Representation Space

The study of representations is widespread across fields, including neuroscience, psychology, and artificial intelligence. While representations are often studied and compared through similarities between stimuli, current methods provide only limited access to the dimensions that shape these representations and are often limited in interpretability. To overcome these challenges, here we introduce Similarity-Based Representation Factorization (SRF), a general computational method for recovering low-dimensional, non-negative, interpretable embeddings from similarity matrices derived from measured data. Across simulations and many neural, behavioral, and computational datasets, SRF recovers interpretable dimensions from diverse forms of representational data, even for very sparsely sampled, incomplete data. The dimensions derived from these datasets match those obtained by task-specific models, predict independent behavioral properties, improve exploratory analysis, and offer higher power for confirmatory hypothesis testing than comparing similarity matrices. Together, these results establish SRF as a general-purpose method with broad applications for uncovering, understanding, and using the dimensions underlying representations.

Korean folk painting (minhwa) is built from a small vocabulary of auspicious symbols, a tiger for protection, a pair of birds for marital harmony, a peony for wealth, that recur across many of its painted genres. This suggests an obvious computational approach, identify which symbols appear in a painting and read the genre from the inventory. Working with a public corpus that pairs whole paintings, eight-field bilingual curatorial captions, and a separate set of expert object crops, we find that this approach does not work. A model given only a list of which symbols a painting contains predicts the genre far worse than a model that fuses the image with the curatorial text, and forcing the genre representation to be object-grounded actively hurts accuracy. The visual evidence on which the genre prediction rests is nonetheless localized and inspectable. A leakage-safe object evidence map projected from a part-level detector is spatially faithful to where curators isolated symbolic objects and to a patch-based surrogate's own gradient saliency. We name this configuration a faithful-but-insufficient dissociation. The part-level explanation is honest about what the part-level model sees, yet the genre target turns on how symbols are arranged rather than on which ones appear. The same lens separates a content label that survives transfer to held-out source institutions, genre, from a style label that does not, era, a prediction we confirm on two further labels in the corpus. We release the multimodal system, a worked-example reading of one painting's evidence map against its catalogue, and a set of evaluation cautions that recur in long-tailed heritage collections.

Robotic systems with redundant degrees of freedom can achieve the same task outcome using multiple configurations, resulting in solution sets that form manifolds in the configuration space. Existing approaches typically exploit such redundancy locally through Jacobian-based techniques to compute individual solutions or trajectories. While effective for solution computation, these methods do not retain a representation of the geometry of the solution set itself. In this work, we adopt a representation-centric approach to estimate the geometric structure of the solution space. We consider solution manifolds induced by general task-defining maps and construct an implicit scalar field over the configuration space, whose zero-level set corresponds to the solution manifold. To this end, we generate samples in the neighborhood of the solution manifold using a Jacobian-guided exploration strategy, which efficiently captures its local and global structure. The resulting implicit representation is defined over the configuration space and naturally induces a continuous, distance field that encodes proximity to the solution manifold. Experiments on a planar three-link robot and a seven-degree-of-freedom Franka manipulator demonstrate the effectiveness of the proposed representation. Furthermore, the framework enables consistent modeling of solution spaces across families of tasks with continuous variation.

An agent must act on the situation before it, learn what it cannot yet represent, and model other agents well enough to coordinate. These faculties are usually realized by separate mechanisms, yet they share a failure mode: the situation can exceed what the agent can currently represent, and the honest response is then a principled refusal that says what was missing. We develop a small cognitive architecture in which these limits arise from a single quantity. An Interpretation-Decision Unit (IDU) interprets a content vector through a family of regimes - local representational frames with private bases - and decides which actions it licenses; a scalar residual of the content against the active regimes' representational scope drives the unit. Low residual with a clean licensing emits an action; otherwise the unit re-interprets, attempts a description-length-justified expansion, or halts with a typed, witnessed terminal. We prove the unit is total and deterministic: for any content and fixed configuration it halts in finitely many bounded-cost steps with a unique terminal witness, so abstention carries its cause by construction. By binding the architecture's open parameters without changing its mechanics, the same residual-against-scope constraint recovers three documented phenomena at three scopes: the typology of not-knowing (typed abstention); a forced misunderstanding between agents, localized to one shared concept and invisible to the agent committing it (bounded empathy); and prerequisite dependence in learning derived from a bounded focus window rather than posited (developmental prerequisites). Each instantiation is worked for a natural and an artificial agent and states a falsifiable prediction, so one constraint can model limits in both human and machine cognition. The account contributes a unification and a notion of accountable abstention, typed and witnessed by construction.

Speech and audio systems operate in inherently non-stationary environments, yet continual learning (CL) research in this domain, especially in the foundation model era, remains fragmented that fail to account for the coupled, geometry-sensitive nature of acoustic representations. Modern speech foundation models operate over highly entangled, continuous representations that jointly encode linguistic, speaker, and paralinguistic factors within a shared latent space. CL is therefore fundamentally about preserving and evolving shared representation structure rather than retaining isolated task knowledge. In this work, we revisit CL for speech from a representation-centered perspective, and introduce a new taxonomy that organizes CL according to how underlying representation geometry evolves under non-stationary acoustic conditions. We further identify key mismatches between current CL assumptions and speech foundation model behavior, and finally outline a set of open challenges and future research directions.

As large language models become a default source of guidance on personal, moral, and existential questions, it matters whether they draw on the religious frameworks that have historically shaped such reasoning, or systematically omit them. In this paper, we ask a deliberately narrow question: when posed an everyday ethical question for which religious perspectives may be valuable, do LLMs invoke religion at all? In contrast to benchmarks that look for the presence of political leanings or social bias, we look for the absence of religious representation as a dimension of value alignment and bias in LLMs. We term this ``omissive bias.'' To measure omissive bias, we contribute the AllFaith Religious Representation Benchmark: 150 ethically and personally salient questions, sourced from in-the-wild chat transcripts and faith-community contributors, paired with an LLM-as-judge rubric that gives full credit for any mention of a religion, a religious practice, or a religious leader. The questions are not themselves about religion--they are open-ended questions about grief, forgiveness, relationships, purpose, and honesty, where religion is one valuable perspective among several. We also run a human-subjects survey to compare LLM behavior against human expectations. Evaluating 27 models, we find that LLMs consistently underrepresent religion relative to human expectations. The omission is asymmetric: models invoke religion more readily for abstract existential questions (meaning, death, truth) than for the practical personal situations--grief, marriage, family conflict, addiction--where many people most rely on it. It is not our purpose to adjudicate which values LLMs should hold. We argue, more modestly, that current LLM responses overlook critical opportunities to reflect religious frameworks that many people draw on when navigating personal and ethical challenges.

What functions can Neural Processes represent? We analyze the representational capacity of popular NP architectures: Conditional Neural Processes (CNPs), Attentive Neural Processes (ANPs), Transformer Neural Processes (TNPs), and their latent variants. We prove these architectures form a strict hierarchy. CNP-representable functions are exactly those depending on finitely many expected features of the context distribution. ANPs strictly generalize CNPs via query-dependent reweighting, enabling kernel smoothers. ConvCNPs and ANPs are incomparable; each contains functions outside the other, separated by stationarity versus translation equivariance. TNPs with $L$ self-attention layers capture $L$-hop context interactions. For latent NPs, we show finite-dimensional latents provide coherent sampling but do not circumvent encoder limitations; matching GP posterior distributions requires latent dimension scaling with context size. These results provide a theoretical foundation for architecture selection based on task structure.

Machine learning deployments in real-world wireless communication tasks face significant generalization challenges due to location and environment-specific signal structure, high diversity in data across different deployments, and limited availability of real-world data. Current approaches for assessing data similarity between training and inference (deployment) distributions, as well as evaluating model transferability, suffer from high computational costs and inconsistent performance, leaving critical model deployment and model life cycle management decisions without a principled foundation. To address this, we introduce a dataset similarity framework built upon the feature space of a pretrained wireless foundation model. Our method, LWM-CDE (Contrastive learning of Dataset Embedding), fine-tunes the dataset embeddings of the foundation model using a combination of contrastive and geometry-shaping losses, creating a structured manifold where distance reliably indicates transferability. Extensive experiments on wireless benchmarks show that LWM-CDE achieves stronger correlation with empirical transfer performance than existing metrics while being more computationally efficient. The learned representation space supports more effective and data-efficient decision-making for tasks like source dataset selection, label-aware augmentation, and budgeted pretraining, demonstrating its broader utility across different wireless communication applications.

