Neural Representations

Implicit neural representation (INR) has achieved remarkable progress in novel view synthesis and image/video coding in recent years.Compared to conventional end-to-end image codecs, INR-based compressors demonstrate significant advantages in decoding complexity. However, their practical application has been hindered by the inferior encoding speed and underutilized decoding efficiency.In this work, we propose a feedforward INR image coding architecture, Practical INR Image Codec (PIC), that computes all the necessary information for INR network in a single forward pass, achieving an encoding speed of 20 FPS. Additionally, we implement a highly optimized decoder that reaches 2000 FPS decoding speed, significantly surpassing JPEG's performance at comparable rate-distortion (RD) performance. To the best of our knowledge, this work presents the first learning-based image codec that simultaneously outperforms or is comparable with JPEG in both RD performance and decoding speed while maintaining practical encoding speed. Code is available at https://github.com/actcwlf/PIC.

Class disentanglement (the separation of a representation's class-conditional point clouds along depth and over training) is usually read off descriptive curves. We measure it as certified topological interaction between labeled point clouds, using the recently introduced Intersection Euler Characteristic Profile: the Euler characteristic of the overlap of the clouds' ball unions as a function of scale, computed by one Alpha-complex sweep with no boundary-matrix reduction. Every number carries a test: exact permutation tests in both directions, a guarded separation certificate, and a paired test for the comparative claims applications make. Across 111 trained networks and 52,650 certified measurements, disentanglement is depth-graded and concentrated in the first epochs, and interaction quotients rank class pairs by confusability (Spearman rho=0.83), on par with cheap separability statistics. In a 96-model factorial population, augmentation is the one training choice that separates classes relative to chance; weight decay compresses the overlap without separating, and depth and width do nothing. The structural finding is one only a k-fold statistic can pose: the joint entanglement of a class triple sits below that of its strongest pair in 97% of triple-layer cells and 99.5% of deep cells, far below a measured null floor, in vision encoders and frozen language models alike. This pairwise dominance is a regularity, not a law: expected from the nesting of overlaps but not forced by geometry, present at initialization and in raw pixels, and manufactured in the last stage alone when a network memorizes random labels. The unnormalized profile mass predicts test accuracy (R^2=0.94), the quotient does not, and neither beats a linear probe. One lesson is reported in full: the paired test must use a scale-free statistic, or it certifies feature-norm dynamics as disentanglement.

Pretrained vision embeddings are increasingly used as general-purpose representations for modelling how people appraise urban scenes, and are validated almost entirely by how well they predict human ratings. High predictive accuracy does not establish that these embeddings organise scenes as human perception does. We test the two properties separately against brain data. Using openly released EEG from 63 adults who viewed and rated 56 Berlin street scenes, we estimate the representational geometry of the scenes over time, the proportion of that geometry that is explainable at all, and its correspondence with seventeen feature spaces spanning language-supervised, self-supervised, category-supervised and dense-prediction training, two orders of magnitude of scale, and interpretable controls. Correspondence is low throughout: the best representation, DINOv2 ViT-B, reaches 29.6% of the lower bound of the noise ceiling, the panel spans 11.0% to 29.6%, and a Gabor energy descriptor is indistinguishable from the best model while outperforming every language-supervised model tested. Within a model, deeper layers still match later neural responses, so the hierarchical correspondence found for object recognition survives even at this low overall level. The same embeddings predict held-out appraisal ratings well, up to r = 0.87, and the two measures do not track each other across models; reweighting features towards the neural geometry lowers appraisal prediction for every model tested, against a control of matched dimensionality. Predicting how a street is appraised is therefore weak evidence that a model represents the street as the brain does. The benchmark uses only public data and requires no training, so evaluating a new representation needs only its embeddings for 55 images.

Representational similarity analysis (RSA) is increasingly used to ask which learning rules give convolutional networks brain-like representations. Because biologically plausible rules such as feedback alignment, predictive coding and STDP do not scale, studies that include them train small networks on small images (typically 32x32 CIFAR) and then compare them to brain responses modeled at much higher resolution. We find that a common result in this setting, that untrained or locally trained networks rival or beat backpropagation at early visual cortex, depends strongly on the resolution at which the network is evaluated. The V1 gap between an untrained network and a backpropagation-trained one widens from -0.001 +/- 0.007 at the 32px training resolution to +0.044 +/- 0.006 at 224px, growing monotonically across six resolutions (n=5 seeds). It holds in human fMRI and, directionally, in single-seed macaque electrophysiology, along the training trajectory, and for an ImageNet ResNet-50 and a Swin-Tiny transformer trained at 224px. Four candidate mechanisms are tested and none accounts for it: train/eval resolution matching, low-level Gabor and pixel structure, the normalization state of the untrained baseline, and convergence of the pooled descriptor toward a global brightness statistic; three are excluded by interventions holding the convolutional weights bit-identical. A fifth experiment locates the effect: capping image detail at the training resolution while letting the pooled positions grow 12-fold removes about 90% of it, so the dependence is carried by image detail rather than by pooling. Separately, a single scalar luminance value per image reaches rho = 0.075 against the V1 RDM, essentially matching the untrained network's 0.076, which bounds what this style of comparison can resolve. The one learning effect that holds across resolution is backprop above untrained, at LOC.

How many directions does a neural representation use to encode a concept? A common answer repeatedly erases probe directions and reports the stopping count or cumulative removed rank. We show that both quantities can change under an information-preserving invertible reparameterization, so neither is intrinsically a concept dimension. We distinguish model-defined population quantities (generating dimension, sufficient linear dimension, and minimum guarding rank) from procedure-defined quantities such as stopping count and cumulative edit rank. In a population Gaussian construction, an invertible shear preserves the prediction problem and all three quantities, yet changes the cumulative Euclidean erasure count from one to two. The separation holds for Moore--Penrose ordinary least squares and every finite nonnegative ridge weight. For a two-output full-QR procedure matching our motivating video analysis, cumulative edit rank similarly changes from two to the ambient dimension four. Conversely, the complete cumulative metric-QR trajectory is affine-equivariant when its positive-definite metric, probe, regularizer, and tie-breaking are transported consistently; exact covariance is one corollary, not a canonical semantic metric. In a known-rank finite-sample Adam/QR calibration, identity mixing stops after one accepted update in all 20 large-sample runs, whereas each tested shear $a\in\{.5,.75,1,1.25,2\}$ accepts at least two updates in all 20 runs. Controlled reparameterizations of frozen V-JEPA2 features preserve rank-zero predictions yet alter later Euclidean trajectories under practical optimization. These visual contact experiments are stress tests, not estimates of contact dimension. Iterative erasure therefore returns a procedure-relative estimand jointly determined by representation geometry and the full measurement procedure, not a semantic dimension by itself.

