a concrete topological counterexample based on the lifting property of $\mathcal{SU}(2)$ to demonstrate that these local representation theorems cannot be globally extended to the entire unitary group $\mathcal{U}(n)$ without encountering fundamental structural obstructions.

Massive Activations (MAs) have been widely observed in Transformer-based models, yet their structure and functional roles in Diffusion Transformers (DiTs) remain insufficiently understood. In this work, we systematically analyze MAs in representative DiTs and find that they are spatially distributed across image tokens while concentrated in a small set of fixed feature dimensions. We further show that these dimensions are closely aligned with AdaLN residual scaling factors and are primarily modulated by the denoising timestep rather than text conditions. This structure leads to two task-dependent effects: for generation, MAs are critical for fine-grained detail synthesis while having limited influence on global semantics; for understanding, their shared high-magnitude directions make raw DiT features overly similar across spatial tokens and weaken dense feature discrimination. Based on these findings, we introduce Eliciting Massive Activation (EMA), a training-free framework that leverages Massive Activations (MAs) as a unified modulation signal to improve both generative and representational capabilities of DiTs. For generation, EMA proposes MA-driven Detail G}uidance (DG), which suppresses MA dimensions to construct a detail-deficient counterfactual prediction and guides sampling toward finer visual details. DG further supports efficient partial-forward inference, integration with classifier-free guidance, and token-level Local DG for refining selected image regions. For understanding, EMA introduces MA-modulated REPresentation extraction (MREP), which uses pretrained AdaLN channel-wise modulation to reduce MA directional dominance and concatenates spatially normalized MA maps to preserve useful spatial structure. Extensive experiments demonstrate that EMA consistently improves both the generation quality and representation capability of DiTs.

While Vision Transformers have achieved remarkable success across computer vision and language applications, the geometric evolution of their internal representations throughout training remains insufficiently understood. Existing analyses primarily focus on attention mechanisms and downstream performance, leaving the evolution of representation geometry largely unexplored. In this work, we present Transformer Geometry Observatory-II (TGO-II), a representation geometry analysis framework designed to investigate how Transformer representations evolve during supervised training. TGO-II analyzes Vision Transformer (ViT-Small/16) representations using Centered Kernel Alignment (CKA), Singular Vector Canonical Correlation Analysis (SVCCA), Two-Nearest Neighbor Intrinsic Dimensionality (TwoNN-ID), and token covariance analysis. Our experiments reveal three key observations. First, both CKA and SVCCA progressively decrease throughout training, indicating increasing representational specialization across Transformer layers. Second, intrinsic dimensionality consistently increases before stabilizing, suggesting progressive expansion of the representation manifold into a larger set of locally accessible degrees of freedom. Third, token covariance and coupling analyses demonstrate that strong token interaction structure persists throughout training, challenging the hypothesis that increasing representational complexity arises primarily from progressive token independence. These findings suggest that representation complexity and layer specialization emerge simultaneously during training. Manifold expansion appears to occur without token decoupling. Together, these observations motivate a new hypothesis in which Vision Transformers increase representational complexity through progressively richer transformations while preserving strong token interaction structure during learning.

We study layered models, including feedforward networks, ResNets, and transformers, by limiting each layer to a width of $d = 3$, i.e., $\mathbb{R}^3$ as representation space. This allows us to track how a neural network changes low-dimensional topological invariants through its layers. Just about any topological structure may be simplified or even trivialized by simply increasing dimension; e.g., any knot is equivalent to an unknot in $\mathbb{R}^4$. By restricting to $\mathbb{R}^3$, we not only isolate the effects of activation and depth from that of width, we work in a space that lends itself to easy visualization. We focus on linking number here, deferring other invariants like link groups, Milnor's $\barμ$-invariants, knot types, ambient cobordisms, to a sequel. We provide full proofs and empirical experiments to justify the following insights: When measured by their power to effect changes in linking numbers, the layer-skipping feature in ResNets is as powerful as the attention mechanism in transformers; both ResNets and transformers are strictly more powerful than feedforward neural networks with monotonic activations, which are in turn more powerful than invertible and flow-based models; but replacing monotonic activation with a nonmonotonic one elevates a feedforward network into the same expressivity class as ResNets and transformers. These results suggest that low-dimensional topology can be a useful tool to guide designs of AI architectures. We also generalize our results from $d = 3$ to arbitrary $d > 3$.

Training in artificial neural networks can be viewed as a trajectory evolving through a high-dimensional loss landscape. However, the large number of trainable parameters makes the direct analysis of these dynamics challenging. In this work, we treat such training trajectories as temporal networks and apply recently proposed strategies for the scalar embedding of temporal networks. We investigate whether such a scalar embedding provides a meaningful low-dimensional representation of neural network training dynamics. Using a multilayer perceptron trained on the MNIST classification task, we show that the embedding preserves the main dynamical features observed in the original parameter space, including the emergence of sensitivity to initial conditions for specific learning rate regimes and an accurate reconstruction of the network's maximum Lyapunov exponent. We then use the embedded scalar trajectory to define a characteristic time, analogous to a Lyapunov time, after which the exponential separation between initially close embedded trajectories saturates. This characteristic time captures the typical decorrelation time between initially close network trajectories in the original high-dimensional system. Finally, we investigate the statistical organization of asymptotic training states through a spacing observable defined in the embedded space. We find that the distributions of rescaled asymptotic spacings collapse onto a common form across initial conditions and are compatible with a skew lognormal distribution. Altogether, our results suggest that scalar low-dimensional embeddings provide a useful framework for studying and visualizing the dynamical properties of neural network optimization trajectories.

Do language models know when they are being tested? This question matters for AI safety: a model that recognises an evaluation context could alter its behaviour strategically, making downstream benchmarks harder to interpret. Using 11 models spanning Qwen 2.5, Gemma 2, and Llama 3.2, we find a systematic size-dependent shift in representational depth: in both Qwen 2.5 and Gemma 2, the layer at which evaluation-awareness is most linearly recoverable moves from late layers in smaller models to early layers in larger ones. This suggests that scale changes not only the strength of evaluation-awareness but also where it is most linearly recoverable in the network. This depth shift helps explain why within-family scaling trajectories are non-monotonic or inverse rather than smooth and family-general, showing that a simple universal power-law account is not supported under denser within-family sampling. Finally, white-box probe signals are consistently stronger than black-box behavioural expression, and the relationship between the two varies by family in ways not predicted by probe AUROC alone.