Scaling recommendation models is a central challenge in recommender systems. Recently, RankMixer has emerged as an effective solution, operating on a unified token representation and alternating between token mixing and per-token feedforward networks (P-FFNs) to achieve scalable performance. However, RankMixer suffers from \textit{embedding collapse}, where learned representations have low effective rank, limiting expressivity and underutilizing the expanded representation space. Through empirical analysis and theoretical insights, we identify rigid token mixing and P-FFN modules as the primary causes of this phenomenon, jointly inducing a \textbf{damped oscillatory trajectory} in effective-rank evolution across layers. To address it, we propose RankElastor, a novel architecture that produces spectrum-robust representations with provable collapse mitigation. RankElastor introduces two components: (i) \textbf{parameterized full mixing}, which enables expressive token mixing with improved spectral robustness; and (ii) \textbf{GLU-improved P-FFNs}, which stabilize representation spectra through GLU-style FFN modules. Extensive experiments on large-scale industrial datasets demonstrate that RankElastor consistently improves recommendation performance, mitigates embedding collapse, and exhibits robust scaling behavior. Code is available at this GitHub repository: https://github.com/vasile-paskardlgm/RankElastor

Neural networks acquire structured representations at specific moments during training, yet identifying these transitions typically relies on retrospective, label-dependent metrics. We introduce a bifurcation theory of representation dynamics to detect these moments in real time. Analyzing a passive GMM probe attached to the evolving encoder, we show the onset of structure corresponds to a supercritical pitchfork bifurcation driven by the loss Hessian. The system exhibits a theoretically predictable zero-crossing ($β_c$) that, compared to the network's current state ($β$), yields a dynamic ratio $β(t)/β_c(t)$: a universal, label-free phase coordinate for representation dynamics, computable entirely from hidden states. We empirically validate four distinct transition regimes predicted by this coordinate across diverse settings: SAEs on language models (Pythia), SSL (CIFAR), and grokking (modular arithmetic). Crucially, under finite dissipation, macroscopic symmetry-breaking can lag the initial zero-crossing by orders of magnitude, which providing a rigorous dynamical account of the delayed escape observed in grokking. Microscopically, the bifurcation creates a shared unstable subspace, forcing collective symmetry breaking. We term this the "feature lottery" in SAE training: a feature's terminal interpretability becomes predictable remarkably early. By only 5% of training, early atom purity robustly predicts final convergence purity, with top-decile early atoms achieving over 12x the baseline purity at convergence. Beyond explaining concept emergence, $β/β_c$ provides a practical early-warning indicator for training health, detecting the onset of usable structure, the crystallization of feature identity, and representational collapse epochs before downstream metrics react.

Parameter-efficient fine-tuning enables fast personalization of text-to-image diffusion models, but composing multiple custom concepts remains challenging due to representation interference. Existing modular methods either rely on expensive post-hoc fusion or freeze adaptation subspaces, which limit expressiveness and concept fidelity. To address this trade-off, we propose Sequential regularized LoRA (SeqLoRA), a constrained continual learning framework that jointly optimizes both LoRA factors via bilevel optimization. Theoretically, we establish strong convergence guarantees for our algorithm and model the residual layer activations as a matrix sub-Gaussian process to derive high-probability bounds on catastrophic forgetting. We further prove that learning the LoRA basis from data minimizes residual interference energy more effectively than frozen-basis methods. Experiments on multi-concept image generation demonstrate that SeqLoRA improves identity preservation and scalability across up to 101 concepts, while avoiding costly fusion and reducing attribute interference in composed generations.

Flow matching with $x$-prediction -- regressing the clean data point rather than the ambient velocity -- is known to exploit low-dimensional manifold structure effectively in pixel space \cite{li2025back}. We ask whether a pretrained representation space, while containing a low-dimensional data manifold of comparable intrinsic dimensionality, offers a distribution more favorable for flow-matching learning. Comparing pixel, SD-VAE, and DINOv2 features along four geometric axes, we find that pixel and DINOv2 share nearly identical intrinsic dimensionalities (both $\hat{d}\!\approx\!33$) yet DINOv2 exhibits $7.3\times$ higher effective rank, $35\times$ better covariance conditioning, $11.5\times$ lower excess kurtosis, and $1.7\times$ lower on-manifold interpolation error; SD-VAE latents are consistently intermediate, indicating that the advantage stems from representation-learning objectives rather than mere compression. These statistical properties render the flow-matching regression well-conditioned and remove the need for the specialized prediction heads or Riemannian transport used by prior DINOv2 diffusion methods. We propose the \emph{Representation Image Transformer} (RiT): a vanilla Diffusion Transformer trained by $x$-prediction on frozen DINOv2 features, augmented only by a dimension-aware noise schedule and joint \texttt{[CLS]}-patch modeling. On ImageNet $256{\times}256$, RiT attains FID 1.45 without guidance and 1.14 with classifier-free guidance, outperforming DiT$^\text{DH}$-XL with $19\%$ fewer parameters (676M vs.\ 839M). The resulting ODE is efficiently solvable at coarse discretizations: with classifier-free guidance, $5$ Heun steps already reach FID 2.0 and $10$ steps reach 1.25, without distillation or consistency training. Code at https://github.com/lezhang7/RiT.

Multi-modality medical vision (MV) foundation models (FM) are fundamentally challenged by pronounced Non-IID feature statistics across heterogeneous imaging modalities. Monolithic self-supervised optimization on such data induces conflicting gradients, driving representations to collapse toward modality-dominant shortcuts. This work reframes this failure as an imbalance between specialization and coordination in emergent modularity, and proposes Director-Experts (DEX), a modular network that explicitly regulates these dynamics in stacked modules. Each DEX module comprises a pool of experts, dynamically adapted by our image-wise activation strategy, autonomously specializing in modality-dominant statistics, together with a director, updated via our group exponential moving average, which distills multi-expert knowledge into a shared space for semantic integration across modalities, thus driving the emergence of modular representations. We curate a new benchmark, Medical Vision Universe, over 4 million images across 10 modalities, which provides a FM-level pre-training with the broadest coverage of distinct imaging modalities to our DEX. Extensive evaluations on 26 downstream tasks demonstrate improved optimization behavior and transferability, indicating DEX as a principled step toward general-purpose multi-modality medical AI. Our code and dataset will be opened at https://github.com/YutingHe-list/DEX.

Deep neural networks have achieved impressive performance on a variety of tasks, but their brittleness to distributional shifts remains a significant barrier to real-world deployment. In this paper, we propose a framework to analyse and quantify the distributional robustness of neural networks by studying the interactions between layer weights and activations. We model these interactions using Bernoulli distributions, using the separation between classes as a diagnostic proxy for robustness. We demonstrate the usefulness of this framework through models trained on CIFAR-10 and ImageNet. We show that our proposed metrics can distinguish between networks that have memorised their training data and those that have not. We also perform analogous experiments in the activation space and find that the same properties do not hold up. Additionally, we investigate the behaviour of our metrics under various distribution shifts and show that these shifts reduce separation under our path-based diagnostics. Our results suggest that this framework provides useful model-level diagnostics of representation structure and robustness.

Mixture-of-Experts (MoE) models are often interpreted by analysing which categories are routed to which experts. However, routing alone does not reveal what each expert actually encodes. We train sparsely-gated convolutional MoE models with a contrastive objective on natural images and characterise expert specialisation using tools from visual neuroscience. Extending from gating-level to expert-level analyses, we measure per-expert category separability, and per-expert tuning using the most exciting inputs. Extending from category-level to feature-level explanations, we interpret tuning via semantic dimensions derived from a dataset of human behavioural judgements (THINGS). Finally, we use tuning and representational similarity analysis to assess the stability of expertise-allocation across independent initialisations. We find that an animate-inanimate distinction dominates expert partitioning, apparent from gating through to expert readout, and is stable across independently trained models. Although routing statistics suggest relatively sparse, categorical preferences, expert analyses reveal broader tuning to continuous visual and semantic dimensions that extend beyond category boundaries. Experts exhibit similar category-separability to one another, despite distinct feature tuning, demonstrating the explanatory benefits of moving beyond category-level analyses. Together, these results show that expert specialisation in vision MoEs extends well beyond category routing and is better understood by probing fine-grained expert-level tuning and representational structure.