Transformer models are most often understood through what they do: their benchmark performance, generation quality, or behavior on downstream tasks. Yet frozen transformer input-embedding spaces may also be examined through their responses to a controlled deterministic probe before contextual computation or task-specific adaptation. Guided by this response-based view, we introduce \emph{ChaosProbe}, a deterministic neurochaos-inspired method for constructing response-based fingerprints of frozen transformer input-embedding spaces. For each prompt-level embedding matrix, ChaosProbe applies a chaotic trajectory-based transformation and summarizes its Firing Rate and Entropy channel responses with complementary representation-level measures, producing a fixed-length signature for each model. In a bounded proof-of-concept study of $80$ neutral prompts and four pretrained models---GPT-2, DistilGPT2, BERT-base-uncased, and RoBERTa-base---Pearson correlation, Spearman correlation, and cosine similarity each recover all four same-family nearest-neighbor assignments and both expected mutual family pairs. Euclidean distance recovers three of the four assignments and one of the two mutual family pairs. Paired bootstrap resampling supports the stability of the Pearson and Spearman pairings over the observed prompt set, and signature-validity checks show that constant or collapsed responses do not dominate the reported fingerprints. These results provide a cohort-dependent proof of concept that deterministic neurochaotic response signatures can expose broad structure among frozen transformer input-embedding spaces.

Decoding visual experience from non-invasive brain activity is central to neuroscience and brain-computer interfaces. Functional magnetic resonance imaging (fMRI) offers fine spatial detail, but its slow hemodynamics and burdensome acquisition limit temporally resolved decoding. Electroencephalography (EEG) and magnetoencephalography (MEG) provide millisecond resolution, making image retrieval compelling: identify the viewed image from one neural response and a fixed candidate bank. Contrastive alignment to pretrained visual representations enables zero-shot retrieval from EEG and MEG, but most systems collapse heterogeneous visual supervision into a single embedding before ranking. This early consolidation imposes one similarity geometry on every candidate order and removes encoder-specific disagreements from the final ranking. We propose CORTIVA, a candidate-score fusion framework that preserves this complementary evidence. Three decoding routes are aligned to heterogeneous visual targets, score the same indexed candidates independently, and combine only their temperature-scaled score vectors before ranking. On the 200-way THINGS-EEG2 benchmark, CORTIVA reaches 73.5% Top-1 and 95.3% Top-5 across ten participants, exceeding the strongest reported baseline by 10.3 and 5.4 percentage points. With a modality-specific neural encoder, the same fusion principle reaches 42.4% Top-1 on THINGS-MEG. Matched route-removal retraining and four weight controls demonstrate that CORTIVA's gain arises from integrating complementary route scores and persists with uniform weighting, without requiring a specialized weighting rule. Independent DINOv2 analyses further reproduce the local error neighborhoods and posterior neural-visual correspondence. These results establish candidate-score fusion as a simple and testable alternative to embedding-level consolidation for neural image retrieval.

Decoders of anesthetic state from cortical activity fail across drug classes, most notoriously ketamine, but reported accuracy cannot say whether the neural representation or only the decision threshold has failed; we separate the two in a controlled preparation with ground-truth labels. We decoded awake versus anesthetized from mouse electrocorticography (the 250-Hz-bandlimited local field potential, sampled at 1875 Hz) under leave-one-anesthetic-out evaluation across five mechanistically distinct anesthetics (isoflurane, dexmedetomidine, ketamine, propofol, and midazolam), comparing a spatially blind band-power decoder, covariance/Riemannian representations, and Riemannian domain adaptation, with all statistics at the session level and mouse-level cluster bootstrapping for the ketamine fold. The representation transfers: band-power ranks awake versus anesthetized at a session AUROC of at least 0.96 on every held-out drug, ketamine included (0.980, cluster confidence interval 0.821 to 1.000). The failure is confined to the threshold: across three representations the ketamine ranking is near-invariant while its balanced accuracy swings from chance to high, and a permutation test is significant for ranking (p = 0.0025) but not for fixed-threshold accuracy (p = 0.3795). Riemannian domain adaptation is net-negative. A causal, label-free threshold anchored to the subject's own pre-induction baseline fixes ketamine (balanced accuracy 0.50 to 0.85) and dominates domain adaptation. Because the ketamine test sessions come from three mice that also contribute training drugs, this is within-subject cross-drug transfer; we do not claim population-level transfer across subjects. In cross-drug state decoding the actionable failure is calibration, not representation.

We propose a neural representation for minimal surfaces. Unlike prior approaches based on discretization or Physics-Informed Neural Networks (PINNs), where meshes or neural fields are optimized to approximate the governing equations, our method builds on an exact representation, similar to the classical Weierstrass--Enneper parameterization, yielding minimal surfaces up to negligible quadrature error in evaluation. We formulate a training objective for the Plateau problem that optimizes over this representation.

The coupled ghost and gluon Dyson--Schwinger equations (DSEs) of four-dimensional Landau-gauge Yang--Mills (YM) theory are solved with a neural representation trained only from renormalized equation residuals. The neural and fixed-point solutions agree at the percent level and remain stable under changes of initialization, network size, integration grid, and infrared boundary condition. Variations of the three-gluon vertex model produce substantially larger effects than the neural error. The MiniMOM ultraviolet running and the sign change of the gluon Schwinger function are also reproduced within the limitations of the truncation.

Implicit neural representation (INR) methods provide continuous coordinate-to-value mappings and integrate naturally with direct volume rendering, making them attractive for representing volumetric data. However, existing INR-based approaches for volumetric data are inherently lossy, and even small reconstruction errors can propagate through rendering and downstream analysis. In this work, we explore Lossless-INR, a lossless INR framework for 3D scientific volumetric data based on bit-plane decomposition. By decomposing each voxel value into binary bit-planes, we reformulate reconstruction as per-bit binary classification, so that exact recovery reduces to predicting every bit correctly. To make this optimization tractable while keeping the representation compact, we combine an octree block-partitioning strategy that adaptively subdivides complex regions with a ternary feature-grid network whose grid entries are parameterized by a ternary set of values. Experiments on diverse volumetric datasets show that this design can achieve zero bit-error rate and bit-exact reconstruction, enabling faithful rendering and downstream analysis with a compact representation. The code is available at https://github.com/TouKaienn/Lossless-INR.

Within Explainable Artificial Intelligence, mechanistic interpretability uses Sparse Autoencoders (SAEs) to extract more interpretable features from neural representations. However, assessing their monosemanticity, and thus explanation quality, remains challenging. Existing metrics require external concept labels or depend on pretrained embedding models, making them sensitive to encoder's geometry. We introduce the Tversky Monosemanticity Score (TMS), a label-free metric that operationalizes monosemanticity as activation-set coherence of binarized SAE latents, and does not require external embedding encoders. We evaluate TMS on SAEs trained on features from pretrained vision and vision-language models (DINOv3, CLIP, BLIP2), two common SAE regimes (TopK, BatchTopK), multiple sparsity levels, and expansion factors. Our results show that TMS is less affected by encoder anisotropy than its embedding-based alternative, while remaining aligned with established monosemanticity indicators. TMS also reveals distinct SAE training dynamics across base models. Moreover, under encoder anisotropy, TMS provides a stronger indication of probe-based concept deletion effectiveness, while being competitive otherwise.

