Representation Space

Recursive architectures such as Tiny Recursive Models (TRMs) perform implicit reasoning through iterative latent computation, yet the geometric structure of these reasoning trajectories remains poorly understood. We investigate the activation manifold of TRMs during recursive unrolling and find that activations occupy an effectively linear, low-dimensional subspace whose principal directions can be tracked dynamically with cheap power iterations. This suggests that weight-sharing concentrates iterative computation along a small number of dominant eigendirections, and we find that this concentration varies sharply across computational sites. We exploit this structure through LASER (Low-Rank Activation SVD for Efficient Recursion), a dynamic compression framework that maintains an evolving low-rank basis via matrix-free subspace tracking with a fidelity-triggered reset mechanism, achieving ${\sim}60\%$ activation memory savings with no statistically significant accuracy degradation. Our analysis raises questions about how recursive architectures allocate representational capacity during implicit reasoning, and whether this concentration can be exploited to improve the efficiency and stability of latent computation.

This paper presents FairNVT, a lightweight debiasing framework for pretrained transformer-based encoders that improves both representation and prediction level fairness while preserving task accuracy. Unlike many existing debiasing approaches that address these notions separately, we argue they are inherently connected: suppressing sensitive information at the representation level can facilitate fairer predictions. Our approach learns task-relevant and sensitive embeddings via lightweight adapters, applies calibrated Gaussian noise to the sensitive embedding, and fuses it with the task representation. Together with orthogonality constraints and fairness regularization, these components jointly reduce sensitive-attribute leakage in the learned embeddings and encourage fairer downstream predictions. The framework is compatible with a wide range of pretrained transformer encoders. Across three datasets spanning vision and language, FairNVT reduces sensitive-attribute attacker accuracy, improves demographic-parity and equalized-odds metrics, and maintains high task performance.

Prior work shows that fine-tuning aligned models on benign data degrades safety in text and vision modalities, and that proximity to harmful content in representation space predicts which samples cause the most damage. However, existing analyses operate within a single, undifferentiated embedding space -- leaving open whether distinct input properties drive the vulnerability differently. Audio introduces a structurally richer problem: a benign sample can neighbor harmful content not only through what is said but through how it sounds, even when its words are entirely innocuous. We present the first systematic study of benign fine-tuning safety in Audio LLMs, evaluating three state-of-the-art models with a proximity-based filtering framework that selects benign audio by embedding-space distance to harmful content. By decomposing proximity into semantic, acoustic, and mixed axes using external reference encoders alongside each model's own internal encoder, we show that benign fine-tuning elevates Jailbreak Success Rate (JSR) from single digits to as high as 87.12%. Crucially, the dominant vulnerability axis and the relative risk of audio versus text fine-tuning are both architecture-conditioned -- determined by how each model's encoder and projector transform audio into the LLM's input space. We propose two defenses: filtering training data to maximize distance from harmful embeddings, and a textual system prompt at inference, both reducing JSR to near-zero without architectural modification. Our mechanistic analysis on two architectures reveals that fine-tuning selectively suppresses the late-layer refusal circuit while the frozen encoder preserves representations, and that even the suppression pattern is architecture-conditioned, mirroring the behavioral asymmetries across modalities. Safety degradation from benign fine-tuning is a qualitatively distinct risk in Audio LLMs.

We introduce the Tan-HWG framework (Hebbian-Wasserstein-Geometry), a geometric theory of Hebbian plasticity in which memory states are modeled as probability measures evolving through Wasserstein minimizing movements. Hebbian learning rules are formalized as Hebbian energies satisfying a sequential stability condition, ensuring well-posed fiberwise JKO updates, optimal-transport realizations, and an energy descent inequality. This variational structure induces a fundamental separation between internal and observable dynamics. Internal memory states evolve along Wasserstein geodesics in a latent curved space, while observable quantities, such as effective synaptic weights, arise through geometric projection maps into external spaces. Simplicial projections recover classical affine schemes (including exponential moving averages and mirror descent), while revealing synaptic competition and pruning as geometric consequences of mass redistribution. Hilbertian projections provide a geometric account of phase alignment and multi-scale coherence. Classical neural networks appear as flat projections of this curved dynamics, while the framework naturally accommodates richer distributional representations, including structural weights and embedding memories, and their spectral extensions in complex internal spaces. Under mild Lipschitz regularity assumptions, including a quasi-stationary "sleep-mode" regime, we establish the existence of continuous-time limit curves. This yields a variational formulation of memory consolidation as a perturbed Wasserstein gradient flow. The framework thus provides a unified geometric foundation for synaptic plasticity, representation dynamics, and context-dependent computation.