Machine unlearning is a fundamental mechanism that enforces the right to be forgotten. Existing unlearning studies that rely on label manipulation or task-gradient reversal often deliver limited unlearning effectiveness. Moreover, they can undermine the original learning objective and typically do not guarantee equivalence to standard unlearning by retraining. In this paper, we propose \textbf{ManiF-SMC} (\textbf{Mani}fold \textbf{F}orgetting with \textbf{S}elf \textbf{M}ode \textbf{C}onnectivity), motivated by the observation that a model retrained on the remaining data tends to classify erased samples by their semantic similarity to the retained data. We begin with systematically recasting the approximate unlearning as pushing each erased sample away from its original learned manifold representation centroid toward its nearest semantic neighbors in the retained data. This reformulation aligns unlearning with retraining behavior and operates purely in representation space, reducing reliance on labels and task-specific gradients. To tackle the manifold representation-based unlearning problem, ManiF-SMC encapsulates the unlearning and representation preservation goals in a margin-based triplet loss. Because finding a suitable margin for unlearning is challenging, we propose a self-mode-connectivity module that rapidly reconstructs the local manifold to guide the adaptive margins generation for each unlearning case. Extensive experiments on four representative datasets show that ManiF-SMC achieves unlearning effectiveness comparable to state-of-the-art approximate methods while operating solely within the model's representation space.

While deep neural networks have achieved remarkable success across a wide range of domains, their underlying mechanisms remain poorly understood, and they are often regarded as black boxes. This gap between empirical performance and theoretical understanding poses a challenge analogous to the pre-axiomatic stage of classical geometry. In this work, we introduce the Pursuit of Subspaces (PoS) hypothesis, an axiomatic framework that formulates neural network behavior through a set of geometric postulates. These axioms, together with their derived consequences, provide a unified perspective on representation, computation, and generalization in both shallow and deep architectures. We show that this framework yields geometric explanations for fundamental questions in deep learning, including representation structure, architectural mechanisms, and generalization behavior, offering a principled step toward a coherent theoretical foundation.

How interpretable are the features of leading vision models? The question is increasingly pressing as these models move from research benchmarks into high-stakes deployments, yet existing methods cannot answer it reliably. We close this gap with a framework for measuring and comparing the human interpretability of vision models, built around two complementary psychophysics protocols: (1) localizability -- can an observer predict where a feature fires on a novel image? -- and (2) nameability -- can an observer accurately describe what the feature represents? Features are recovered via sparse autoencoders, and a chance-anchored scoring function places every model on a common scale. Applying the framework to six vision transformers -- two supervised ViTs and four foundation models (DINOv2, DINOv3, CLIP, SigLIP) -- we collected more than $15{,}000$ behavioral responses, analyzing the $13{,}400$ responses from the $377$ participants who passed our pre-specified quality checks. Foundation models are consistently less interpretable than their supervised counterparts, and the gap is not a capability tradeoff: interpretability does not correlate with downstream task performance on any benchmark we examine. What does correlate is the locality of a feature's activations and coarse-grained semantic alignment with humans -- models with focal activations and representations that reflect the world's broad categorical structure produce more interpretable features, whereas fine-grained perceptual alignment does not. The two protocols yield strongly correlated rankings and share the same predictors, establishing interpretability as an independent, measurable dimension of representation quality -- and, surprisingly, one on which every foundation model we tested falls below the supervised baselines that came before. Capability alone cannot close that gap; locality and coarse-grained alignment can.

Neural operators have achieved strong performance in learning solution operators of partial differential equations (PDEs), but their inherently continuous representations struggle to capture discontinuities and sharp transitions. Existing approaches typically approximate such features within continuous function spaces, often requiring increased model capacity and high-resolution data. In this work, we propose Cut-DeepONet, a two-stage training framework that explicitly models discontinuities while reducing learning complexity. Our approach reformulates the problem via a lifting strategy, partitioning the domain into smooth subregions while representing discontinuities as boundaries in a higher-dimensional space. This separation aligns the operator learning task with the inductive bias of neural networks and avoids directly approximating discontinuities. An additional network predicts input-dependent discontinuity locations for unseen inputs, which are then used to guide the neural operator in generating smooth components within each region. Experiments on benchmark PDEs show that Cut-DeepONet outperforms state-of-the-art methods, even when trained on low-resolution datasets. The method excels on problems with discontinuities and sharp transitions, while using fewer trainable parameters. Our results highlight the benefits of changing the representation of operator learning rather than increasing model complexity.

Single-cell perturbation modeling is fundamental for understanding and predicting cellular responses to genetic perturbations. However, existing approaches, from causal representation learning to foundation models, often struggle with an overlooked challenge: gene expression is dominated by perturbation-invariant information, while perturbation-specific signals are intrinsically sparse. As a result, learned representations either entangle invariant and perturbation-specific information, leading to spurious and non-generalizable predictors, or suppress perturbation-specific signals altogether, rendering them ineffective for prediction. To address this, we propose PerturbedVAE, a general framework designed to resolve this signal imbalance. The framework explicitly separates perturbation-specific information from dominant invariant structure and recovers causal representations to effectively utilize such information for prediction. We further provide an identifiability analysis that characterizes the conditions under which sparse perturbation effects can be reliably recovered, thereby clarifying how the framework can be concretely specified under such conditions. Empirically, PerturbedVAE achieves state-of-the-art performance on a widely used benchmark across multiple evaluation settings, yielding significant gains on out-of-distribution combinatorial predictions and uncovering interpretable perturbation-response programs.

We study the problem of recovering a globally consistent Euclidean embedding of data, given only a local distance graph and propose a method that optimally represents these distances. The method operates solely on a neighborhood graph weighted by pairwise distances, without requiring any prior vector representation of the data. The embedding is obtained by solving a variational problem that matches local, on-graph distances to the Euclidean metric, induced by the differentials of the embedding functions. The resulting Euler-Lagrange equations are derived in a coordinate-free form, enabling direct evaluation of all operators from the distance graph alone. Though non-linear and missing an explicit expression for their non-linearity, these equations are shown to be resolved as an iteratively updated sparse linear problem. The main contributions of the proposed approach are (a) the derivation of the functional equations governing the optimal Euclidean embedding in the continuum, (b) a representation-free formulation that requires only a neighborhood distance graph and no feature vectors and (c) an estimation procedure based exclusively on local graph operations. We experimentally evaluate the resulting non-parametric algorithm on synthetic manifolds and real datasets, demonstrating consistent preservation of local metric structure and neighboring relations, while approximating the global isometric embedding.

Infrared object detection is crucial for perception in autonomous driving and surveillance but remains vulnerable to physical adversarial attacks. Unlike in the RGB domain, where attacks rely on color texture, infrared attacks must manipulate thermal signatures, making the geometry shape of heat-blocking materials the primary adversarial information carrier. Current shape-based methods suffer from a fundamental trade-off between representational capability and optimization power, limiting their attack effectiveness.In this work, we overcome this dilemma by introducing learnable Fourier shapes to the infrared domain. We utilize an end-to-end differentiable framework where a compact set of Fourier coefficients, defining the shape boundary, is analytically mapped to a pixel-space mask via the winding number theorem. This enables efficient gradient-based optimization to generate potent shapes that cause human targets to evade detection. Extensive digital and physical experiments provide a comprehensive evaluation and validate our superior performance. Our resulting physical patch achieves striking robustness, successfully evading detectors across diverse distances, angles, poses, and individuals, and achieves over 88% attack success rate at distances greater than 25m (conf.=0.5). Code is available at https://github.com/Yongyx99/Fourier-shape-attack.