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

In this paper, we investigate preprocessing techniques aimed at improving the efficiency of accessing models of propositional formulas represented in conjunctive normal form (CNF). We focus on three fundamental tasks: uniform sampling, direct model access, and model enumeration. Our analysis reveals that most state-of-the-art preprocessors, when they do not preserve formula equivalence, are generally unsuitable for these tasks. In contrast, we demonstrate that preprocessors which preserve model counts can be effectively leveraged, provided relevant preprocessing information is maintained. To validate our approach, we perform extensive experiments on a diverse suite of benchmarks from multiple domains. The experimental results show that our preprocessing methods are both efficient and robust, yielding significant performance improvements for model access queries when CNF formulas are compiled into d-DNNF representations.

Neural networks increasingly guide decisions in high-stakes domains such as medical diagnosis, credit approval, and energy bidding. Audit in these settings requires case-level evidence: which training cases support an action and what outcomes they carried. Case-based decision theory (CBDT) formalizes this reasoning by aggregating outcome support from remembered cases. We show that an OLS action readout fitted on a fixed neural representation admits an exact case-based decomposition. Each action score is a weighted sum of training-case returns, with coefficients determined by empirical Gram geometry. We identify a sufficient regime for CBDT similarity semantics; outside it, the coefficients should generally be treated as signed Gram-geometric influence. The decomposition yields audit signals that trace scores to training cases, measure action coherence, and identify weak support. Across synthetic CBDT, PJM, Adult Income, and Default Credit tasks, the method recovers case-level preference structure and achieves the highest mean Top-30 consistency among compared attribution baselines, while remaining competitive on support reconstruction. The audit requires only fitting an OLS top-layer probe, without retraining the representation or accessing the original optimization trajectory; probe fidelity is measured by score reconstruction.

Dimensionality reduction has proven powerful for identifying neural manifolds, which are low-dimensional structures underlying high-dimensional neural activity. These low-dimensional representations have improved the interpretability of population-level coding. Yet whether such low-dimensional representations are biologically relevant and confer functional advantages in learning systems, or merely reflect neuron-level activity, remains contested in neuroscience. We show that an explicit information bottleneck forcing a recurrent neural network to learn a low-dimensional representation is necessary for rotational and out-of-distribution generalisation in a time-series prediction task. Using information-theoretic measures of causal emergence, we characterise the dynamics of this representation across the memorisation-to-generalisation transition, finding a non-monotonic trajectory which shows an initial decrease, a minimum, and a subsequent rise to a maximum, even as prediction loss falls monotonically. This trajectory scales with task complexity, and the magnitude of emergent structure reliably predicts generalisation performance. Analysis of CA1 hippocampal activity in mice learning an alternating maze task reveals analogous non-monotonic emergence dynamics that track behavioural performance. Together, these findings indicate that the ability of neural networks to learn compact, distributed and emergent representations confers a functional advantage for generalisation, supporting a causal role for learned representations in cognition.

In computational vision science, Convolutional Neural Networks (CNNs) have emerged as a popular model of biological vision because of the alignment they can exhibit with neural and behavioral data in humans and animals. However, it remains unclear to what extent this alignment persists for visual tasks that extend beyond the canonical object recognition paradigm based on well defined semantic content. In this study, we diverge from the common object-centric view by focusing on another aspect of vision: texture perception. We consider textures of different complexity generated with three different algorithms from the same source images. Using a rank-based statistic, we quantify the information encoded in the internal representations of a CNN and three Vision Transformers (ViTs), and we compare the similarity of these representations to those inferred from human psychophysics data. We find that the representation of textures is aligned in different ViTs, but not between the ViTs and the CNN; that ViTs form similar representations for textures of different complexity; that human performance in recognizing textures can be better predicted from ViTs representations rather than CNN representations. Taken together, these results suggest that ViTs may capture more faithfully than CNNs how texture patterns are visually processed by humans, and that the representations of texture stimuli in computational models may be driven by the network architecture.

How concepts are represented in neural networks is a fundamental question in machine learning. The dominant view treats concept representations as stationary geometric objects. Yet concepts appear in context, and context transforms them. Drawing from neural population geometry, we formalize concept representations as point-cloud manifolds and contextual transformations as vector fields, and instantiate this framework in large language models. Across six model families of varying scales, we find that context moves each concept differently. The variance in these displacements is semantically organized, correlating with lexical concreteness and density. Importantly, both the concepts being transformed and this variance structure are shared across models: displacement structure transported from one model predicts held-out displacements in others significantly above chance. Together, these findings show that models share a common geometry not only in how concepts are represented, but more importantly in how context transforms them, a structure with richer organization than prior work has recognized.

Trial-to-trial variability limits how reliably neural population geometry can be estimated, while comparisons across populations depend on neuron and trial counts, response quality, and clustered sampling. We quantified within-session geometric reliability using Shesha, the Spearman correlation between representational dissimilarity matrices estimated from independent trial subsets, in all 39 Steinmetz Neuropixels sessions and in olfactory bulb and piriform cortex recordings from Bolding and Franks. Steinmetz analyses matched neurons and repetitions, compared observed reliability with a stationary residual-bootstrap expectation, and used mouse-level or mouse-clustered inference. Mean matched reliability was 0.0402 across 312 area-by-session recordings. Regional differences and reliability above the stationary benchmark did not survive correction. Temporal effects received the strongest support: interleaving early and late trials increased reliability relative to blocked allocation ($Δ=0.02666$, $q=0.001953$), and RDM similarity declined with within-session lag (mean mouse-level slope $=-0.01912$, $q=0.001953$; $n=10$ mice). Outer-cross-fitted reliability was not associated with choice-direction coupling or stimulus or response-direction decoding after correction. Olfactory comparisons remained descriptive because few paired sessions and no animal identities were available. In held-out simulations, associative recurrence outperformed feedforward subspace denoising but not divisive normalization. Representational geometry became less reproducible with temporal separation within a session, and comparisons across neural populations require sampling calibration and independent inference.