While Low-rank adaptation (LoRA) enables highly efficient fine-tuning by constraining task-specific updates to fixed low-rank subspaces, this rigid design limits representational flexibility and often results in overconfident predictions and miscalibrated uncertainty, especially in low-data regimes. Recent Bayesian LoRA variants improve uncertainty estimation by modeling posterior distributions over adaptation parameters. However, these approaches typically rely on fixed or heuristically determined ranks, overlooking the inherently context-dependent nature of adaptation capacity. In this paper, we propose BaRA, a Bayesian Adaptive Rank Allocation framework for parameter-efficient fine-tuning. Drawing inspiration from probabilistic topic models, BaRA dynamically allocates adaptation capacity by activating a sparse, context-dependent subset of disentangled latent factors, enabling instance-wise variation in effective rank. This Bayesian formulation provides principled, data-driven capacity control, mitigating over-parameterization while preserving expressiveness. Beyond the modeling contribution, we provide a complexity-theoretic generalization analysis showing that the generalization gap of BaRA depends on the learned joint effective rank $\bar{s}_{Φ,θ}$ induced by the global-local gate, rather than the maximum rank $r$. This result explains why sparse adaptive rank allocation can reduce the effective hypothesis complexity while preserving input-dependent expressiveness. Extensive experiments on diverse natural language benchmarks demonstrate that BaRA consistently improves predictive performance, robustness, and uncertainty calibration compared to standard LoRA and existing Bayesian LoRA variants.

Current approaches to enhance Large Language Model (LLM) reasoning, such as Chain-of-Thought and "Wait" prompts, primarily encourage models to think more, yet often fail to guide them toward Truth. While Representation Editing (RepE) offers a intrinsic control, its application to dynamic reasoning trajectories remains underexplored. In this work, we bridge this gap by investigating the geometry of truth within unfolding reasoning chains. We uncover three critical insights: (1) Truth is encoded at the sentence level and is entangled with latent reasoning patterns; (2) Effective intervention follows an Uncertainty Principle and a Decay Effect, requiring localization to early, high-entropy forks; (3) Naive steering vectors suffer from noise, risking collateral damage to correct trajectories. Based on these findings, we propose DynaSteer, a dynamic RepE framework. DynaSteer employs pattern clustering to disentangle reasoning manifolds and utilizes Fisher-LDA to project purified truth. By dynamically monitoring lookahead entropy, it selectively steers and rolls back trajectories only when necessary. Comprehensive experimental results on several MATH benchmark verify the effectiveness of DynaSteer, and experiments on out-of-domain coding tasks further confirm its generalization ability. Our code is publicly available at https://github.com/tianlwang/DynaSteer.

Discrete audio representations have become increasingly popular for building multimodal text-audio systems and integrating audio capabilities into Large Language Models (LLMs). However, numerous studies report performance degradation on various downstream tasks due to information loss during discretization. To address this, we propose a novel approach combining temporally compressed discrete tokens with dimensionality-reduced continuous residuals. Our framework consists of a hybridized discrete-continuous focal modulation codec and a hybrid Transformer. This architecture performs autoregressive inference in the discrete domain, coupled with non-autoregressive prediction and continuous residual upsampling. Experimental results show that our approach significantly improves the retention of speaker characteristics compared to discrete-only methods, while simultaneously reducing the number of required autoregressive steps.

Adapting CLIP for open-vocabulary video recognition necessitates a delicate balance between newly acquired video knowledge and the pretrained generalization. While existing studies pursue this generalization-specialization trade-off with additional regularizations or constraints, we argue that they overlook the deviation of representations beyond the fine-tuning data distribution, resulting in suboptimal adaptation effects. We believe such deviation is inherited from the inconsistency between the fine-tuning and evaluation objectives, where model optimization is restricted to the known training distribution but evaluated on unseen ones. In this paper, we introduce \emph{TACO}, a simple yet effective framework to mitigate the potential negative effects induced by this inconsistency. Our key insight is that adaptation should preserve OOD-relevant alignment beyond the training distribution. To this end, we propose \emph{Relative Structure Distillation}, which regularizes the relative geometry of the representation space and suppresses harmful alignment shift during training. We further decouple the representation space from the optimization space with a lightweight specialization projection, allowing task-specific adaptation without directly overspecializing the representations used at test time. \emph{TACO} establishes state-of-the-art performance on diverse benchmarks under cross-dataset and base-to-novel settings. Code will be released at https://github.com/ZMHH-H/TACO.

Ordered bottlenecks aim to provide utility at flexible budgets by assigning coarse information to early tokens and task-relevant detail to later ones. Prior work, including tail dropping (TD), typically enforces ordering by means of a masking-based ordering pressure (MBOP): Late tokens are masked more frequently than early tokens and are therefore encouraged to store less essential fine details. We introduce predictive residual inference for ordered representations (PRIOR), a framework designed to address inherent weaknesses of MBOP. MBOP is prone to weak late-token utility because it lacks an explicit refinement objective and uses gradient exposure as a proxy for importance. Furthermore, representations may become particularly brittle in optimization-sensitive settings, such as when using discrete or quantized token representations. PRIOR replaces activation-rate control with log2-scaled levels and level-wise predictors. These predictors separate already explained from unexplained information, focusing each level on residual error. We compare PRIOR against MBOP-TD and independent tail-biased dropout (MBOP-ITD) in contrastive learning and image reconstruction tasks. Unlike the baselines, PRIOR learns well-ordered representations across experiments: low budgets provide coarse descriptors, while high budgets add refinements. Simultaneously, full-budget performance with PRIOR is higher in all but one experimental setting, where performance remains comparable. MBOP baselines are severely limited in discrete and quantized settings, while PRIOR approaches the performance of continuous counterparts. Taken together, these findings establish PRIOR as an effective framework for ordered representation learning.

User modeling in industrial recommender systems typically produces dense embeddings, which suffer from representational constraints inherent to fixed-dimensional vectors. An emerging alternative for discrete user representation -- using LLMs to generate text-based user tokens -- captures topical co-occurrences rather than deep sequential behavior dynamics and produces outputs that are difficult to ground to item attributes. Meanwhile, Semantic ID (SID) based item tokenization has proven effective for improving generalization in generative recommendation, yet discrete SID-based representations for users remain largely unexplored. We propose TokenMinds, an industrial-scale system that extends the PLUM framework from item retrieval to user modeling, generating both discrete SID-based user tokens and dense user embeddings via an encoder-decoder architecture adapted from pre-trained LLMs. This dual-output design provides the complementary benefits of discrete, semantically grounded user representations while maintaining compatibility with existing downstream models that rely on dense embeddings. Additionally, the shared SID vocabulary naturally extends to cross-scenario modeling: by unifying long-form and short-form video behaviors into a single model, we substantially reduce training and serving costs. We validate TokenMinds through extensive offline experiments and live launches on multiple YouTube surfaces, served on full user traffic (billions of users) via an asynchronous infrastructure that decouples representation generation from downstream scoring. Focusing on ranking as the primary downstream use case, our results confirm the practical viability of SID-based user tokens at industrial scale and demonstrate that tokens and dense embeddings provide complementary value across different production ranking systems.

Large automatic speech recognition (ASR) models such as Whisper must be deployed across hardware with widely varying memory and inference-speed constraints. We present a compression framework that jointly parametrizes Whisper deployment along \emph{six} dimensions: model size, temporal resolution, encoder token stride, low-rank adaptation capacity, weight precision and sparsity pattern. All axes are jointly optimized using NSGA-III with respect to three deployment objectives: word error rate (WER), inference FLOPs, and memory footprint. Across 50 of the 1,680 candidate configurations evaluated, we characterize the conditional effect of each axis and identify compression combinations that dominate naive single-axis scaling, while finding that 1:4 structured sparsity fails to recover acceptable accuracy under the tested recovery budgets. We report measured WER and resident memory, use analytical EffFLOPs as the search-time compute surrogate, and separately validate representative inference configurations using measured real-time factor (RTF).