Current fMRI foundation models primarily rely on a limited range of brain states and mismatched pretraining tasks, restricting their ability to learn generalized representations across diverse brain states. We present Brain-DiT, a universal multi-state fMRI foundation model pretrained on 349,898 sessions from 24 datasets spanning resting, task, naturalistic, disease, and sleep states. Unlike prior fMRI foundation models that rely on masked reconstruction in the raw-signal space or a latent space, Brain-DiT adopts metadata-conditioned diffusion pretraining with a Diffusion Transformer (DiT), enabling the model to learn multi-scale representations that capture both fine-grained functional structure and global semantics. Across extensive evaluations and ablations on 7 downstream tasks, we find consistent evidence that diffusion-based generative pretraining is a stronger proxy than reconstruction or alignment, with metadata-conditioned pretraining further improving downstream performance by disentangling intrinsic neural dynamics from population-level variability. We also observe that downstream tasks exhibit distinct preferences for representational scale: ADNI classification benefits more from global semantic representations, whereas age/sex prediction comparatively relies more on fine-grained local structure.

Self-supervised pre-training, where large corpora of unlabeled data are used to learn representations for downstream fine-tuning, has become a cornerstone of modern machine learning. While a growing body of theoretical work has begun to analyze this paradigm, existing bounds leave open the question of how sharp the current rates are, and whether they accurately capture the complex interaction between pre-training and fine-tuning. In this paper, we address this gap by developing an asymptotic theory of pre-training via two-stage M-estimation. A key challenge is that the pre-training estimator is often identifiable only up to a group symmetry, a feature common in representation learning that requires careful treatment. We address this issue using tools from Riemannian geometry to study the intrinsic parameters of the pre-training representation, which we link with the downstream predictor through a notion of orbit-invariance, precisely characterizing the limiting distribution of the downstream test risk. We apply our main result to several case studies, including spectral pre-training, factor models, and Gaussian mixture models, and obtain substantial improvements in problem-specific factors over prior art when applicable.

We investigate whether graded states of mind form spectrum-like structure in transformer representation spaces. To do so, we construct a dataset of 636 short natural-language sentences annotated with both a continuous score from $-5$ to $5$ and one of seven ordered tiers, ranging from collapsed or scarcity-driven expressions to more coherent, reflective, and integrative ones. We evaluate five frozen transformer representations: four sentence-embedding models and one decoder-only residual-stream representation. Across all representations, simple probes reliably recover both the continuous score and the discrete tier labels, and permutation tests show that performance significantly exceeds shuffled-label baselines. Additional analyses reveal a consistent geometric pattern: UMAP projections show low-to-high organization, confusion matrices concentrate errors between neighboring tiers, and directional ablation identifies a prominent score-aligned component. These results suggest that transformer representations contain statistically significant, spectrum-like organization aligned with the annotated state-of-mind structure. The annotations are used only as an operational framework for representation analysis, not as a clinical or diagnostic measure.

Large language models (LLMs) often produce incorrect or outdated content after being employed. Efficient and accurate knowledge updates without costly retraining are a major challenge. This problem is particularly challenging in lifelong settings, where complex, unstructured knowledge must coexist without interference. We introduce RILKE (Representation Intervention for Lifelong KnowledgE Control), a robust and scalable method that treats knowledge control as interventions within the model's representation space. Leveraging representation-space expressiveness, we identify two key properties enabling RILKE to achieve fine-grained control over complex, unstructured knowledge while maintaining general utility with frozen base weights. During training, RILKE learns paraphrase-robust and edit-localized modules that limit each update to a low-dimensional subspace to minimize cross-edit interference. At inference, a query-adaptive router selects the appropriate module to guide the model's generation. Across LLaMA and Qwen models, RILKE scales effectively to large-scale benchmarks, demonstrating high edit success and strong paraphrase generalization while preserving general utility with modest memory overhead. These results show RILKE is an effective and scalable solution for lifelong knowledge control in LLMs.