Zero-shot visual decoding from electroencephalography (EEG) aims to infer visual semantics from non-invasive neural recordings, but remains challenging due to the low signal-to-noise ratio, non-stationarity, and limited spatial resolution of EEG. Existing EEG-vision alignment methods often rely on holistic EEG embeddings, which can obscure the complementary temporal, spectral, and spatial structure underlying visual perception. We introduce a unified multiview EEG representation learning framework for aligning brain responses with visual semantic embeddings. Our method builds an EEG encoder that jointly models three complementary views: input-conditioned state-space temporal dynamics, learnable wavelet-based spectral decomposition for sample-adaptive frequency modeling, and attention-modulated graph learning for structured electrode interactions. The resulting multiview EEG embeddings are fused and aligned with pretrained visual representations in a shared semantic space using contrastive learning with EEG-specific regularization, enabling 200-way zero-shot visual classification. Experiments on THINGS-EEG benchmark show that our method achieves state-of-the-art performance, with 54.8% Top-1 and 85.6% Top-5 accuracy in the within-subject setting and 15.3% Top-1 and 45.4% Top-5 accuracy in the cross-subject setting. We further present the first systematic cross-session EEG-image decoding evaluation, achieving 40.8% Top-1 and 78.0% Top-5 accuracy. These results suggest that explicitly modeling multiview neural structure improves both semantic alignment and generalization in EEG-based visual decoding.

Deep neural networks transform input data into latent representations that support a wide range of downstream tasks. These representations can be characterized along information-theoretic and geometric dimensions, but their relationship remains poorly understood. A central open question is whether low mutual information (MI) between inputs and representations necessarily implies geometrically compressed latent spaces and vice versa. We investigate this question using class-wise clustering as a measure of geometric compression and theoretically sound MI estimation in conditional entropy bottleneck (CEB) networks and continuous dropout networks. We evaluate the interplay between MI, geometric compression, and generalization on classification tasks under controlled noise injection schemes. Our findings show that low MI does not reliably correspond to geometric compression, and that the connection between the two is more nuanced than often assumed. Indeed, our experiments reveal a negative and nonlinear relationship that can reverse when varying training setup. Our results put forward a hypothesis that generalization acts as a potential confounder in this connection rather than being their direct consequence.

Neurosymbolic semantics is fragmented: classical, fuzzy, probabilistic and neural systems each define truth by their own inductive rules. NeSyCat, extending ULLER, subsumes them under a single inductive definition of truth, parametric in a strong monad and an aggregation structure on truth-values. NeSyCat has so far lacked an account of predicates and functions learned by neural networks. We provide NeSyCat Torch as the missing link and interpret computational symbols via neural networks, implementing the framework in probabilistic programming and tensor-based backends. We use the distribution monad for reference semantics and metric evaluation, and complement it by a monad for numerically stable, differentiable training: the lazy log-tensor monad over the log-semiring. For efficient training in batches, we furthermore employ a batch monad. The axioms are the source code: written once in monad-based do-notation, monadic bind performs marginalisation, lazily pruning unneeded branches. On MNIST addition, our HaskTorch, JAX, and PyTorch implementations outperform LTN and DeepProbLog in speed and accuracy, while achieving nearly the accuracy of DeepStochLog. However, unlike DeepStochLog, we stay in a uniform framework that applies to many first-order NeSy approaches. Namely, the construction is parametric in the monad; instantiating it with, e.g., the Giry monad extends the approach to continuous probability (working out a neural representation here is left for future work).

Oppenheim and Lim (1981) showed that natural images stay recognizable when reconstructed from their Fourier phase alone, while the magnitude carries little of their identity. We ask whether trained image classifiers reproduce this asymmetry inside their hidden layers, and we test it causally: given two images, we transplant the phase of one onto the magnitude of the other at a chosen layer and record which image the prediction follows. In PRISM2D, GFNet, and ViT-B/16 the prediction follows the phase or sign donor, and deleting all image-specific magnitude barely moves accuracy, so identity rides on phase while image-specific magnitude is largely dispensable to the readout. ResNet-50 at first seems to break the pattern, because transplanting sign after its ReLUs does nothing; a fair intervention before the ReLU reveals a strong latent sign code in the late blocks, and a DC-only control shows the readout consumes a channel-wise spatial average. Controls rule out the trivial case in which magnitude simply stops depending on the image. The architectures therefore share a phase/sign identity code but expose it in different bases, set by rectification and readout geometry, which gives a mechanistic account of the texture--shape gap between CNNs and attention models.

Memory formation is fundamental to intelligence, yet whether deep neural networks preserve identifiable memory traces analogous to biological memory units remains an open question. This work introduces a geometric framework to identify such "AI engrams" by formalizing the neuroscientific criteria of specificity, reactivation, sufficiency, and necessity into a constrained inverse problem. We derive a closed-form estimator that isolates individual memory traces from globally entangled parameters, and show that this biologically-derived solution corresponds to a natural gradient update on the parameter manifold. AI engrams enable surgical manipulation of learned knowledge: any subset of memories can be composed or erased through linear arithmetic, without iterative optimization. Experiments ranging from simple MLPs to LLMs demonstrate the causal validity and substantial scalability of AI engrams. Together, these results bridge theories of biological memory and artificial representation learning and offer geometric insight into how deep networks simultaneously support functional specificity within distributed storage.

Transformer models are increasingly used as computational models of cognition and neural representation, so the mechanism implemented by self-attention is of interest beyond engineering performance. A complementary tradition in cognitive science models coordination, binding, and memory through dynamical interactions such as oscillator synchrony; we bring this mechanism into self-attention by introducing the Kuramoto Attention layer, whose value update is a synchronization step. Each token carries a bank of phase oscillators, so its hidden state lives on a high-dimensional torus. The attention weights form an adaptive coupling graph, and using the raw phase states as values makes the value update exactly the Kuramoto coupling direction for fixed attention weights. The softmax selects which oscillators couple, while the value path moves each token toward the attention-weighted circular mean of the tokens it selects. We train Kuramoto Attention on enwiki8 and CodeParrot against parameter-matched RoPE and SwiGLU transformers. At 5M parameters on CodeParrot, it improves on the transformer by both median and mean, with mean gaps of 0.012 validation and 0.010 test bits per byte. At 5M on enwiki8, all six runs have lower validation/test medians than the transformer and all-seed means within 0.01 BPC; five of six also form a tight lower-mean cluster. At 1M, it trails by about 0.02 BPC on enwiki8 and by 0.013-0.015 bits per byte on CodeParrot. Ablations and phase diagnostics show how the layer's synchronization and geometry-motivated components shape model performance. The result is a self-attention mechanism whose learned computation can be read directly as adaptive synchronization on phase states.