Deep unrolled networks (DUNs) integrate physical forward models with learned regularization in cascaded network architectures, achieving exceptional performance in inverse problems while maintaining interpretability. While most DUNs operate in the object domain (e.g., image space), recent variants explored representation spaces for improved information flow. However, these methods rely on heuristic methods for data consistency (DC), sacrificing fidelity with measurements. In this work, we introduce DUNE (Deep Unrolled Networks in rEpresentation space), a framework that maintains exact adherence to physical measurements while operating in learned representation spaces. By deriving the DC gradient via the chain rule and implementing it through the Vector-Jacobian Product (VJP), we enable exact backpropagation of measurement residuals into the representation space. This formulation supports diverse architectural backbones, including pre-trained encoders to guide the iterative process. We assess DUNE against state-of-the-art baselines on accelerated MRI reconstruction tasks, demonstrating that exact VJP-based gradients yield superior reconstruction quality and structural fidelity across both single-channel portable low-field and multi-channel clinical high-field MRI acquisitions. The code will be available upon publication at https://github.com/EfeIlicak/DUNE.

Continual learning (CL) systems often forget previously acquired knowledge, yet the mechanisms driving forgetting remain hard to isolate in practice because real datasets entangle many factors. We present a controlled, toy-world framework that makes these mechanisms observable and testable. Using a synthetic generator-separator pipeline, we define ground-truth latent features, build tasks with tunable sparsity and overlap, and introduce measurable quantities for representation strength and superposition (directional overlap among features). We then study retention dynamics-the temporal change of representation strength by fitting sparse dynamical relations (via SINDy) between retention, superposition, and exposure history. A complementary task-level analysis based on effective rank characterizes how representational capacity is allocated across tasks. Our controlled experiments yield three takeaways. (1) Superposition tends to increase over time with transient dips at task boundaries, suggesting boundary-specific interference rather than steady drift. (2) Higher feature sparsity induces more superposition yet does not inevitably cause forgetting; when representations remain strong, forgetting can be reduced despite overlap. (3) Task-level effective rank grows with sparsity, indicating broader capacity usage under sparse regimes. Together, these results nuance the common intuition that more superposition leads to more forgetting by showing that overlap interacts with representation strength and capacity allocation. Our toy analysis provides falsifiable hypotheses and diagnostic tools for CL.

MultiModal Federated Graph Learning (MM-FGL) offers a natural collaborative training paradigm, but its practical deployment is challenged by two granularities of modality imbalance. Client-level imbalance occurs when certain clients lack entire modalities, while node-level imbalance occurs when individual nodes exhibit missing visual or textual attributes. While several relevant studies exist, our investigation reveals that they predominantly target graph-agnostic or centralized scenarios, rendering them difficult to adapt directly. To address these challenges, we formalize modality-imbalanced MM-FGL as an implicit graph-aware latent semantic representation synthesis problem. This paradigm recovers missing modal semantics directly within the representation space, thereby maximizing alignment with the original data's semantic distribution and mitigating the high variance induced by missing modalities. To this end, we propose FedMGS (Federated Modality-aware Graph Synthesis), which integrates three core components. The availability-aware graph encoder prevents missing modalities from contaminating local structural propagation. The prototype-guided latent semantic synthesizer establishes cross-client semantic anchors for unavailable modalities. The reliability-calibrated semantic fusion mechanism regulates the impact of recovered latent representations prior to predictive readout. Extensive experiments on four tasks show that FedMGS consistently outperforms competitive baselines with gains up to 17.41% with best efficiency-performance tradeoff.

Scaling end-to-end autonomous driving to complex, open-world environments requires perceptual models that generalize to anomalous scenarios and planners that produce kinematically valid trajectories. Existing paradigms face a distinct dichotomy between representational efficiency and generalization capacity. Dense models (e.g., occupancy networks), while geometrically robust, incur critical computational bottlenecks and struggle with high-level semantic reasoning. Conversely, sparse, query-based planners are efficient but reliant on closed-set definitions, rendering them vulnerable to out-of-distribution (OOD) events. Although recent Vision-Language-Action (VLA) models offer open-vocabulary reasoning, their autoregressive, discrete token generation fundamentally conflicts with the continuous, high-frequency control requirements of vehicle dynamics. To address this, we propose Lagrange, an open-vocabulary, computationally sparse driving framework based on Masked Latent Fields (MLF). Rather than relying on dense volumetric reconstructions or closed-set query mechanisms, Lagrange exploits Vision-Language Models (VLMs) to encode class-agnostic object proposals into continuous semantic visual tokens. We introduce an intent-driven masked cross-attention module that temporally filters irrelevant entities, decoding the attended tokens into an implicit continuous energy field defined over spatial coordinates. By framing decision-making as a Lagrangian action minimization problem spanning this energy field, we enforce strict compliance with vehicle kinematics while executing collision avoidance. Extensive offline evaluations on both standard (nuScenes) and long-tail (CODA) benchmarks demonstrate that Lagrange establishes a promising framework for robust, interpretable, and kinematically feasible open-world autonomy.

Despite the widespread adoption of Vision Transformers (ViTs) and their success across numerous computer vision applications, the fundamental understanding of their dimensional and representational geometry remains relatively underexplored. To address this gap, we introduce Transformer Geometry Observatory (TGO), a systematic framework of experiments and analysis pipelines designed to investigate the representational geometry and dynamics of Vision Transformers. TGO-I, the first installment of the framework, focuses on the spectral geometry of ViT representations. Using a ViT-Small/16 model trained on ImageNet-100, we analyze Effective Rank, Stable Rank, Participation Ratio, Spectral Entropy, Spectral Flatness, Spectral Anisotropy, covariance structure, eigenspectra, and singular value spectra throughout training. Our results reveal a consistent increase in dimensional utilization, accompanied by decreasing anisotropy, increasing spectral entropy, increasing participation ratio, and progressively flatter eigenspectra. Contrary to the common intuition that training should concentrate information into a small number of dominant directions, we observe a progressive redistribution of variance across representational dimensions. This phenomenon is particularly pronounced in the final CLS token representation, which exhibits the highest effective dimensionality and lowest anisotropy within the network.

Decoder-only Transformers compute attention over the KV cache of preceding tokens. Keys (and Values) are typically represented with the same dimensionality, regardless of its distance from the prediction target. In natural language, however, the next word is most strongly influenced by the immediately preceding tokens. We hypothesize that local and distant tokens impose asymmetric demands on representational capacity: local tokens are more critical for predicting immediate outputs and thus require richer representations, whereas distant tokens primarily serve as long-range memory, for which lower-dimensional representations may suffice. We formalize this idea as Distance-Adaptive Representation (DAR), implemented in a controlled setting that preserves full-dimensional representations within a local context window while assigning reduced-dimensional representations (e.g. 1/4 of the original dimensionality) to tokens beyond that window. Across multiple pretraining scales (70M to 410M parameters), as well as continued supervised fine-tuning on a 1B-scale model, this approach closely matches the performance of full-dimensional baselines. In contrast, uniformly reducing dimensionality across all token positions leads to worse performance. These results challenge the common assumption that key and value dimensionality should be uniform across token positions. Our findings suggest a new direction for designing attention architectures that adaptively allocate representational capacity across sequences, enabling further reductions in KV cache during inference.