Generative text-to-image models are typically trained on large-scale web-scraped datasets that include diverse visual content such as copyrighted and stylistically distinctive artworks, raising concerns about ownership, attribution, and the unintended reuse of protected visual expressions. A key issue is that models can learn stylistic patterns from this data and reproduce them in generated outputs without any explicit reference in the prompt. We refer to this phenomenon as The Silent Brush, where such learned styles reappear even when they are not requested. Existing evaluation methods mainly focus on near-duplicate retrieval or membership inference and do not account for this form of unintended stylistic resurfacing across prompts. To address these gaps, we first formulate guiding principles for evaluation of The Silent Brush. We then introduce Art Arena, an evaluation protocol that measures how strongly artworks are encoded, how they interact, and how frequently their stylistic traits reappear in generated outputs without explicit mention in prompts. We evaluate Art Arena on widely used text-to-image diffusion models, including Stable Diffusion v1.5, Stable Diffusion XL (SDXL), and SANA-1.5, and design it to generalize across text-to-image generative systems. Our results show that The Silent Brush arises from differences in representational strength and interaction dynamics between artworks, leading to asymmetric blending in model generations. Code and evaluation resources are available at: https://anonymous.4open.science/r/ArtArena-EBE4.

Large language models (LLMs) exhibit remarkable flexibility: they can adapt to novel tasks from in-context examples without any parameter updates, a capability known as in-context learning (ICL). Prior work on synthetic tasks has shown that ICL can implement specific algorithms, demonstrating architectural competence, and mechanistic analyses have identified key circuits that support this behavior. However, because in-context computation -- regardless of its algorithmic form -- relies on transformations in high-dimensional representation space, it remains unclear how the geometry of that space shapes ICL effectiveness. Motivated by the neuroscience view of classification as the untangling of neural representations, we hypothesize that ICL depends on the successful online untangling of task-relevant representations. To test this idea, we study how LLMs classify in-context examples whose labels are defined by the model's own internal representations with known structure. We show that ICL performance correlates systematically with the representational structure of the underlying classification task and that successful ICL is accompanied by geometric reorganization that increases online separability. We further find that LLM behavior is well described by a prototype-like algorithm that integrates evidence while reshaping representations to support classification. These findings offer a geometric account of ICL in pretrained LLMs, establish representational geometry as a mechanistic constraint on ICL, and quantify the gap between what pretrained representations afford and what in-context learning can exploit.

In language models, what a representation encodes is determined by the geometry of its representation space: distances, not activations, carry meaning. Existing tools characterize the shape of this geometry but do not ask what that shape is organized for. We introduce Subspace PGA, a metric that tests whether a layer's distance structure aligns with the readout subspace of the unembedding matrix $W_U$ more than with random subspaces of equal size. Across seven Pythia models (70M--6.9B) and three cross-family models, intermediate geometry is significantly organized for prediction (peak $z = 9$--$24$), but the degree is scale-dependent: small models ($d \leq 1024$) progressively lose it at late layers during training -- even as loss keeps improving -- while large models ($d \geq 2048$) preserve it throughout. We trace this to a capacity trade-off: a few dominant directions migrate away from $W_U$'s readout, masking rather than destroying the predictive structure beneath, and removing them restores alignment. Neither spectral metrics nor loss curves capture this distinction. Scale thus determines not only how well a model predicts, but how its representation geometry is organized to do so.

We study behavioral alignment and representation dynamics of large language model (LLM) agents in financial decision environments. TradeArena, an auditable trading-agent testbed with risk reports, execution simulation, memory, and replayable trajectories, lets us analyze how rationales, positions, and interventions evolve under market stress. Code and data artifacts are available through the \href{https://github.com/weich97/TradeArena.git}{TradeArena repository}. We find pre-failure signatures: planning embeddings drift from normal centroids, fused plan-risk representations separate normal from pre-drawdown states, and local manifolds exhibit effective-rank contraction. Across 80 rolling failure anchors and eight LLM trajectories, this pattern persists across hash, LSA, Transformer, and white-box hidden-state probes. Stress tests with CoT-free target weights, lexical controls, OHLCV noise, and false audits show that rationale-level contraction can vanish without rationales, while intent-space and fused signatures remain informative. Structured risk feedback can act as an external alignment signal without fine-tuning, but not as a universal performance enhancer: true audit feedback improves calibration for some models, returns for others, and exposes cases where placebo or hidden feedback has higher short-horizon return but weaker alignment diagnostics. A 51-stock intraday experiment reveals a correlation blind spot: LLM rationales justify exposure to coupled assets that the risk layer clips. Finally, a financial-audit task suite shifts comparison from ``which model trades best'' to whether models can audit trajectories, respect execution boundaries, reproduce artifacts, and avoid claim overreach. These results support a research claim, not a profitability claim: auditable risk feedback and representation trajectories reveal when LLM financial reasoning is aligning, drifting, or failing.

Multimodal pre-training demonstrates strong generalization performance, but this paradigm is often impractical in domains where paired data are scarce. A promising alternative is post-hoc multimodal alignment, which aligns separately pre-trained unimodal encoders using a limited number of paired examples. However, existing methods focus primarily on aligning global representations, missing patch-token relations. This may hinder transfer to tasks that require fine-grained cross-modal matching beyond coarse sample-level semantics. To address this issue, we propose a post-hoc alignment method that learns token-level cross-modal structure using relative representations. Specifically, we represent images and texts through their token-level similarities to a set of learnable anchors in each modality space, which are trained to induce consistent cross-modal similarity patterns for matched pairs. Despite learning only the anchors without heavy projection layers, our approach consistently outperforms existing methods in zero-shot classification, cross-modal retrieval, and zero-shot segmentation by a substantial margin. This highlights the importance of modeling fine-grained cross-modal structure for effective post-hoc multimodal alignment with limited paired data.

Contrastively pre-trained Vision-Language Models (VLMs) serve as powerful feature extractors. Yet, their shared latent spaces are prone to structural anomalies and act as repositories for non-semantic, multi-modal noise. To address this phenomenon, we employ spectral decomposition of covariance matrices to decompose the VLM latent space into a multi-modal semantic signal component and a shared noise subspace. We observe that this noise geometry exhibits strong subgroup invariance across distinct data subsets. Crucially, pruning these shared noise dimensions is mainly harmless, preserving or actively improving downstream task performance. By isolating true semantic signals from artifactual noise, this work provides new mechanistic insights into the representational structure of modern VLMs, suggesting that a substantial fraction of their latent geometry is governed by shared, architecture-level noise rather than task-relevant semantics alone.

Continual learning requires models to adapt to new data while preserving previously acquired knowledge. At its core, this challenge can be viewed as principled one-step adaptation: incorporating new information with minimal interference to existing representations. Most existing approaches address this challenge by modifying model parameters or architectures in a supervised, task-specific manner. However, the underlying issue is representational: tasks require distinct yet structured representations that can be selectively updated without disrupting representations, while structure should reflect intrinsic organization in the data rather than task boundaries. In sequential data, time-delayed dependencies provide a natural signal for uncovering this organization, revealing how fundamental representations give rise to more specific ones. Inspired by the modular organization of the human brain, we propose MoRe, a framework that identifies modularity in the representation itself rather than allocating it at the architectural level. MoRe decomposes knowledge into a hierarchy of fundamental and specific modules with identifiability guarantees, enabling principled module reuse, alignment, and expansion during adaptation while preserving old modules by construction. Experiments on synthetic benchmarks and real-world LLM activations demonstrate interpretable hierarchical structure, improved plasticity-stability trade-offs, suggesting MoRe as a principled foundation for continual adaptation

Language models often generate long chain-of-thought traces, but it remains unclear how much of this reasoning is necessary for preserving the final prediction. We study this through the lens of overcomplete reasoning traces: generated traces that contain more intermediate steps than are needed to support the model's answer. We define the minimal core as the smallest subset of steps that preserves either the final answer or predictive distribution, and introduce metrics for compression ratio, redundancy mass, step necessity, and necessity concentration. Across six deliberative reasoning benchmarks spanning arithmetic, competition mathematics, expert scientific reasoning, and commonsense multi-hop QA, we find substantial overcompleteness: on average, 46% of steps are removable under greedy minimal-core extraction while preserving the original answer in 86% of cases. We also find that predictive support is concentrated: the top three steps account for 65% of measured necessity mass on average. Beyond compression, minimal cores expose a cleaner geometry of reasoning: compared with full traces, they improve correct-incorrect trace separation by 11 points, reduce estimated intrinsic dimensionality by 34%, and transfer across model families with 85% off-diagonal answer retention. Theoretically, we establish existence of minimal sufficient subsets, local irreducibility guarantees for greedy elimination, and certificates of overcompleteness and sparse necessity. Together, these results suggest that full reasoning traces are often verbose and overcomplete, while minimal cores isolate the effective support underlying language-model predictions.