The Euler Characteristic Curve (ECC) records the Euler characteristic of a linearly embedded cell complex as a function of filtration height in a given direction, and the Euler Characteristic Transform (ECT) is the injective shape descriptor obtained by collecting ECCs over many directions. How the ECT is encoded for a neural network is itself an inductive bias, conventionally fixed by discretizing each ECC. We introduce a continuous encoding: for each direction and each vertex it records the net Euler-characteristic change attributed to that vertex, producing a per-direction token sequence that a small transformer maps to a feature vector. We separate the resulting pipeline into two stages on orthogonal axes: an ECC encoder that acts within each direction, mapping its curve to a fixed-length vector, and an ECT representation that acts across directions, aggregating the per-direction vectors into one. We study six ECT representation architectures spanning a range of inductive biases, from a structure-agnostic feedforward baseline to convolutional and complex-valued models that preserve equivariance under planar rotations. Across six classification benchmarks covering point clouds, graphs, cubical complexes, and meshes, the continuous encoding improves accuracy on five of six datasets, and control experiments attribute the gain to the tokenization itself rather than to the added transformer capacity. The representation architecture matters less than the encoding, and the payoff from its inductive biases depends on the encoding: a feedforward network performs best under continuous encoding but is less robust under discretization than convolutional architectures.

Understanding how transformer representations evolve across layers, not merely what they encode, remains an open problem in mechanistic interpretability. We recast the transformer forward pass as a discrete population trajectory through a high-dimensional representation manifold, drawing on geometric tools from computational neuroscience. Rather than probing for pre-specified features, we characterize trajectory geometry using five metrics computed directly in the ambient space: trajectory length, curvature, a semantic convergence index, layerwise cosine similarity, and representational stability. Across three model families (GPT-2, TinyLlama, Qwen2.5) and five controlled prompt families, we report four findings. First, semantically related prompts converge significantly in middle-to-late layers (peak CI 0.41--0.58, p<0.001, Mann-Whitney U), consistent with attractor-like dynamics. Second, reasoning tasks produce trajectories of greater curvature than lexical variations (0.71--0.83 rad vs. 0.27--0.31 rad), suggesting curvature encodes computational complexity. Third, ambiguous tokens exhibit trajectory bifurcation with up to 5.6x representational separation by the final layer, absent in unambiguous controls. Fourth, layerwise cosine similarity reveals a universal three-phase structure: encoding, elaboration, and output preparation, consistent across all three architectures. All four effects vanish under shuffled-layer and random-embedding controls. We release a fully open-source, model-agnostic pipeline and argue that trajectory geometry constitutes a principled, probe-free lens for mechanistic interpretability.

While neural collapse (NC) predicts that a $K$-class-balanced classifier should organize terminal representations as a $(K-1)$-dimensional simplex equiangular tight frame (ETF), modular addition consistently enters a different regime: networks compress to a two-dimensional cyclic geometry in which both classifier weights and token embeddings lie on circles. We refine the explanation of this phenomenon in three directions. First, we formalize a layerwise non-uniform training mechanism: downstream classifier weights are driven by dense cross-entropy gradients into a rank-2 equiangular configuration before upstream embeddings fully reorganize, and once this classifier plane forms, backpropagated feature gradients constrain embedding motion to the same plane while weight decay suppresses orthogonal components. Second, after this subspace locking, the induced in-plane dynamics admit an entropy-regularized transport interpretation on $S^1$; combined with modular-addition labels, this reduces embedding formation to phase alignment, whose minimizers are single-frequency characters of $\mathbb{Z}/P\mathbb{Z}$ and hence equal-angle points on a circle. Third, we quantify why this solution prevails over NC: a simplex ETF gains only an $O(1)$ advantage in cross-entropy, whereas the cyclic rank-2 solution enjoys a $Θ(K)$ advantage under Schatten or weight-decay surrogates, yielding a critical threshold $λ_{\mathrm{crit}} = Θ(1/K)$. Our results explain both why classifier weights move first and why embeddings subsequently align with them, showing that grokking on modular arithmetic is governed not by maximal separation alone but by a task-structured trade-off between separation, symmetry, and complexity.

Modern neural classifiers commonly rely on linear readouts, yet predictive metrics alone do not characterize the class-wise geometry of the representations on which such readouts operate. We introduce the directional linear separability measure (LSM), a finite-sample diagnostic for one-sided affine separability. For a target class A and a competing set B, LSM searches over affine halfspaces that contain all samples in A and measures the smallest competing-sample intrusion that must remain on the target side, normalized by |A|. The resulting quantity is asymmetric, class-wise, target-normalized, and applicable to finite representations extracted from neural networks. We establish its supporting-hyperplane characterization, relate it to optimal affine classification accuracy, and prove invariance under full-rank linear embeddings. These results separate changes caused by linear reparameterization from those caused by information loss or nonlinear geometric transformations. We also give a penalty-based affine search for estimating class-wise LSM in high-dimensional features, with reported values computed from the original discrete preservation and violation criterion. Finally, we analyze coordinatewise gated nonlinearities as finite-sample geometric operators and empirically use LSM to diagnose class-wise intrusion across common deep-learning components and architectures.

We propose a unified mathematical framework for a geometric understanding of concept learning and neuron interpretation in sparse autoencoders (SAEs). While SAEs improve interpretability of neural networks by learning sparse feature representations, a principled definition of ''concept'' and ''learning'' remains unclear. We formalize concepts as sets of data points and cast concept learning as a set-alignment problem between human-defined and model-induced concepts. This formulation distinguishes three increasingly strong notions of learning -- detection, separation, and approximation -- and yields geometric conditions, error bounds, and capacity constraints for when concepts can be represented by individual neurons or multi-neuron units. It also provides a set-theoretic account for common SAE phenomena, including feature splitting, feature absorption, feature families, and hierarchical concepts. Finally, we connect concept learning and neuron interpretation through formal concept analysis, showing that the two directions need not agree and that their many-to-many structure can be organized by concept lattices. Experiments on synthetic data with ReLU and Top-$K$ SAEs illustrate the theory and reveal the effects of SAE size and sparsity on concept learning.

A central goal of cognitive neuroscience is to characterize the features that are represented by human language cortex. Artificial language models (LMs) have emerged as a powerful tool to address this challenge, but studies relating biological and artificial representations are often criticized as relating one black box to another. The present work introduces Augmented Sparse Encoding Models, an encoding framework that replaces dense LM hidden states with hierarchically-organized sparse autoencoder (SAE) features, while explicitly including surprisal as a predictor. Using this approach, we (i) produce interpretations of neural responses and (ii) test whether model-brain alignment reflects primary or idiosyncratic variation in LM representations. Using a high-field 7T fMRI dataset of eight participants listening to 200 linguistically diverse sentences, we first validate our modeling framework by recovering previous interpretations of voxel populations tuned to processing difficulty and meaning abstractness. We then interpret a previously-uncharacterized (but reliable) voxel population and find that it is tuned to people-related content. Next, we show that the fronto-temporal human language network is predicted by a common set of features across its constituent regions, but find that frontal regions are relatively well-explained by surprisal alone, even in the absence of LM-based features. Finally, we show that brain responses during language processing are not merely predictable from an arbitrary set of LM features. Rather, brain responses are best explained by the features that tend to capture the most general information encoded in LM representations, suggesting a nontrivial correspondence between brain and LM language representation.