We study latent geometry as an explicit component of representation quality in data-scarce learning. For an encoder (φ), we define (Q_{β,γ}(φ)=I(φ(X);Y)-β\mathcal C(φ)-γd_{\mathrm{int}}(φ)), combining task-relevant information with penalties for curvature and intrinsic latent dimension. Thus geometry becomes part of the bottleneck criterion, not only a post hoc diagnostic. Under smooth-manifold, loss-transfer, and estimator-concentration assumptions, we derive non-asymptotic low-label generalization bounds where intrinsic dimension and covering complexity enter explicitly. We characterize the information--geometry frontier and prove empirical-surrogate consistency. The analysis links encoder geometry to learning through latent covering numbers, loss-class entropy, and uniform deviation. We instantiate the theory as \texttt{V-GIB}, adding curvature and dimension penalties to variational bottleneck training. Real low-label benchmarks compare \texttt{V-GIB} with ERM, VIB, and ablations across (1%)--(20%) label fractions. Results show improved performance and reduced geometric complexity in several regimes, especially FashionMNIST and CIFAR-10, while confirming that no fixed regularizer is universally dominant.

Few-shot 3D point cloud semantic segmentation aims to segment novel categories using a minimal number of annotated support samples. However, prototypes derived from the limited non-structural point cloud support set are often misaligned and have a small capacity, hindering effective gen eralization to novel categories. This stems from two core issues: i) the prototype possess limited representational capacity fails to cover the full intra-class diversity of a novel category, and ii) the prototypes suffer from misalignment with the query space due to the inter-set inconsistency between support and query sets. To address these issues, our work focuses on leveraging the few support samples to construct a well-aligned big-capacity prototype. Motivated by the powerful generative capabilities of diffusion models, we re-purpose its pre-trained conditional encoder to provide rich feature components for prototype ex pansion. Subsequently, a push-pull force aligns this expanded prototype towards the query feature space. Under this setup, we introduce the Prototype Expansion Network (PENet), a framework that constructs aligned big-capacity prototypes from two complementary feature sources. Specifically, PENet employs a dual-stream learner architecture: it retains a conventional fully supervised Intrinsic Learner (IL) to distill representative features, while introducing a novel Diffusion Learner (DL) to provide rich generalizable features. The resulting dual prototypes are then processed by a Prototype Assimilation Module (PAM), which adopts a push-pull attention block to align the prototypes with the query space. Furthermore, a Prototype Calibration Mechanism (PCM) regularizes the final big-capacity prototype to prevent semantic drift. Extensive experiments on the S3DIS and ScanNet datasets demonstrate that PENet outperforms state-of-the-art methods across various few-shot settings.

Explanations of cognitive behavior often appeal to computations over representations. What does it take for a system to implement a given computation over suitable representational vehicles within that system? We argue that the language of causality -- and specifically the theory of causal abstraction -- provides a fruitful lens on this topic. Drawing on current discussions in deep learning with artificial neural networks, we illustrate how classical themes in the philosophy of computation and cognition resurface in contemporary machine learning. We offer an account of computational implementation grounded in causal abstraction, and examine the role for representation in the resulting picture. We argue that these issues are most profitably explored in connection with generalization and prediction.