Large language model (LLM) unlearning aims to remove specific data influences from pre-trained model without costly retraining, addressing privacy, copyright, and safety concerns. However, recent studies reveal a critical vulnerability: unlearned models rapidly recover "forgotten" knowledge through relearning attacks. This fragility raises serious security concerns, especially for open-weight models. In this work, we investigate the fundamental mechanism underlying this fragility from a representation geometry perspective. We discover that existing unlearning methods predominantly optimize along dominant components, leaving minor components largely unchanged. Critically, during relearning attacks, the modifications in these dominant components are easily reversed, enabling rapid knowledge recovery, whereas minor components exhibit stronger resistance to such reversal. We further provide a theoretical analysis that explains both observations from the spectral structure of representations. Building on this insight, we propose Minor Component Unlearning (MCU), a novel unlearning approach that explicitly targets minor components in representations. By concentrating unlearning effects in these inherently robust directions, our method achieves substantially improved resistance to relearning attacks. Extensive experiments on three datasets validate our approach, demonstrating significant improvements over state-of-the-art methods including sharpness-aware minimization.

Average-reward reinforcement learning requires estimating the gain and the bias, which is defined only up to an additive constant. This makes direct distributional analogues ill-posed on the real line. We introduce a quotient-space formulation in which state-indexed bias laws are identified up to a common translation, together with a categorical parameterization that respects this symmetry. On this quotient-categorical space, we define a projected average-reward distributional operator and show that it is well-defined, non-expansive in a coordinate Cramér metric, and admits fixed points. We then study sampled recursions whose mean-field maps are asynchronous relaxations of this operator. In an idealized centered-reward setting, a one-state temporal-difference update enjoys almost sure convergence together with finite-iteration residual bounds under both i.i.d. and Markovian sampling. When the gain is unknown, we augment the recursion with an online gain estimator, and prove non-expansiveness and Markovian convergence of the resulting coupled scheme. Finally, we show that synchronous exact updates are gain-independent at the quotient-law level, isolating a structural contrast between ideal quotient distributions and practical fixed-grid categorical representations.

Embedded numeric anchors on images systematically bias Vision-Language Model quality judgments across six VLMs from five architectural families (ANOVA eta^2 = 0.18-0.77, all p < 0.001). Anchor effects are 2.5x larger than severe image quality degradation, confirming bias is not reducible to visual changes. Layer-wise probing reveals consistent dissociation: layers where anchor classification saturates (L12-L34) are suboptimal for quality prediction, with optimal layers deeper (R^2 = 0.69-0.91). Fusion analysis identifies architecture-dependent integration -- instant fusion at L1-L2 in two models versus partial or no fusion in three others. These results establish a causal account of visual anchoring bias, linking behavioral susceptibility to representation dynamics.

Knowledge graph embedding (KGE) models typically represent each relation as an operator on entity embeddings. In this work, we identify three structural axioms that any principled relation operator must satisfy, linearity, trace preservation, and complete positivity, and show that they characterize a Kraus channel structure via the Kraus representation theorem. The completeness constraint defining this family is equivalent to these axioms, providing a principled foundation rather than an externally imposed condition. Under this formulation, most existing operator-based KGE models are recoverable as special cases with Kraus rank $κ= 1$ under specific embedding choices. We further generalize this characterization to arbitrary metric geometries by introducing \mbox{w-Kraus} channels, which satisfy completeness by construction within their respective spaces. Building on this theory, we propose \textsc{KrausKGE}, a principled KGE model that naturally handles $1$-to-$N$ and $N$-to-$N$ relations, supports $k$-hop reasoning without requiring explicit path encoders, and eliminates the need for norm constraints on entity embeddings. Additionally, our framework yields the first theoretically grounded per-relation complexity measure in the KGE literature, with a provable lower bound in terms of the empirical relation matrix rank. Empirical evaluation demonstrates that \textsc{KrausKGE} consistently outperforms strong baselines on $N$-to-$N$ relations, with performance gains that increase monotonically with relation fan-out, in alignment with theoretical predictions.

While researchers are finding concepts represented as linear directions in language models, a bag of linear directions fails to capture relational structure. To better understand this dichotomy, we study a model with known linear representations, but trained in a highly structured domain -- the board game Othello. While the model's internal board-state representation is linearly decodable, we find additional structure in the form of tensor product representations (TPRs). We train TPR probes to recover shared structure amongst the linear probes, yielding a factorization into square-embeddings, color-embeddings, and a binding matrix that composes them to construct the model's board-state representation. We find geometric signatures within the weights of our TPR probe that align with the structure of the board, but perhaps more importantly, that the linear probes can be recovered directly from the parameters of our TPR probe. Our findings suggest that directional representations may be projections of more structured underlying representations.

Latent representations learned by neural networks often exhibit semantic structure, where concept similarity is reflected by geometric proximity in embedding space. However, comparing such spaces across models remains difficult: changes in architecture, pretraining data, objective, or random seed can yield embeddings with similar content but incompatible geometry. This latent space alignment problem is central to interpretability, transfer and multimodal learning, federated systems, and semantic communication; however, progress remains limited by the lack of large-scale, model-diverse, and metadata-rich benchmarks. To address this gap, we introduce SEMASIA, a large-scale collection of latent representations extracted from approximately 1,700 pretrained vision models across eight standard image-classification benchmarks. SEMASIA pairs embeddings with structured metadata describing architectures, training regimes, pretraining sources, and model scale. We demonstrate three applications of the resource. First, we analyze the conceptual organization of individual latent spaces, showing consistent prototype-like clustering and hierarchical semantic neighborhoods across models and datasets. Second, we benchmark supervised alignment mappings between latent spaces using reconstruction error and downstream task performance. Third, we perform a large-scale regression analysis of how pretraining-data complexity, specialization, transfer learning, augmentation, and model scale relate to geometric and probing properties of embeddings. By coupling representational scale with standardized metadata, SEMASIA provides a reproducible foundation for studying latent geometry, evaluating alignment methods, and developing next-generation heterogeneous and interoperable AI systems.

Sparse autoencoders (SAEs) decompose transformer residual streams into interpretable feature dictionaries, yet the relationship between SAE width and causal influence on model output has not been systematically characterised. We introduce causal dimensionality kappa(L, M, T), defined as the effective rank of the expected Jacobian outer product at layer L, and show it can be estimated via the SAE width sweep paired with attribution patching. Across seven SAE widths from 16,384 to 1,048,576 features on Gemma-2-2B layer 12, representational capacity grows 15.6x while causal capacity grows only 4.35x: a robust separation we term the representational-causal wedge. A saturating fit yields kappa-hat approximately 1,990 with kappa-hat / d_model = 0.86 and participation-ratio lower bound kappa_PR approximately 280. Crucially, kappa is invariant to model scaling: Gemma-2-9B and Gemma-2-2B yield identical N_causal = 328 at the same SAE width despite a 3.46x parameter increase (the count is forced to 2% of SAE width by calibration; the substantive empirical claim is shape invariance of the AtP score distribution under matched seq=512 conditions). Across eight network depths kappa is constant while the absolute attribution threshold drops 20x from layer 1 to layer 23. Five controls (architecture invariance, threshold robustness, geometric privilege, synthetic ground-truth recovery, and a four-cell encoder/decoder ablation) pin down what kappa measures and what it does not. Our findings establish kappa as a measurable, model-intrinsic property of transformer layers: sub-linearly recoverable by SAE width, invariant to model scaling, and structured across network depth.