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

Catastrophic forgetting is commonly interpreted as the irreversible erasure of previously acquired knowledge during sequential learning. In this work, we investigate an alternative perspective: that forgetting may arise not from complete destruction of task representations but from a loss of accessibility to preserved information. We introduce a three-level framework separating knowledge storage, representation, and accessibility, and evaluate each component through a series of continual-learning experiments on sequential CIFAR-100 classification using ResNet-18. Our analysis combines checkpoint persistence, linear probing, representation geometry, classifier-reset recovery, and layer-wise recoverability experiments. We observe complete behavioral forgetting of earlier tasks, with task accuracy collapsing from 54.8% to 0%, while linear probe performance retains approximately 76% of the original representational information. Furthermore, retraining only the final classifier restores 75.7% of the original task performance without modifying the backbone network. Layer-wise analysis reveals that early and intermediate layers preserve highly recoverable task information despite severe degradation at later stages. Projection-energy and principal-angle analyses indicate that retained knowledge persists as distributed high-dimensional representations rather than through preservation of a small dominant subspace. These findings suggest that catastrophic forgetting is better characterized as an accessibility failure than complete representational erasure, and that substantial task-relevant information remains embedded within neural representations even after functional forgetting has occurred.

The ability to train spiking neural networks is essential for modeling biological neural networks as well as for neuromorphic computing. However, for the extensively used leaky integrate-and-fire (LIF) neurons, arbitrarily small parameter changes can induce spike (dis)appearances that disrupt subsequent activity, leading to unstable neural representations and permanently silent neurons during exact spike-based gradient descent. Recent work shows that a class of neuron models, which includes the quadratic integrate-and-fire (QIF) neuron, avoids these discontinuities and enables continuous and even smooth spike-based gradient descent. However, it remains unclear whether these advantages translate into practice. Here, we demonstrate that they do so via a controlled comparison between networks of LIF and QIF neurons on the popular Spiking Heidelberg Digits dataset. Specifically, in a first step, we perform a thorough hyperparameter search to optimize both models, revealing a clear performance advantage of QIF neurons. In a second step, we visualize the loss and gradient landscapes. Consistent with their inferior performance, we find that the loss landscapes of LIF neurons, which are discontinuous, appear more fragmented and the related gradients more erratic. An analysis of the landscapes of single samples indicates that these features arise from changes in the temporal order of spikes, which often cause disruptive spike (dis)appearances. Overall, our results advocate replacing LIF neurons with neuron models exhibiting continuous spiking dynamics, such as QIF neurons, for gradient descent training.

Understanding how structured internal structure emerges during neural network training is central to the study of deep learning. We investigate this phenomenon through the group composition task, where a two-layer neural network is trained to predict $g_1 \star g_2$ for elements of a finite group $G$. By lifting the projected gradient flow to the Fourier domain, we demonstrate that the training dynamics are governed by a Riemannian gradient ascent on a representation-theoretic energy functional. We prove that, under random initialization, this flow drives each neuron to converge almost surely toward a single irreducible representation, while the cross-layer Fourier coefficients achieve a rotational rank-one alignment. This framework provides a representation-theoretic account of feature learning and characterizes a novel low-rank compression phenomenon for matrix-valued group representations. Moreover, for Abelian groups, we provide a complete population-level description: random initialization promotes uniform diversification across nontrivial representations and induces Haar-uniform phases, jointly approximating the indicator via a majority-vote mechanism. We further prove that both phase alignment and representation competition emerge with exponential convergence rates.

Deep neural networks learn representations where individual features often lack interpretable meaning; a single neuron may activate for scattered, unrelated inputs. We introduce coherence, a geometric property inspired by neural coding in the brain, where neurons like grid cells and head direction cells respond to contiguous regions of state space. A non-negative matrix is coherent if each row (sample) attends to geometrically clustered columns (features) and vice versa, and in addition every sample is well described by some feature and every feature is needed by some sample. We prove that coherent matrices induce a bounded interleaving between the Vietoris-Rips filtrations of samples and features, guaranteeing that both spaces share compatible topological structure. This geometric constraint facilitates interpretability. For example, if data lies on a circle, coherent features must tile that circle into contiguous arcs. We introduce Coh, a differentiable objective function based on Fréchet variance that enforces coherence during training. Unlike sparsity, which bounds how many samples a feature activates on, coherence bounds which samples, requiring geometric connectivity rather than only rarity. This yields not just interpretable features but an interpretable feature space. We validate Coh in an auto-encoder using synthetic and rotated MNIST datasets and in a token embedding of BERT using language data.

Implicit neural representations (INRs) model a signal as a continuous coordinate-to-value function. For terrain elevation data, this supports analytic derivatives, arbitrary-resolution decoding, and a smooth surface model of the underlying heightfield. However, fitting and storing a separate INR for every tile does not scale to large terrain datasets. Amortized neural representations reduce this cost with a shared network: a new tile is mapped to a compact per-tile payload, and a shared decoder reconstructs the heightfield from it. Most such methods are hypernetworks that predict the payload in a single forward pass, while others recover it through a short per-tile optimization. These methods were developed primarily for natural images, and their suitability for terrain heightfields remains unclear. We introduce a controlled benchmark on a 1 m/pixel terrain dataset and evaluate three representative methods under a unified protocol. Observing a clear cross-domain gap, we propose HUVR+SIREN, a hypernetwork that adapts the strongest benchmarked method (HUVR) by replacing its coordinate decoder with a smooth, analytically differentiable one. It attains the best height and derivative fidelity on the benchmark with no additional per-tile storage and lower decode cost, and tolerates aggressive post-training quantization with negligible quality loss, giving a compact terrain neural format. Ablations and diagnostics further identify which design choices transfer to terrain and show that the per-tile bottleneck is already near its useful limit, leaving the remaining gap in the shared hypernetwork's architectural design.

Random, untrained neural networks consistently match or exceed trained networks in representational similarity to early visual cortex. This puzzling finding challenges the assumption that learning improves brain alignment. We investigate it by tracking representational similarity analysis (RSA) alignment to human fMRI data across training for four learning rules: backpropagation (BP), feedback alignment (FA), predictive coding (PC), and spike-timing-dependent plasticity (STDP). Using 720 object images from the THINGS database and fMRI data from three subjects across six visual ROIs, we measure Spearman correlations between model and brain representational dissimilarity matrices at eight training checkpoints (epochs 0-40). We find that (1) a single epoch of training reduces V1 alignment by 25-90%, depending on the learning rule; (2) backpropagation reduces V1 alignment most severely (delta r = -0.080), while predictive coding and STDP preserve substantially more (delta r ~ -0.04); and (3) a weaker, opposite tendency appears in object-selective cortex (LOC), where BP shows the largest increase in alignment during training, although the absolute change is small. These results suggest that untrained architectures capture low-level visual statistics through inductive biases alone, and that global error signals (BP) reshape early representations more aggressively than local learning rules (PC, STDP), which better preserve brain-like structure.