Self-supervised learning has become an increasingly important paradigm in the domain of machine intelligence. Furthermore, evidence for self-supervised adaptation, such as contrastive formulations, has emerged in recent computational neuroscience and brain-inspired research. Nevertheless, current work on self-supervised learning relies on biologically implausible credit assignment -- in the form of backpropagation of errors -- and feedforward inference, typically a forward-locked pass. Predictive coding, in its mechanistic form, offers a biologically plausible means to sidestep these backprop-specific limitations. However, unsupervised predictive coding rests on learning a generative model of raw input (akin to "generative AI" approaches), which entails predicting a potentially high dimensional input; on the other hand, supervised predictive coding, which learns a mapping between inputs to target labels, requires human annotation, and thus incurs the drawbacks of supervised learning. In this work, we present a scheme for self-supervised learning, specifically for an emerging research sub-domain that we label as neuroscience-informed self-supervised learning (NeuroSSL), within a neurobiologically plausible framework that appeals to the free energy principle, constructing a new form of predictive coding that we call meta-representational predictive coding (MPC). MPC sidesteps the need for learning a generative model of sensory input (e.g., pixel-level features) by learning to predict representations of the input across parallel streams, resulting in an encoder-only learning and inference scheme. This formulation notably rests on active inference (in the form of sensory glimpsing) to drive the learning of representations, i.e., the representational dynamics are driven by sequences of decisions made by the model to sample informative portions of its sensorium.

This review examines neural operator architectures for learning solution operators of parametric partial differential equations (PDEs), with an emphasis on conceptual clarity and practical implementation. The work analyzes key models, including DeepONet, PCANet, and the Fourier Neural Operator, highlighting their underlying representations, computational structures, and comparative performance. These architectures are demonstrated on three canonical PDE problems: the Poisson equation, a linear elasticity problem, and a hyperelasticity problem. To make the presentation self-contained, key foundational topics are introduced, including finite-dimensional representations of function spaces, singular-value decomposition, and sampling from infinite-dimensional function spaces. Beyond forward modeling, the review discusses the use of neural operators as surrogate models within a Bayesian inverse-problem framework, including prior specification, forward-map approximation, and posterior computation. The performance of the three neural-operator architectures is evaluated on in-distribution samples, out-of-distribution samples, and Bayesian inference tasks. The review also discusses challenges related to prediction accuracy and generalization, outlining emerging strategies such as residual-based error correction and multi-level training. The review concludes by positioning neural operators within broader scientific-computing workflows and by identifying directions for reliable, scalable operator learning.

In contrast to RNNs, which compress their history into a single hidden state, Transformers can attend to all past tokens directly. However, standard Transformers rely solely on the hidden state from the previous layer to represent the entire context. We show that this design creates pressure toward representation collapse and can degrade performance. To address this issue, we introduce Layer-Integrated Memory (LIMe), a lightweight extension that leverages existing key-value buffers and learns per-head, per-layer routing weights to integrate representations from previous layers. Across language modeling, synthetic reasoning, and deep architectures, LIMe improves perplexity per FLOP in the studied regimes and yields strong gains on synthetic tasks while preserving higher value-vector entropy and token separability. Finally, learned routing weights reveal systematic reuse of local and long-distance features, showing how LIMe enriches attention-time memory without increasing hidden-state size. Code is available at https://github.com/corl-team/lime.

We propose a computational tropical geometry framework for the symbolic analysis of neural networks with tropical activations. The number of linear regions of a neural network has been actively studied as a measure of the expressivity of a given architecture. To study these, we work in the setting of tropical geometry---a combinatorial and polyhedral variant of algebraic geometry---where there are known connections between tropical rational maps and feedforward neural networks. We expand this connection by developing concrete computational tools for studying the linear regions of neural networks. We present an algorithm, together with a proof of correctness, which computes the linear regions of a neural network as explicit unions of polyhedra. We further relate the computation of the number of linear regions of a tropical expression to the number of monomials that appear in it, and show how tropical expressions can often be pruned to remove redundant monomials. We introduce the Hoffman constant of a neural network's tropical expression, a geometric quantity that controls the distance from any point in the input space to the farthest linear region. We provide the open source Julia library TropicalNN.jl, which is built on top of the OSCAR computer algebra system and implements the algorithms mentioned above to analyze neural networks symbolically using their tropical representations. We present a set of proof-of-concept computational examples to demonstrate how our tropical geometric theory can be applied to reveal insights on the expressivity of a network architecture.