Temporally contrastive representation learning induces a latent structure capable of reducing long-horizon planning to inference in a low-dimensional linear system. However, existing contrastive planning work learns a single latent geometry which cannot distinguish multiple valid behaviors trading task efficiency against risk exposure for the same start-goal query. We introduce MoMo, a preference-conditioned contrastive planner allowing a scalar user preference to continuously modulate plan conservativeness at inference time, without retraining. MoMo learns a joint conditioning of the representation geometry and latent prediction operator via Feature-Wise Linear Modulation and low-rank neural modulation, respectively. We show that our formulation preserves the probability density ratio encoded in the representation space that is required for inference-driven contrastive planning, further retaining its inference-time efficiency. Across six environments, MoMo smoothly adapts plan safety according to user preferences, yielding improved temporal and preferential consistency over state augmentation baselines.

The scaling behavior, in which test performance often improves as model size and data increase, is a central empirical phenomenon in modern deep learning, yet its theoretical basis remains incomplete. In this paper, we study depth expansion in normalized residual networks: starting from a trained model in an old hypothesis class, we insert a new residual block at an intermediate layer and ask when such an expansion can yield a provable improvement in test risk. We develop a unified framework that decomposes this question into representational gain, optimization gain, and generalization transfer. First, under a first-order descent condition near zero initialization, we prove that the expanded hypothesis class contains an auxiliary jumpboard model with strictly smaller population risk than the original model. Second, under norm control tailored to post-normalized residual architectures, we establish a norm-based Rademacher complexity bound for the expanded model class. These ingredients lead to two complementary test-risk guarantees: one route passes through population risk and is tighter when a positive population margin is available, while the other works directly at the train/test level, avoids Hoeffding transfer, and is more robust in degenerate regimes. Together, these results provide a theorem-driven mechanism under which residual depth expansion can improve test performance in normalized residual networks. More broadly, they suggest that scaling is inherently joint: depth creates new improving directions, width enhances the finite-sample observability of weak signals, and data determines whether the statistical cost of expansion can be controlled.

Layer pruning efficiently reduces Large Language Model (LLM) computational costs but often triggers sudden performance collapse. Existing representation-based analyses struggle to explain this mechanism. We propose studying pruning through decision representation. Focusing on multiple-choice tasks, we introduce two metrics, Decision Margin and Option Frequency, and an Iterative Pruning method to analyze layer-wise decision dynamics. Our findings reveal a sharp decision transition that partitions the network into two stages: a Silent Phase, where the model cannot yet predict the correct answer, and a Decisive Phase, where the correct prediction emerges. We also find that pruning the Decisive Phase has minimal impact, whereas pruning the Silent Phase triggers immediate performance collapse, highlighting its extreme sensitivity to structural changes. Therefore, we conclude that pruning-induced collapse stems from disrupting the Silent Phase, which prevents the critical decision transition from occurring.

Mixture-of-Experts (MoE) architectures achieve scalable capacity through sparse routing, yet the geometric structure of expert specialization remains poorly understood. We introduce a unified Jacobian-PCA-Grassmann framework for analyzing MoE layers in both function space and representation space. Across pretrained MoE Transformers (Mistral, Qwen), we find a consistent structural asymmetry: experts exhibit strong functional decorrelation (consistently low, near-zero cross-expert Jacobian alignment) while their routed representations occupy distinct but partially overlapping subspaces. This indicates that functional decorrelation and representation overlap coexist rather than coincide in MoE specialization. Controlled routing experiments further indicate that routing sparsity appears to be a key factor shaping this geometry: top-k routing induces sharper functional separation and larger subspace divergence, whereas fully soft routing yields more entangled expert structure. Together, these results suggest a geometric interpretation in which MoE layers may be viewed as implementing locally decorrelated operators over overlapping submanifolds on a shared representation manifold, and provide a general diagnostic framework for studying conditional computation in modern Transformer architectures.

Mobile devices face diverse resource constraints and non-IID data class distributions, requiring fast on-device inference for local in-distribution (ID) classes and on-demand remote support for client-specific out-of-distribution (OOD) classes. Hybrid split federated learning (Hybrid SFL) couples personalized client-side front ends (supporting early exit) with a generalized server-side backend for fallback inference, balancing accuracy and cost. However, under client architectural heterogeneity, the existing hybrid SFL suffers from representation skew, where features from customized extractors fail to align in the shared space, leading to a sharp degradation in the server model responsible for OOD prediction. We propose HARMONY, the first hybrid SFL framework to support heterogeneous client architectures. HARMONY modifies meta-learning to simulate diverse extractors across parameters and architectures, and to learn to personalize. To mitigate representation skew, HARMONY conducts server-side contrastive learning to align extracted features, neither sacrificing clients' personalization nor sharing raw labels. Compared to the state of the art across multiple datasets and model families, HARMONY improves test accuracy by up to 43.0%/28.3% without/with OOD, respectively, while maintaining acceptable latency.

A hallmark of life on Earth is the ability of agents to exert causal power and be drivers of subsequent events. This is key to cognition at all scales. Causal emergence, measuring the degree to which an agent exerts unique predictive power on its future, is one consequence of causal power. Indeed, recent discoveries have shown that biological agents, even minimal ones, increase their causal emergence after learning new memories. However, there is a major knowledge gap regarding how causally emergent artificial agents are. We focused on Reinforcement Learning (RL) of neural-network agents across an array of environmental conditions, encompassing different algorithms, agent architectures, and six environments arranged on a complexity spectrum. For consistency, we computed the causal emergence of their latent-space representations over their lifetimes. We used the recently proposed ΦID to estimate causal emergence and tested how it related to learning performance. Our results suggested a Causally Emergent Alignment Hypothesis: successful agents exhibited causal emergence that was consistently predictive of final reward early in training and whose representational dynamics aligned with reward improvement in most tasks. This idea suggests that causal emergence may be a previously undisclosed axis of reorganization of neural representations in RL agents, with the potential to establish causal relationships and interventions that will lead to better RL agents. Our work also highlights the alignment between causal emergence and learning as another way biological and artificial creatures compare.

The main XAI attribution methods for deep neural networks -- GradCAM, SHAP, LIME, Integrated Gradients -- operate on separate theoretical foundations and are not formally comparable. We present GRALIS (Gradient-Riesz Averaged Locally-Integrated Shapley), a mathematical framework establishing a representation theory for attributions: every additive, linear, and continuous attribution functional on L^2(Q,mu) admits a unique canonical representation (Q, w, Delta), proved necessary by the Riesz Representation Theorem. This class encompasses SHAP, IG, LIME and linearized GradCAM, but excludes nonlinear functionals such as standard GradCAM or attention maps. Seven formal theorems provide simultaneous guarantees absent in any individual method: (T1) necessary canonical form; (T2) exact completeness; (T3) Monte Carlo convergence O(1/sqrt(m))+O(1/k); (T4) exact Shapley Interaction Values; (T5) Hoeffding ANOVA decomposition; (T6) Sobol sensitivity generalization; (T7) multi-scale extension (MS-GRALIS) with minimum-variance weights. An algebraic appendix justifies the GRALIS-SIV correspondence via the Mobius transform without circularity. GRALIS satisfies 13.5/14 axiomatic properties vs. 2.5-6/14 for individual methods, including completeness, sensitivity, locality, order-k interactions and optimal multi-scale aggregation simultaneously. Preliminary validation on BreaKHis (1,187 histology images, DenseNet-121) reports deletion faithfulness AUC +0.015 (malignant), 96% class-conditional consistency, SAL = 0.762+/-0.109 and sparsity index 0.39. Extended comparison with baseline XAI methods is planned for a companion paper.