Large language models (LLMs) have emerged as powerful representation learners whose internal features increasingly align with human cognition. We study whether modern LLMs can serve as a lens for understanding neural representations in the human brain, focusing on emotional valence in EEG. We first build a one-dimensional valence direction, the V-axis, from modern LLMs using only nine emotion-evocative sentences. We validate it through zero-shot transfer to sentiment benchmarks and cross-model consistency across fourteen LLMs. We then show that this LLM-derived direction maps onto human neural activity. On a public EEG cohort of 123 subjects watching affective videos, a single linear projection on EEG features tracks the V-axis position of each stimulus. Moreover, 36 EEG emotion classifiers trained without exposure to the V-axis spontaneously rediscover the same direction in their internal representations, suggesting that the same valence structure emerges in both language models and human electrophysiology. Yet this convergence does not provide an effective training signal. We test twenty-five alignment strategies, including knowledge distillation, representational similarity, contrastive, and topographic losses; none improve decoding, and sixteen significantly reduce accuracy. We formalize this result as the saturation regularity: once task labels alone drive a brain-decoding network onto the target direction, additional supervision mainly distorts an already-saturated basin, while the load-bearing within-class residual receives little useful gradient. This regularity also indicates where improvement should come from: the residual subspace unreachable by supervision. Motivated by this insight, we ensemble across residual diversity rather than supervising the basin, improving balanced accuracy by 10.5% over the prior best on FACED, with the same effect replicated on SEED-V.

Visual decoding from brain signals is a key challenge at the intersection of computer vision and neuroscience, requiring methods that bridge neural representations and computational models of vision. We introduce a tri-modal contrastive framework for EEG-based visual decoding that aligns EEG, visual, and textual representations within a unified latent space. Our approach follows a two-stage design. First, we pre-train an EEG encoder via masked reconstruction on unlabeled trials, learning spatio-temporal regularities that transfer robustly to downstream tasks. Second, we jointly align EEG, image, and LLM-generated textual descriptions through contrastive learning, where text supervision acts as a semantic regularizer that injects linguistic structure into the shared space without overwhelming the primary EEG-image signal. The encoder integrates subject-specific adaptation, graph-attention over channels, and temporal-spatial convolutional embeddings. On the Things-EEG2 200-way zero-shot benchmark, our framework achieves 54.1% Top-1 and 83.4% Top-5 accuracy, substantially exceeding the strongest prior baseline (32.4% / 64.0%), with paired Wilcoxon tests confirming significance (p < 0.01) over all in-subject baselines. We validate generalization on Things-MEG. Analysis reveals that compact embedding geometries (CN-CLIP) outperform much larger backbones, and that decoding aligns with established neurophysiology of visual processing. This work is a critical step towards robust, semantically-grounded visual decoding from non-invasive temporal neural signals. The source code is publicly available in https://github.com/anon-eeg/eeg_image_decoding.

What can representational similarity matrices (RSMs) tell us about a neural code? As the popularity of these summary statistics grows, so too does the need for a more complete characterization of their properties. Here, we show that symmetries in network inputs can confound RSM-based analyses. Stimulus symmetries render many representations functionally equivalent, but these different configurations can lead to different RSMs. These different RSMs reflect qualitatively different representational geometries. We show that stochastic gradient descent or energetic regularization can generate sparse, drifting codes, leading in turn to drifting RSMs. Moreover, we demonstrate that these phenomena are present in networks trained to encode image data, where the symmetry is latent. Our results illustrate the challenges inherent in comparing nonlinear neural codes, when functionally-equivalent representations are not related by a simple rotation.

We develop a representability-aware and interpolable neural network (NN) framework for predicting two-particle reduced density matrices (2-RDMs). The NN incorporates a subset of representability conditions through its architecture and loss function, and can operate on different momentum meshes, enabling evaluating the representability conditions across multiple meshes, which we call interpolated representability condition. The framework can be used either to predict 2-RDMs on large momentum meshes by interpolating exact results from small meshes, or as a variational 2-RDM ansatz optimized by energy minimization on arbitrary meshes. We apply this approach to the fractional Chern insulator in the one-band projected model of twisted bilayer MoTe$_2$ at twist angle $3.89^\circ$ and hole filling $2/3$. Trained on exact-diagonalization (ED) 2-RDMs from meshes with $12$ or $18$ momentum points using six different NN architectures, the best NN is the residual multilayer perceptron, which predicts the $6\times6$ 2-RDM with $97.07\%-98.18\%$ accuracy relative to the ED 2-RDM but predicts an energy $77.353$ meV above ED ground-state energy. We then variationally optimize the NN on several meshes including $6\times6$, predicting a $6\times 6$ energy of just $0.104$ meV below ED while maintaining $98.94\%-98.96\%$ accuracy. Compared with the conventional boundary-point semidefinite programming, which gives an energy $5.560$ meV below ED with $96.40\%-98.94\%$ accuracy, the NN achieves a more accurate energy and similar accuracy while using only less than 1/20 as many parameters. Eventually, we add a symmetric mesh of $48$ momentum points to the variational optimization of the NN, and provide a prediction of the many-body ground-state energy and the many-body quantum metric on that mesh.

Biological visual systems learn from limited experience, unlike deep learning models that rely on millions of training images. What learning principles make this possible? We tested whether efficient coding, the idea that neural representations capture the statistical structure of natural inputs, can build a hierarchy of human-aligned visual features from limited data. We developed an unsupervised learning procedure in which each layer of a deep network compresses its inputs onto the dominant modes of variation in natural images, using only local statistics and no labels, tasks, or backpropagation. This unsupervised procedure yields features that progress from edges and colors to textures and shapes. The features of this deep efficient coding model are readily recognized by human observers and are predictive of image-evoked fMRI responses in human visual cortex. Furthermore, a hybrid learning procedure that combines efficient coding with supervised fine-tuning yields better brain alignment in low-data settings and more rapid category learning. These findings suggest that efficient coding may shape representations across the entire visual hierarchy and help explain the data efficiency of biological vision.