The full potential of artificial intelligence in tibial plateau fracture characterisation remains unrealised, constrained by a fundamental dependency on labelled datasets whose consistency cannot be guaranteed: conventional classification schemes such as Schatzker and AO/OTA suffer from inter-observer variability, causing supervised models to learn human disagreement rather than stable fracture morphology. We design, implement, and validate a label-agnostic framework that eliminates this constraint by learning fracture representations directly from imaging data without observer-assigned labels. A RadImageNet-pretrained ResNet-50 encoder is fine-tuned on 154 cleaned knee radiographs using the SimCLR contrastive objective, preceded by a data cleaning protocol and followed by UMAP dimensionality reduction and k-means clustering to discover four imaging-derived phenotypes. Phenotype validity is assessed through a blinded expert review protocol administered to two independent clinicians. The four phenotypes demonstrate robust stability (bootstrap ARI = 0.319 +/- 0.041), strong internal cohesion (silhouette = 0.511), and coherence ratings of 3-5/5 from both reviewers under blinded conditions; one phenotype was unanimously identified as exhibiting comminution -- a high-complexity feature isolated without any supervisory signal. Inter-partition comparison against Schatzker labels yields ARI = 0.013, confirming orthogonality to conventional classification boundaries. Notably, expert reviewers anchored to established classification vocabularies perceived imaging-derived groups as heterogeneous precisely where Schatzker alignment was lowest, suggesting that Schatzker-trained perception and label-agnostic embedding geometry measure orthogonal dimensions. These findings establish label-agnostic SSL phenotyping as a reproducible and clinically interpretable complement to conventional classification.

Understanding the meaning of negated sentences remains one of the challenges for language models, even in the era of large language models (LLMs). We analyze systematicity regarding LLM understanding of negation from two perspectives: behavioral systematicity and representational systematicity. For behavioral systematicity, we confirm that through demonstrations and in-context learning, LLMs can recognize negation expressions and scope within sentences to some extent, but they fail to achieve perfect performance. In particular, the difficulty of the negation scope recognition for models varies depending on the output format. For representational systematicity, we analyze the extent to which function vectors can be robustly constructed from in-context examples for tasks that are essential to understanding negation. The experiments suggest that while function vectors can be composed for negation cue extraction tasks, extracting function vectors for recognizing scope is more challenging.

Large language models (LLMs) are widely used in text-to-image (T2I) systems, but they are typically limited to text encoding, while denoising is handled by newly trained generative backbones. The emergence of representation autoencoders (RAEs) shifts the generation target toward semantically structured visual representations, creating a latent space that is more compatible with pretrained LLM priors. Inspired by multimodal LLMs (MLLMs), where an MLP projector is sufficient to align clean visual representations with a pretrained LLM, we repurpose the MLLM itself as a noisy representation encoder, extending this mechanism from clean to noisy inputs. We present RepFusion, which uses the resulting MLLM outputs as the conditioning signal for a diffusion transformer. In controlled comparisons at similar inference budgets, RepFusion outperforms baselines that devote comparable capacity to newly initialized denoisers. These results demonstrate that MLLMs provide strong priors for denoising visual representations and that, by conditioning on evolving noisy representations, test-time compute can be productively spent on repeated MLLM conditioning in modern T2I systems.

Contact-rich manipulation requires world models to capture complex interaction dynamics from heterogeneous visual and tactile observations, yet the representation properties that enable reliable predictive planning remain poorly understood. We present ContactWorld, a systematic study of vision-tactile representations across 12 contact-rich manipulation tasks. Through controlled evaluation within a unified world-model and planning framework, we find that representations preserving spatial structure and temporal continuity consistently support more accurate prediction and stronger planning performance. Point-cloud observations increase average success from 20.7% and 22.0% with wrist- and front-view RGB, respectively, to 32.1%. Tactile sensing provides further gains only when its representation is compatible with the visual modality, with point clouds and tactile force fields achieving the highest overall success rate of 36.1%. These advantages become more pronounced at increasing goal offsets, where prediction errors and contact uncertainty accumulate. Controlled representation studies and real-world experiments across four manipulation tasks further support these trends. Together, our results establish spatial structure, temporal continuity, and cross-modal compatibility as key principles for designing vision-tactile world models for contact-rich robotic manipulation.

Vision-language models (VLMs) can answer image-based questions confidently, and often correctly, even when no image is provided. This mirage behavior inflates benchmark scores without reflecting visual grounding. Prior work treats this as a single failure mode. We argue it is two. Using Mirage Probes, a contrastive probing framework that pairs paraphrased question variants with matched mirage and non-mirage labels on the same image, we show that mirage behavior is linearly decodable from internal activations across residual stream, MLP, post-attention, and attention-head sites in two open-source VLMs. We demonstrate that a Naive Bayes text baseline cannot recover this signal, ruling out surface lexical confounds. Cross-benchmark separability patterns, together with a novel Prior Harnessing Index (PHI) measuring how much a model can answer from text alone, expose two distinct regimes: textual biases, where the model answers from language priors without engaging visual representations, and spurious images, where it constructs false visual content in latent space and answers as if grounded. The distinction has direct mitigation consequences: text-distribution cleaning can address the first regime but cannot reach the second, since spurious-image mirages live in the model's visual representations rather than its text. Faithful visual grounding will require interventions at the representational level.

Self-consistency improves LLM reasoning by sampling multiple outputs and selecting the most consistent answer, but existing formulations largely rely on exact matching and therefore remain limited to tasks with categorical outputs. In this work, we study self-consistency in open-ended generation tasks such as code synthesis and text summarization. We hypothesize that consistency can be understood as a geometric property of the generation space, where semantically compatible generations concentrate in similar regions of representation space. To study this hypothesis, we introduce Embedding-Based Agreement (EBA), a simple training-free operationalization that estimates agreement by clustering sampled generations in embedding space. Through experiments on mathematical reasoning, code generation, and summarization, we show that agreement in representation space provides a robust and scalable signal of self-consistency for open-ended tasks. In particular, EBA consistently outperforms random selection and exhibits more stable scaling behavior than recent selection approaches based on LLM evaluation or uncertainty estimation. We further show that these agreement signals remain stable across model families and embedding spaces, even with native hidden representations. Finally, our analysis shows that the geometric location occupied by sampled generations is strongly correlated with generation quality: generations concentrated near central regions of representation space tend to correspond to more reliable outputs, whereas peripheral generations are substantially less accurate. Overall, our findings support viewing self-consistency as a property of the geometric organization of sampled generations rather than exact symbolic overlap.