Transformer architectures have been widely adopted for time series forecasting, yet whether the representational mechanisms that make them powerful in NLP actually engage on time series data remains unexplored. The persistent competitiveness of simple linear models such as DLinear has fueled ongoing debate, but no mechanistic explanation for this phenomenon has been offered. We address this gap by applying sparse autoencoders (SAEs), a tool from mechanistic interpretability, to probe the internal representations of PatchTST. We first establish that a single-layer, narrow-dimensional transformer matches the forecasting performance of deeper configurations across commonly used benchmarks. We then train SAEs on the post-GELU intermediate FFN activations with dictionary sizes ranging from 0.5x to 4.0x the native dimensionality. Expanding the dictionary yields negligible downstream performance change (average 0.214%), with large portions of overcomplete dictionaries remaining inactive. Targeted causal interventions on dominant latent features produce minimal forecast perturbation. Across all evaluated settings, we observe no empirical evidence that the analyzed FFN representations rely on strong superposition. Instead, the representations remain sparse, stable under aggressive dictionary expansion, and largely insensitive to latent interventions. These results demonstrate that superposition is not necessary for competitive performance on standard forecasting benchmarks, suggesting they may not demand the rich compositional representations that drive transformer success in language modeling, and helping explain the persistent competitiveness of simple linear models

Stage-wise audio-visual encoders propagate fused intermediate states across layers, making the formation of later representations depend on the readiness of earlier fusion states. Strong local audio-visual agreement provides useful correspondence evidence, yet a fused state also needs sufficient cross-layer and cross-modal support before it can reliably guide later fusion. This paper studies this issue through propagation-aware representation readiness and formulates premature perceptual commitment as a readiness-deficiency problem, where local plausibility, propagation influence, and support insufficiency jointly appear at an intermediate stage. We propose the Delayed Perceptual Commitment Network (DPC-Net), an encoder-level framework that estimates an observable readiness-deficiency surrogate, localizes the intervention-sensitive bottleneck, and applies support-aware correction with cross-layer and cross-modal evidence. DPC-Net preserves task-specific heads, losses, decoding modules, and evaluation protocols, making it applicable to different audio-visual tasks through encoder-side intervention. Experiments on audio-visual speech separation, audio-visual event localization, and audio-visual speech recognition show consistent improvements across reconstruction, localization, and recognition regimes. Further analyses on component contribution, selection criteria, counterfactual intervention, and readiness trajectories support the effectiveness of readiness-guided bottleneck correction.

Spectral graph sparsification is a classical tool for reducing graph complexity while preserving Laplacian quadratic forms. In graph neural networks (GNNs), sparsification is often used to accelerate computation while maintaining predictive performance. In this work, we study a complementary representation-level question: does sparsification preserve the geometry of learned embeddings? For polynomial-filter GNNs, we prove that any $ε$-spectral sparsifier induces $O(ε)$ perturbations in polynomial graph filters, multilayer hidden representations, and their Gram matrices. These guarantees imply stability of squared pairwise distances, class means, and covariance structure in embedding space. We further establish finite-time training stability: under smoothness and boundedness assumptions, gradient descent on dense and sparsified graphs produces weight trajectories whose separation grows at most proportionally to the sparsification distortion. Empirically, effective-resistance sparsification validates the predicted perturbation chain on synthetic graphs and preserves hidden representation geometry on real datasets. In our experiments, the gram matrix and training dynamics show low divergence even under substantial sparsification, consistent with the predicted stability under spectral sparsification. Hidden Gram preservation strongly predicts neighborhood preservation and class-centroid stability across FashionMNIST, Cora, and Paul15. Together, these results show that spectral sparsification preserves not only graph operators, but also the representation geometry that supports downstream use of GNN embeddings for interpretability.

Beckmann & Butlin's (2026) ontological framework for the LLM individuation problem inherits an unargued cross-regime co-reference assumption from the persona-vectors literature: that the same direction picks out the same content under prompt-conditioning, gradient-descent fine-tuning, and inference-time steering. We present four empirical wedges from persona-topology experiments on Qwen3-4B-Instruct and Mistral-7B-Instruct-v0.2 - non-collinearity of prompt-extracted vectors and fine-tune basins; fictional personas displacing the model along real-anchor directions more strongly than real anchors do; contradictory-valenced mixtures biased toward a training-history-determined attractor; and asymmetric compositional algebra under inference-time arithmetic versus fine-tune-time chimera training - that jointly undermine the assumption. We propose regime-indexed individuation: the identity unit for representational content is a (vehicle, regime) pair, not a vehicle alone. Under this framework, Beckmann & Butlin's three candidate positions describe three different regime-internal objects rather than competing for the same referent; the same diagnosis applies to Mollo & Millière, Chalmers, and Cerullo.

Multi-task learning (MTL) has emerged as a pivotal paradigm in machine learning by leveraging shared structures across multiple related tasks. Despite its empirical success, the development of likelihood-based efficiently solvable algorithms--even for shared linear representations--remains largely underdeveloped, primarily due to the non-convex structure intrinsic to matrix factorization. This paper introduces a first-order algorithm that jointly learns a shared representation and task-specific parameters, with guaranteed efficiency. Notably, it converges in $\widetilde{\mathcal{O}}(1)$ iterations and attains a \emph{near-optimal} estimation error of $\widetilde{\mathcal{O}}(dk/(TN))$, \emph{improving} over existing likelihood-based methods by a factor of $k$, where $d$, $k$, $T$, $N$ denote input dimension, representation dimension, task count, and samples per task, respectively. Our results justify that likelihood-based first-order methods can efficiently solve the MTL problem.

Recent advances in large-scale visual representation learning have significantly improved performance in plant species and plant disease recognition tasks. However, state-of-the-art models, often based on high-capacity vision transformers or multimodal foundation models, remain computationally expensive and difficult to deploy in resource-constrained environments such as mobile or edge devices. This limitation hinders the scalability of automated biodiversity monitoring and precision agriculture systems, where efficiency is as critical as accuracy. In this work, we investigate knowledge distillation as an effective approach to transfer the representational capacity of large pretrained models into smaller, more efficient architectures. We focus on plant species and disease recognition, and conduct an extensive empirical study on two challenging benchmarks: Pl@ntNet300K-v2 and Deep-Plant-Disease. We evaluate four representative architectures, including two ConvNeXt models and two vision transformers, under multiple training regimes: from-scratch training and pretrained initialization, each with and without distillation. In total, we train and evaluate 70 models. Our results show that knowledge distillation consistently improves performance across tasks and architectures. Distilled models are able to match the performance of significantly larger models while maintaining substantially lower computational cost. These findings demonstrate the potential of knowledge distillation techniques to enable efficient and scalable deployment of plant recognition systems in real-world environmental applications.

We investigate variational quantum classifiers (VQCs) for land-cover classification from multispectral satellite imagery, adopting a feature-map perspective in which the quantum circuit defines a nonlinear data embedding while the readout determines how this representation is exploited. Using the EuroSAT-MS dataset, we perform a systematic one-vs-one evaluation across all class pairs under a controlled experimental protocol, comparing classical baselines (logistic regression, SVMs, neural networks) with VQCs employing both linear readout and quantum-kernel SVM strategies. Our results show that, while VQCs with linear readout do not outperform strong classical baselines such as RBF-SVM, the same trained quantum feature map can significantly improve performance when reused within a kernel-based decision framework. A qubit-count sweep further reveals saturation effects consistent with the mismatch between exponential Hilbert space dimension and linear parameter scaling. Overall, our findings highlight that the effectiveness of quantum models depends critically on the interplay between representation and readout, and that meaningful gains may arise from combining learned quantum feature maps with classical decision mechanisms rather than seeking direct replacement of classical models.

We present Density Field State Space Models (DF-SSM), a framework for compressing SSMs to a 1-bit scaffold with int8 low-rank correction. Applied to Mamba-2 1.3B, we achieve a 278 MB model (9.7x smaller than the 2.7 GB FP16 teacher) that runs at 21.4x faster inference on GPU (batch=1, relative to the mamba-ssm reference implementation) while maintaining downstream task performance within 2-4 percentage points of BitMamba-2, a 1.58-bit model trained from scratch on 150B tokens. The distillation itself requires only 32M tokens and 6 hours on a single A100 GPU, though it presupposes a pretrained FP16 teacher. We develop an optimized inference pipeline combining cuBLAS INT8 tensor cores for the scaffold matmul, custom CUDA kernels for stateful SSM and convolution operations, and an AVX-512 CPU backend for efficient deployment on both GPU and CPU. Beyond compression, we investigate the internal knowledge organization of the resulting model, discovering three distinct processing phases: intent classification (layers 0-3, operating in an abstract space with no vocabulary alignment), knowledge retrieval (layers 25-35, where factual associations localize to a 5-layer window), and output formatting (layers 36-47, where category structure dissolves). Through systematic analysis of 445 factual prompts across 19 categories, we find that early-layer classification is syntactic (driven by template structure) rather than semantic, and that the model exhibits well-organized knowledge representations despite weak factual recall--suggesting that representational structure may precede factual strength.