Learning compatible representations aims to learn feature representations that can be used interchangeably over time whenever a model undergoes updates. In this paper, we demonstrate that stationary representations learned by d-Simplex fixed classifiers imply compatibility as in its formal definition. This result establishes a foundation for future works and can be directly exploited in practical learning scenarios. We address the challenge of learning compatibility using $d$-Simplex fixed classifiers when the model is sequentially fine-tuned. Learning according to a d-Simplex fixed classifier with the cross-entropy loss aligns feature distributions at the first-order statistics. Consequently, it may not fully capture higher-order dependencies in the representation between model updates. To address this issue, we demonstrate that training the model using a $d$-Simplex fixed classifier through a convex combination of the cross-entropy loss and a contrastive loss not only captures higher-order dependencies, but is also equivalent to learning with the cross-entropy under the compatibility constraints. We confirm our findings with extensive experiments also considering a new scenario where a pre-trained model is sequentially fine-tuned and occasionally replaced with an improved model. We show that stationary representations enable uninterrupted retrieval services (without reprocessing gallery images) while improving performance during model updates and replacements, achieving state-of-the-art. Code at https://github.com/miccunifi/iamcl2r.

Diffusion models have demonstrated remarkable generative capabilities and have also emerged as powerful self-supervised representation learners, yet the connection between these two abilities remains less explored. Drawing inspiration from self-supervised learning (SSL), we introduce a framework for jointly evaluating the representation and generation capabilities of diffusion models. Specifically, we decompose features into invariant and residual components and derive the Invariant Contamination Ratio (ICR), a Fisher-based metric that quantifies how residual variation contaminates invariant signal in feature space. We use this framework to analyze both discriminative and generative behavior of diffusion models. On the representation side, we find that invariance peaks at intermediate noise levels, which also yield the best downstream classification performance. On the generative side, we study how training transitions from genuine generalization to memorization in data-limited regimes, and show that ICR serves as a sensitive training-time indicator of early learning: increasing residual energy along Fisher directions marks the onset of memorization, detectable from training features alone without external evaluators or held-out test sets. Overall, our results show that diffusion models can be monitored from a self-supervised perspective through the geometry of their learned representations.

Large Language Models (LLMs) and Vision-Language Models (VLMs) are increasingly evaluated on table reasoning tasks, but the role of table representation remains under-explored. In practice, the same table content may appear in different structural formats, such as HTML, Markdown, and LaTeX, or as rendered images. However, existing evaluations often let content, format, layout, and modality vary together, making it difficult to isolate representation effects. We introduce TABVERSE, a controlled multimodal table benchmark that aligns the same table content across multiple structural formats and rendered images, with question category and difficulty tags. This design enables systematic evaluation of representation effects while holding table content fixed. We evaluate LLMs and VLMs across three tasks: Question Answering (QA), Structural Understanding Capability (SUC), and Structure Reconstruction (SR). Our results show that representation choice substantially affects table understanding. Models generally perform better with structured text than with rendered images, but the size of this gap depends on the task, model, and format. HTML is often the most robust text format, while row-sensitive structural tasks and syntactically usable LaTeX reconstruction remain challenging. These findings show that table representation is a key factor in reliable table evaluation.

We introduce CHIAR-Former (CHIAroscuro Attention-based tRansFormer), an efficient transformer that routes each token to either DCT spectral mixing (O(d log d), sub-quadratic) or full self-attention (O(n^2 d), quadratic in sequence length n) based on per-token spectral entropy H(x) in [0,1], which measures the frequency-domain complexity of each token embedding x. We make three contributions: (1) we discover routing collapse -- a three-operator system collapses to DCT+Attention, revealing the optimal operator subset; (2) we propose a learned task-level MetaRouter g = sigma(Linear(x-bar)) in [0,1], where x-bar is the batch-mean embedding and g soft-blends spectral and identity paths end-to-end; and (3) we demonstrate 35-40% FLOP reduction at 400M parameters with a 3.93 PPL cost on WikiText-103 (Test PPL 27.51 vs. 23.58). Under mixed-dataset training, CHIAR-Former dramatically outperforms full attention on small corpora, confirming the regularisation value of spectral mixing. The MetaRouter stabilises at g ~ 0.22, indicating that at scale the model reaches a robust compute-quality equilibrium: attention layers absorb representational complexity while spectral preprocessing efficiently anchors low-frequency structure.

Large Language Model (LLM) safety has often been evaluated at the behavior level, which provides limited evidence of internal robustness, as these evaluations target outputs rather than representation-level vulnerability under intervention. We formalize this discrepancy as the audit gap: the difference between behavioral safety and robustness under intervention. To study this gap, we construct dissociated models that preserve safe outward behavior while remaining vulnerable in the latent space. We introduce an intervention-based evaluation framework to test model robustness through soft interventions in parameter and latent spaces, including harmful fine-tuning and layer-wise latent perturbations. To formalize the evaluation, we propose the Latent Vulnerability Score (LVS) to measure how easily harmful behavior can be elicited by bounded latent perturbations. Using this evaluation framework, we show that behavioral safety metrics are insufficient measures of representation-level robustness across multiple safely and unsafely aligned state-of-the-art models. Notably, dissociated models show substantially elevated LVSs despite comparable refusal behavior under harmful intervention, with intermediate representations being the most sensitive to intervention. Our results suggest that behavioral safety evaluation alone provides an incomplete picture of model robustness, motivating representation-aware audits of latent vulnerability and observable behavior.

Topological Data Analysis (TDA) offers a principled, intrinsic lens for comparing neural representations. However, existing paired topological divergences (e.g., RTD) are limited by heuristic asymmetry and, more critically, unbounded scores that depend on sample size, hindering reliable cross-scenario benchmarking. To address these challenges, we develop a unified topological toolkit serving two complementary needs: fine-grained structural diagnosis and robust, standardized evaluation. First, we complete the RTD framework by introducing Symmetric Representation Topology Divergence (SRTD) and its efficient variant SRTD-lite. Beyond resolving the theoretical asymmetry of prior variants, SRTD consolidates diagnostic information into a single, comprehensive cross-barcode signature. This allows for precise localization of structural discrepancies and serves as an effective optimization objective without the overhead of dual directional computations. Second, to enable reliable benchmarking across heterogeneous settings, we propose Normalized Topological Similarity (NTS). By measuring the rank correlation of hierarchical merge orders, NTS yields a scale-invariant metric bounded between -1 and 1, effectively overcoming the scale and sample-dependence of unnormalized divergences. Experiments across synthetic and real-world deep learning settings demonstrate that our toolkit captures functional shifts in CNNs missed by geometric measures and robustly maps LLM genealogy even under distance saturation, offering a rigorous, topology-aware perspective that complements measures like CKA.

We develop a high-dimensional statistical theory of low-rank adaptation (LoRA) in attention models, capturing the interplay between pre-training and fine-tuning. We introduce a solvable framework in which a single-head attention layer is first pre-trained on a data-abundant task and subsequently adapted via a rank-one LoRA update on limited data. In the high-dimensional limit, both stages admit a sharp asymptotic characterization in terms of a finite set of order parameters, yielding explicit predictions for test errors and representation alignment. Our analysis shows that the impact of pre-training on LoRA is summarized by an effective noise term, from which we derive prescriptions for the optimal pre-training procedure. We also demonstrate a regime with a mismatch between the value of the test error and representation quality, and propose an application of our theory to active fine-tuning.

Across contemplative, philosophical, and psychological accounts, human consciousness is often described along a similar spectrum, ranging from reactive and self-focused patterns to more integrative and coherent ones. Understanding whether language models encode such a structured, human-interpretable consciousness spectrum in representation space is important for model guidance, evaluation and alignment. In this work, we study the geometric structure and dynamics of patterns along this spectrum in transformer embedding spaces. We show that embeddings exhibit a globally organized geometry aligned with this spectrum: sentences associated with similar states cluster into locally coherent regions, forming a structured manifold. In particular, higher-level and lower-level regions exhibit convexity-like stability, while intermediate regions form a transition corridor. Dynamically, both utility-guided and geometry-only greedy trajectories consistently traverse from lower- to higher-level regions, passing through intermediate tiers, indicating that navigability is an intrinsic property of the representation space, guided but not dictated by a global directional signal. These results suggest that embedding spaces encode structured and navigable geometry aligned with a hypothesized consciousness-spectrum taxonomy, broadly inspired by recurring structural descriptions of human consciousness across contemplative traditions, philosophy, and modern psychology, providing a representation-level perspective for analyzing and guiding model behavior.

Knowledge benchmarks for LLMs face three issues: scaling-driven designs that do not operationalize disciplinary representativeness; flat-payment annotation that permits lazy consensus; and unaudited ranking instability under bounded test budgets. We introduce KINA, an 899-item benchmark across 261 fine-grained disciplines, with two formal results. First, we cast representativeness as a coverage-style objective over expert-elicited anchors and operationalize disciplinary representativeness through a proxy, yielding a (1-1/e) greedy approximation (Proposition 1); the guarantee applies to the proxy, not to population representativeness. Second, we prove a bonus-on-bar tournament weakly FOSD-dominates flat payment in released-review quality, with incentive-compatibility threshold B > Delta C / Delta p_min (Theorem 1). Evaluating 42 models from 13 labs, the top model, Gemini-3.1-Pro-Preview, reaches 53.17%, followed by Claude-Opus-4.6 at 49.92% and GPT-5.4 at 48.55%, leaving substantial headroom below saturation. The full leaderboard shows a tiered structure rather than a smooth total order: a small frontier tier lies above 48%, a dense strong-model tier spans roughly 38-45%, and low-performing models remain only modestly above the 10% chance baseline. Tool augmentation adds up to 5.17 points across the five tool-use evaluations, with gains varying substantially across models. We report bootstrap ranking-stability statistics to make bounded-budget variance explicit and to discourage over-interpretation of adjacent ranks.

Many striking phenomena in deep learning, such as linear mode connectivity and the structured behavior of training dynamics, are closely tied to parameter symmetries: transformations that leave the realized function unchanged. Despite growing attention to parameter symmetries, the exact interplay between parameters, data, and representations remains underexplored. To investigate this, we develop a theoretical framework of effective function classes, i.e., the set of functions a neuron can realize on its input support, and the norm cost of realizing them. We then formalize effective symmetry breaking via neuron identifiability across independent training runs. Our analysis shows that neural networks can admit large families of approximately equivalent solutions even in structurally asymmetric models. We further show that neuron identifiability enables representation merging without prior alignment, and characterize when such merging admits a linear low-loss path. These findings highlight the role of effective function classes in affecting the loss landscape.

Formal languages have proven to be effective conduits to understand the inner mechanisms of transformers. Past work has shown that transformers trained on next token prediction over counter languages learn representations consistent with an underlying stack structure. Beyond representational analysis, this paper investigates the causal role of these representations. Linear probes are trained to predict the stack depth at each token from the model's hidden states, and a principal representation direction is extracted from the probe. Ablation of this direction from the model causes sequential accuracy to collapse to near 0%, providing strong empirical evidence that the stack representation is not just learned, but is causally necessary for model performance.

Self-supervised learning (SSL) has emerged as a powerful paradigm for learning meaningful representations from unlabeled data. However, the standard protocol for evaluating these representations, linear probing, is computationally expensive, sensitive to hyperparameters, and provides limited insight into the geometric structure of the representation space. In this work, motivated by connections between neural network generalization and intrinsic dimension (ID) we propose IdEst, a method for estimating the ID of SSL representations via the Minimum Spanning Tree dimension estimator ($\mathrm{dim}_\mathrm{MST}$). Across diverse datasets, architectures, and SSL pretraining objectives, we show that IdEst strongly correlates with downstream linear probe performances. Furthermore, we demonstrate that IdEst enables efficient hyperparameter selection, significantly reducing the computational cost compared to supervised alternatives. Our results highlight intrinsic dimensionality as a principled geometric proxy for assessing SSL representations, complementing standard supervised probing protocols.

Foundation models have sparked a revolution via a pretraining-adaptation paradigm, with recent efforts extending this success to graphs. Unlike other modalities, graphs contain rich structural patterns, yet their structural transferability remains poorly understood. Prior studies consider common substructures in the discrete realm, and we are motivated by a fundamental question: Are common substructures transferable? The underlying theory is largely underexplored. In this work, we shift toward learning transferable structures through the lens of functional behavior. Theoretically, we connect transferable substructures to intrinsic geometry of the representation space. However, characterizing such intrinsic geometry has rarely been touched. Grounded in Riemannian geometry, we develop a graph intrinsic geometry learning framework called Neural Vector Bundle, which enables parsing intrinsic geometry with local coordinates. Building on this, we design GAUGE, a pretrainable neural architecture that constructs the vector bundle, flattening geometrically compatible local coordinates, and a new Dirichlet loss, which also measures the transfer effort. We empirically validate its superior expressiveness in challenging tasks including zero-shot link prediction and graph isomorphism.

Model dimension ($d_{model}$) is a fundamental hyperparameter in transformer language models, yet its role in setting the geometric limits of feature representation remains under-explored. Grounded in the Linear Representation and Superposition Hypotheses - which propose that models encode features as near-orthogonal directions in latent space - we develop a framework for estimating how many such directions a model can support. We first establish the embedding matrix as a measurable proxy for near-orthogonality constraints across the latent space: the boundary between meaningful token relationships and incidental similarity in the pairwise cosine similarity distribution gives a concrete estimate of the model's accepted deviation $\varepsilon$ from perfect orthogonality. Applying this metric across dozens of open-source models reveals two classes: models with high $\varepsilon$ whose embeddings lack near-orthogonal structure, and models with low $\varepsilon$ that maintain it. We then show that the standard Johnson-Lindenstrauss lemma greatly underestimates the packing efficiency of trained representations, and derive an adjusted capacity formula in which the number of near-orthogonal directions depends on the ratio of vectors to dimensions ($k/d$) rather than the raw count - a single modification that cuts prediction error by two orders of magnitude with no extra parameters. Combining these results, we define representational capacity as an upper bound on the number of distinguishable directions available for features and embeddings in a model's latent space. Capacity is exponentially sensitive to $\varepsilon$, and larger models favor tighter orthogonality constraints over maximizing raw capacity - a pattern compatible with several explanations (a stability-capacity trade-off, a ceiling on usable concepts, or confounds with model scale) that we leave to future work.