There has been growing interest in studying the complexity of Rectified Linear Unit (ReLU) based activation networks. Recent work investigates the evolution of the number of piecewise-linear partitions (linear regions) that are formed during training. However, current research is limited to examining the complexity of models trained in a supervised way. Self-Supervised Learning (SSL) differs in that it directly optimises the representation space using a loss function to enhance the model's performance across multiple downstream tasks. This study investigates the local distribution of linear regions produced by SSL models. We demonstrate that the evolution of linear regions correlates with the representation quality by utilising SplineCam to extract two-dimensional polytopes near the data distribution. We track the number, area, eccentricity, and boundaries of regions throughout training. The study compares supervised, contrastive, and self-distillation methods over two standard benchmark datasets, MNIST and FashionMNIST. The analysis of the experimental results shows that self-supervised methods create substantially fewer regions to achieve comparable accuracy to supervised models. Contrastive methods rapidly expand regions over time, whereas self-distillation methods tend to consolidate by merging neighbouring regions. Lastly, we can detect representation collapse early within the geometric space of linear regions. Our analysis suggests that polytopal metrics can serve as reliable indicators of representation quality and model performance.

A central problem in unsupervised domain adaptation is determining what to transfer from labeled source domains to an unlabeled target domain. To handle high-dimensional observations (e.g., images), a line of approaches use deep learning to learn latent representations of the observations, which facilitate knowledge transfer in the latent space. However, existing approaches often rely on restrictive assumptions to establish identifiability of the joint distribution in the target domain, such as independent latent variables or invariant label distributions, limiting their real-world applicability. In this work, we propose a general domain adaptation framework that learns compact latent representations to capture distribution shifts relative to the prediction task and address the fundamental question of what representations should be learned and transferred. Notably, we first demonstrate that learning representations based on all the predictive information, i.e., the label's Markov blanket in terms of the learned representations, is often underspecified in general settings. Instead, we show that, interestingly, general domain adaptation can be achieved by partitioning the representations of Markov blanket into those of the label's parents, children, and spouses. Moreover, its identifiability guarantee can be established. Building on these theoretical insights, we develop a practical, nonparametric approach for domain adaptation in a general setting, which can handle different types of distribution shifts.

Large language models (LLMs) are increasingly used for text generation tasks from everyday use to high-stakes enterprise and government applications, including simulated interviews with asylum seekers. While many works highlight the new potential applications of LLMs, there are risks of LLMs encoding and perpetuating harmful biases about non-dominant communities across the globe. To better evaluate and mitigate such harms, more research examining how LLMs portray diverse individuals is needed. In this work, we study how national origin identities are portrayed by widely-adopted LLMs in response to open-ended narrative generation prompts. Our findings demonstrate the presence of persistent representational harms by national origin, including harmful stereotypes, erasure, and one-dimensional portrayals of Global Majority identities. Minoritized national identities are simultaneously underrepresented in power-neutral stories and overrepresented in subordinated character portrayals, which are over fifty times more likely to appear than dominant portrayals. The degree of harm is amplified when US nationality cues (e.g., ``American'') are present in input prompts. Notably, we find that the harms we identify cannot be explained away via sycophancy, as US-centric biases persist even when replacing US nationality cues with non-US national identities in the prompts. Based on our findings, we call for further exploration of cultural harms in LLMs through methodologies that center Global Majority perspectives and challenge the uncritical adoption of US-based LLMs for the classification, surveillance, and misrepresentation of the majority of our planet.

Artificial intelligence is increasingly deployed to synthesize large-scale public input in policy consultations and participatory processes. Yet no formal framework exists for auditing whether these summaries faithfully represent the source population, an accountability gap that existing approaches to AI explainability, grounding and hallucination detection do not address because they focus on output quality rather than input fidelity. Here, participatory provenance is introduced: a measurement framework grounded in optimal transport theory, causal inference and semantic analysis that tracks how individual public submissions are transformed, filtered or lost through AI-mediated summarization. Applied to Canada's 2025-2026 national AI Strategy consultation ($n = 5{,}253$ respondents across two independent policy topics), the framework reveals that both official government summaries underperform a random-participant baseline ($-9.1\%$ and $-8.0\%$ coverage degradation), with $16.9\%$ and $15.3\%$ of participants effectively excluded. Exclusion concentrates in clusters expressing dissent, scepticism and critique of AI ($33$-$88\%$ exclusion rates). Brevity, semantic isolation and rhetorical register independently predict representational outcome. An accompanying open-source interactive tool, the Co-creation Provenance Lab, enables policymakers to audit and iteratively improve summaries, establishing genuine human-in-the-loop oversight at scale.

Graph neural networks (GNNs) are the standard for learning on graphs, yet they have limited expressive power, often expressed in terms of the Weisfeiler-Leman (WL) hierarchy or within the framework of first-order logic. In this context, topological neural networks (TNNs) have recently emerged as a promising alternative for graph representation learning. By incorporating higher-order relational structures into message-passing schemes, TNNs offer higher representational power than traditional GNNs. However, a fundamental question remains open: what is the logical expressiveness of TNNs? Answering this allows us to characterize precisely which binary classifiers TNNs can represent. In this paper, we address this question by analyzing isomorphism tests derived from the underlying mechanisms of general TNNs. We introduce and investigate the power of higher-order variants of WL-based tests for combinatorial complexes, called $k$-CCWL test. In addition, we introduce the topological counting logic (TC$_k$), an extension of standard counting logic featuring a novel pairwise counting quantifier $\exists^{N}(x_i,x_j)\, \varphi(x_i,x_j),$ which explicitly quantifies pairs $(x_i, x_j)$ satisfying property $\varphi$. We rigorously prove the exact equivalence: $\text{k-CCWL} \equiv \text{TC}_{k{+}2} \equiv \text{Topological }(k{+}2)\text{-pebble game}.$ These results establish a logical expressiveness theory for TNNs.

As a means to balance the growth of the AI industry with the need for privacy protection, machine unlearning plays a crucial role in realizing the ``right to be forgotten'' in artificial intelligence. This technique enables AI systems to remove the influence of specific data while preserving the rest of the learned knowledge. Although it has been actively studied, most existing unlearning methods assume that unlearning is performed only once. In this work, we evaluate existing unlearning algorithms in a more realistic scenario where unlearning is conducted repeatedly, and in this setting, we identify two critical phenomena: (1) Knowledge Erosion, where the accuracy on retain data progressively degrades over unlearning phases, and (2) Forgetting Reversal, where previously forgotten samples become recognizable again in later phases. To address these challenges, we propose SAFER (StAbility-preserving Forgetting with Effective Regularization), a continual unlearning framework that maintains representation stability for retain data while enforcing negative logit margins for forget data. Extensive experiments show that SAFER mitigates not only knowledge erosion but also forgetting reversal, achieving stable performance across multiple unlearning phases.

Static concreteness ratings are widely used in NLP, yet a word's concreteness can shift with context, especially in figurative language such as metaphor, where common concrete nouns can take abstract interpretations. While such shifts are evident from context, it remains unclear how LLMs understand concreteness internally. We conduct a layer-wise and geometric analysis of LLM hidden representations across four model families, examining how models distinguish literal vs figurative uses of the same noun and how concreteness is organized in representation space. We find that LLMs separate literal and figurative usage in early layers, and that mid-to-late layers compress concreteness into a one-dimensional direction that is consistent across models. Finally, we show that this geometric structure is practically useful: a single concreteness direction supports efficient figurative-language classification and enables training-free steering of generation toward more literal or more figurative rewrites.

Recent studies reveal striking representational alignment between artificial neural networks (ANNs) and biological brains, leading to proposals that all sufficiently capable systems converge on universal representations of reality. Here, we argue that this claim of Universality is premature. We introduce the Umwelt Representation Hypothesis (URH), proposing that alignment arises not from convergence toward a single global optimum, but from overlap in ecological constraints under which systems develop. We review empirical evidence showing that representational differences between species, individuals, and ANNs are systematic and adaptive, which is difficult to reconcile with Universality. Finally, we reframe ANN model comparison as a method for mapping clusters of alignment in ecological constraint space rather than searching for a single optimal world model.