Generating objects with specific symmetries is essential in various real-world scenarios. However, adapting existing 2D continuous representations to enforce planar group symmetry remains a challenge, as the transformation of non-reflective group elements may disrupt continuity. To overcome this limitation, we propose a symmetrization framework for arbitrary planar groups. Our method transforms any 2D continuous representation into a symmetric one while preserving continuity. We provide the mathematical formulation of this representation, demonstrate its approximation capability for symmetric functions, and detail the construction methodology. We validate our approach through three visual design tasks (pattern design, paper-cutting design and stylized topology design) and one material design task. Experiments confirm that our representation enables effective symmetry control and demonstrate its broader applicability.

Learning a good action embedding space is fundamental to scalable robot policy learning, yet existing methods treat action latents as task-specific intermediates rather than first-class representations. The resulting latents are unstructured, embodiment-specific, and weakly tied to motion semantics, limiting interpretability, controllability, and transferability across robots. We position the action embedding space itself as a first-class design target, with downstream policy quality emerging from representation quality. Exploiting motion's intrinsic periodicity, we factorize it into a phase manifold that captures cyclic structure via FFT-parametric coefficients, together with a pose branch that conditions the manifold on non-periodic configuration detail. Combined with motion-semantic distillation, this factorized structure yields a cross-embodiment motion manifold that is interpretable and embodiment-agnostic by design. Anchoring multiple humanoid robots to a shared human-pretrained manifold then produces a unified action embedding space across diverse platforms, achieving strong cross-embodiment retrieval and consistent gains on downstream robot tasks.

Large language models (LLMs) have excelled in processing multiple languages through pre- and post-training on multilingual data, even though English dominates the training data. Prior work focusing on token representations has revealed how those LLMs process non-English text. Although these analyses have provided insightful findings, they fail to capture a structural view, which is an inherent property of language. In this study, we explore the multilinguality of LLMs through representational structural analysis. Our findings reveal that low-resource languages are structurally more different from English than high- and mid-resource languages, and that language-specific post-training alters their structures while preserving inter-language relationships.

Biological and neuromorphic recurrent neural networks (RNNs) are subject to spatial and temporal locality constraints on the information that can plausibly be used during learning. A common strategy to satisfy these constraints is to modify gradient descent by neglecting non-local terms to varying degrees, as in random feedback local online (RFLO) learning and truncated backpropagation through time (tBPTT). However, the learning dynamics of these algorithms, and how they compare with BPTT, remain poorly understood. We apply dynamical systems theory to data-aligned linear RNNs -- whose dynamics can be separated into orthogonal modes -- to compare stationary solutions, stability properties, and convergence rates, finding qualitatively distinct behaviour for RFLO versus BPTT and one-step tBPTT. We further observe that the solutions learned by RFLO are restricted to low-rank perturbations of initial parameters, a result which holds beyond the data-aligned setting. Our work provides analytical insight into how locality constraints shape learning dynamics, with implications for neuroscientific models of learning and alternative optimization approaches for RNNs.

Unified multimodal models (UMMs) aim to handle perception and generation in a single model. Yet existing UMMs still rely on a frozen, separately pretrained VAE for image generation, imposing a structural bottleneck. Naively removing it introduces a quality gap, as the model must learn both high-level structure and low-level details from raw pixels. In this paper, we propose Representation Forcing (RF), a technique that closes this gap by making representation prediction a native capability of the model. Concretely, RF forces the decoder to autoregressively predict visual representations as intermediate tokens before pixels; these tokens then stay in context to guide pixel diffusion within the same backbone. By turning representations from perception outputs into generation targets, RF eliminates the need for any external generative latent space. We find that RF benefits both understanding and generation. On image generation, our pixel-space model with RF matches state-of-the-art VAE-based unified models. On image understanding, pixel-space RF generally outperforms its VAE-based variant. Together, these results offer an effective step toward end-to-end, bottleneck-free UMMs.

Many languages are written in multiple scripts, requiring large language models (LLMs) to generate equivalent linguistic content in distinct orthographic forms. While prior work suggests that LLMs route information through shared latent representations, how they internally mediate script variation remains poorly understood. We study this question by first examining per-layer output distributions with the logit lens, which reveals consistent latent romanization during transliteration, and then through representational and mechanistic analyses of script generation. At the representational level, we show that scripts of the same language become increasingly separable across layers and that a simple linear steering direction can flip a model's output script while largely preserving semantic content. The vector generalizes to writing systems unseen during construction: it flips non-Latin output to Latin, but maps Latin output into varied non-Latin scripts. At the mechanistic level, we localize a small set of late-layer attention heads that causally mediate script choice. These heads transfer across unrelated languages and writing systems, suggesting that script routing is implemented by language-agnostic components. Across both analyses, we observe a consistent directional asymmetry: non-Latin output is produced by a compact, identifiable gate, while Latin-script output emerges from diffuse contributions across the model. Overall, our findings hint that LLMs organize script variation around shared latent representations while exhibiting a privileged substrate toward Latin script.

Best-of-$N$ sampling is widely used to construct pairwise preference data: $N$ candidates are drawn from a base distribution, and the best is paired with a rejected response. Despite its widespread use, what Bradley--Terry (BT) reward learning extracts from such data, and how to choose $N$ and the base distribution, remain unclear. We specialize a recent analysis of preference data via its induced conditional distribution to Best-of-$N$. For independent-reference variants, we derive closed-form reward targets as explicit functions of $N$ and the base distribution, and show that they preserve the latent reward ranking. For the practical Best-vs-Random and Best-vs-Worst variants, chosen and rejected responses are coupled through the same candidate set, so exact BT representability generally fails; nevertheless, bounded-class minimizers approach the reference targets as $N$ grows. Although margin and connectivity are known to govern sample efficiency in pairwise preference learning, Best-of-$N$ couples them through $N$ in opposing directions: larger $N$ widens pairwise margins but reduces connectivity. This trade-off yields two design principles: use larger $N$ when preference labels are the bottleneck, smaller $N$ when generation is the bottleneck; and shape the base distribution to place mass between the responses whose comparison matters most at test time. Experiments on synthetic and real preference data support the predicted dependence on sample size and base-distribution shape.

Independently trained neural models typically converge to incompatible latent representations, creating a fundamental barrier to highly modular AI systems. While Relative Representations (RR) address this by mapping absolute coordinates to a shared space defined by similarities to common anchor points, traditional implementations rely on randomly sampled anchors and cosine similarity, which frequently fail to capture the anisotropic geometries of modern architectures like Transformers. In this work, we propose a robust framework for cross-model communication based on two improvements. We learn anchors as robust semantic prototypes and utilize a geometry-aware similarity metric which preserves discriminative magnitude information and is invariant to affine shifts. Our approach demonstrates significant gains in performance and consistency across vision and language tasks. Notably, it enables nearly lossless information transfer and stable zero-shot communication even between highly heterogeneous architectures, such as small language models of varying scales.

Printed circuit board (PCB) schematic design defines nearly all electronic hardware, but it remains manual and expertise-intensive. While generative AI has advanced digital and analog IC design, PCB schematic generation from natural-language intent is largely unexplored. This paper presents SchGen, the first large language model that generates editable PCB schematics from natural-language requests. The key challenge lies in the lack of an LLM-suited representation and a large-scale dataset. Current schematic formats are dominated by verbose, tool-specific syntax and geometry-heavy descriptions, making them difficult to generate reliably. We introduce a semantically grounded code representation that encodes schematic editing primitives with relative placement and pin-name-based wiring, transforming a geometry-driven generation problem into a semantics-driven matching task amenable to LLMs. We further construct a large-scale dataset of PCB schematics paired with user prompts via a human-agent collaborative pipeline that converts open-source hardware designs into our representation. Experiments show that SchGen significantly outperforms alternative representations and even larger general-purpose LLMs on wire connectivity accuracy and functional correctness. Our results highlight the critical role of representation design in enabling generative models for complex hardware design tasks.

Understanding how Vision-Language-Action (VLA) models transform multimodal knowledge into embodied control remains an open challenge. We present VLA-Trace, a progressive diagnostic framework that analyzes VLA models through a unified evidence chain from representation dynamics to causal control attribution and behavioral manifestation. It specifically combines cross-modal and checkpoint-drift centered kernel alignment (CKA) to trace representation evolution, attention knockout interventions to identify modality-specific control pathways, and rollout-level behavioral probes to examine grounding, shortcut dependence, and semantic following. Experiments on $π_{0.5}$ and OpenVLA reveal three key findings. First, the two models exhibit distinct modality-specific adaptation dynamics during VLA finetuning. Second, they rely on different multimodal routing strategies and layer-wise dependencies during action decoding. Third, although VLA policies excel at visually grounded trajectory generation, they remain limited in fine-grained semantic following. These findings highlight future directions for representation-preserving adaptation, causal VLA circuits, and compositional semantic control.

Cross-Reynolds generalisation in neural PDE solvers remains poorly characterised. On the canonical forced 2D Navier-Stokes benchmark, a trained Fourier Neural Operator reaches 46.68% relative L2 error under a 10x Reynolds-number shift, yet zero-forward-model retrieval baselines already improve to 41-42%. This suggests representation geometry as a major organising variable among the tested methods. We test this hypothesis through ConvAE-Relay, which matches states in a source-trained convolutional autoencoder latent space and borrows dynamics from a source-regime database, achieving 38.34+/-0.07% using only a source-regime database and no target-regime fitting, labels, or database entries. A 2x2 ablation isolates matching quality as dominant over the update rule. Oracle experiments confirm that source-regime dynamics directions remain transferable (cosine similarity ~0.84) when matching stays on-manifold; autoregressive drift is the primary bottleneck (~12 percentage points). From the learned-prediction side, a U-Net with multi-scale skip connections achieves 34.72+/-0.60%, consistent with the retrieval-side finding that local, multi-scale representations organise cross-Reynolds transfer among tested methods. All claims are scoped to this benchmark.

Class-Incremental Learning (CIL) is important in building real-world learning systems. In CLIP-based CIL, the model performs classification by comparing similarity between visual and textual embeddings obtained from template prompts, e.g., ``a photo of a [CLASS]''. This seemingly monolithic matching process can be decomposed into two conceptually distinct stages: attribute extraction and attribute aggregation. For example, a model may recognize cat using attributes such as fur texture and whiskers. When learning a new class like car, the model must extract additional attributes like wheels and adjust how they are aggregated in the shared representation space. However, since only data from the current task is available, incremental updates can bias both attribute extraction and aggregation toward new classes, leading to catastrophic forgetting. Therefore, we propose AREA for attribute extraction and aggregation in CLIP-based CIL. To stabilize extraction, we anchor class-level visual and textual attributes on the hyperspherical embedding space via principal geodesic analysis. To stabilize aggregation, we learn lightweight task-specific experts with scoring and residual refinement, regularized by a variational information bottleneck objective. During inference, we perform routing over task attribute manifolds via optimal transport for more concise prediction. Experiments show that AREA consistently outperforms SOTA methods. Code is available at https://github.com/LAMDA-CL/ICML2026-AREA.

In real-world scenarios of linguistic steganalysis, tested texts usually come from unseen domains with different vocabularies, topics, writing styles, and steganographic generation patterns, which can significantly degrade the detection performance. Although existing cross-domain steganalysis methods can effectively alleviate this problem through distribution alignment, domain-invariant feature learning, etc., the detection performance is not satisfactory. In this paper, we propose a post-training representation editing method for cross-domain linguistic steganalysis. Specifically, the detector is first trained on source-domain data, and then the feature extractor and classifier are kept frozen, and the intermediate representations are deterministically edited before classification. For domain adaptation, we construct a domain-offset vector from marginal source and target representations. For domain generalization, we derive a source-domain cover-to-stego direction to guide sample-specific editing. Experimental results show that compared with the advanced methods, the proposed method can achieve high cross-domain detection performance, especially in terms of F1-score, while requiring no architecture modification or parameter updates after source-domain training.

Landscape feature representations play a central role in automated algorithm selection and meta-learning for black-box optimization, yet little is known about how different representations agree (or disagree) in the structures they impose on problem spaces. This paper presents a systematic unsupervised evaluation of four state-of-the-art representations (ELA, DeepELA, TransOptAS, and DoE2Vec) using a diverse set of affine combinations of BBOB functions (MA-BBOB). By applying extensive clustering analyses, coverage-based stability measures, and cross-representation similarity assessments, we show that each representation organizes the same problems in markedly different ways: ELA and TransOptAS form compact geometric structures, DeepELA provides a balanced intermediate view, and DoE2Vec achieves strong semantic alignment but with substantial fragmentation. Our results reveal that no single representation dominates; rather, they capture complementary aspects of the underlying landscapes. These findings highlight the importance of multi-view analyses for understanding representation behavior and offer guidance on selecting or combining representations in downstream meta-learning and algorithm selection tasks. In addition, across two different algorithm families (Differential Evolution and Particle Swarm Optimization), we show that landscape representations face an inherent trade-off in how well they align structural landscape descriptions with observed performance, indicating that no single representation can fully capture algorithm performance.

Diffusion models have emerged as powerful tools for high-quality image generation and editing, but guiding these models to produce specific outputs remains a challenge. Conventional approaches rely on conditioning mechanisms, such as text prompts or semantic maps, which require extensively annotated datasets. In this preliminary work, we explore diffusion models conditioned on representations from a pre-trained self-supervised model. The self-conditioning mechanism not only improves the quality of unconditional image generation, but also provides a representation space that can be used to control the generation. We explore this conditioning space by identifying directions of variations, and demonstrate promising properties in terms of smoothness and disentanglement.