Leveraging the universal representations of pre-trained LLMs and MLLMs offers a promising path toward brain foundation models. However, visually-evoked EEG datasets remain scarce, leading existing methods to align neural signals mainly with abstract text, a lossy translation that may discard fine-grained perceptual information encoded in brain activity. We propose Generative Visual Grounding (GVG), a framework that visualizes the invisible by using an EEG-to-image generative model as a visual translator. Instead of forcing EEG into text alone, GVG hallucinates instance-specific proxy images for non-visual EEG, providing structured visual contexts that allow MLLMs to exploit their visual priors for clinical-state interpretation. We validate this idea on two MLLM backbones, GVG-X-Omni and GVG-Janus. Image-only alignment is already competitive: the lightweight GVG-X-Omni matches 1.7B-parameter text-aligned baselines while tuning only 170M parameters on a frozen 7B backbone. We further extend GVG-Janus with trimodal Image+Text alignment, where text supplies categorical semantic anchors and visual proxies enrich neural representations with perceptual details. Experiments show consistent gains in EEG understanding and visual generation, suggesting visual proxy grounding as an effective complement to textual alignment.

A widespread assumption in local feature research holds that classical handcrafted descriptors are accuracy-limited relics best replaced by learned alternatives. We show this is wrong. Through an 8-configuration ablation spanning four benchmarks (HPatches, ROxford5K, IMC Phototourism, MegaDepth), we demonstrate that classical SIFT with DSP multi-scale pooling outperforms neural descriptor and orientation replacements (HardNet, OriNet) on every accuracy metric--while running 2--18$\times$ faster--and that learned matchers (LightGlue) complement rather than supersede classical features. The conclusion reframes a decade of work: not "replace SIFT" but "compose with SIFT," classical extraction paired with learned matching only where geometric context demands it. This finding was invisible because no prior GPU SIFT kept the complete pipeline in VRAM or offered modularity for controlled classical-vs-learned ablations. We present PySIFT, the first fully GPU-resident SIFT, implemented in CuPy/Numba CUDA kernels with DLPack zero-copy handoff to downstream DL frameworks--submillisecond O(1) metadata swap regardless of keypoint count. On a laptop-grade NVIDIA RTX 3050 (4 GB VRAM), PySIFT achieves: (i) higher Mean Matching Accuracy (MMA) than OpenCV SIFT on HPatches, (ii) 383 ms faster per pair on high-resolution MegaDepth, (iii) higher geometric accuracy on cross-dataset benchmarks (+5.6 pp AUC@10${}^\circ$ on MegaDepth, more inliers on IMC Phototourism), and (iv) bitwise deterministic output--identical keypoints and descriptors across runs, with detection reproducing identically even across GPU architectures: a guarantee that learned extractors cannot match without significant performance sacrifice, and cannot achieve at all across GPU architectures due to cuDNN's architecture-dependent algorithm selection. PySIFT is open-source, requiring no C++ compilation.

Diffusion geometry is a manifold learning framework that uses random walks defined by Markov transition matrices to characterize the geometry of a dataset at multiple scales. We use diffusion geometry for neural representations, incorporating tools from multi-view learning into this field for the first time. Our key technical observation is that a broad class of similarity measures based on representational similarity matrices (RSMs) admits a closed-form equivalent formulation in terms of row-stochastic Markov matrices, opening the door to manipulations from diffusion geometry. As a first application, we develop multi-scale variants of Centered Kernel Alignment and Distance Correlation, which utilise the $t^{th}$ power of the underlying transition matrix to probe the data geometry at adjustable diffusion scales. Going further, we introduce variants of these measures which fuse the Markov matrices of several layers via alternating diffusion into a single operator that captures the network's joint sample geometry, allowing similarity to be computed across multiple layers and shifting the comparison from layer-to-layer to network-to-network. We perform extensive numerical experiments, evaluating our measures on the Representational Similarity (ReSi) benchmark comprising 14 architectures trained on 7 datasets across three different domains. Our methods achieve SoTA results in accuracy and output correlation for both language and vision tasks across different models. We furthermore show SoTA performance on an additional benchmark evaluating on out-of-distribution data.

A central goal in understanding human vision is to uncover the visual features that drive neuronal activity. A growing body of work has used artificial neural networks as encoding models to predict cortical responses to natural images, revealing the visual content that activates category-selective regions. However, existing approaches are largely correlational and treat the encoder as a black box, leaving open which image features drive each voxel's response. We introduce Mechanistically Interpretable Neural Encoding (MINE), a framework that opens this black box by applying mechanistic-interpretability tools to localize the features within natural images that drive millimeter-scale (voxel-level) activity. MINE predicts each voxel's response using language-aligned image representations, and produces semantically interpretable descriptions of the features critical for the voxel's activation. We further generalize these per-image features into per-voxel functional profiles. To validate the per-image descriptions, we show they are sufficient to generate images that elicit voxel responses matching the responses to the original images, more accurately than images generated from random or low-attribution controls. Moreover, counterfactually inserting or removing the predicted features from images shifts activation in the expected direction, providing causal evidence. Counterfactual editing guided by the per-voxel activation profiles produces even stronger activation shifts, indicating that the profiles faithfully capture each voxel's selectivity. Finally, we apply MINE to well-studied category-selective brain regions, showing it recovers their known categorical preferences while revealing fine-grained unique voxel structure within each region. Overall, our results establish mechanistic interpretability as a path to discover and causally validate fine-grained hypotheses about neural function.

Data augmentation is widely recognized for improving generalization in deep networks, yet its impact on the geometry of learned representations remains poorly understood. In this work, we characterize how different data augmentation strategies reshape internal representations in neural networks. Using tools from shape analysis, we embed network hidden representations into a metric space where distance is invariant to scaling, translation, rotation and reflection. We show that increasing augmentation strength leads to well-behaved trajectories in this space, and that different augmentation types steer representations in distinct directions. Moreover, we investigate how neural representation shapes are distorted along data augmentation trajectories, and show that insights from neural geometry can predict which representations provide the most improvement when ensembling models. Our results reveal shared geometric patterns across architectures and seeds, and suggest that analyzing shape-space trajectories offers a principled tool for understanding and comparing data augmentation methods.

Deep neural networks trained with different architectures, objectives, and datasets have been reported to converge on similar visual representations. However, what remains unknown is which visual properties models actually converge on and which factors may underlie this convergence. To address this, we decompose the object similarity structure of 162 diverse vision models into a small set of non-negative dimensions. To determine universal versus model-specific dimensions, we then estimate how often each dimension reappears across models. In contrast to model-specific dimensions, universal dimensions are more interpretable and more strongly driven by conceptual image properties, indicating the relevance of interpretability and semantic content as implicit factors driving universality across models. Differences in architecture, objective function, training data, model size, and model performance do not explain the emergence of universal dimensions. However, models with more universal dimensions also better predict macaque IT activity and human similarity judgments, suggesting that universality reflects representations relevant to biological vision. These findings have important implications for understanding the emergent representations underlying deep neural network models and their alignment with biological vision.

Neural representations are not unique objects. Even when two systems realize the same downstream computation, their hidden coordinates may differ by reparameterization. A probe family intended to reveal structure already present in a representation should therefore be stable under the relevant representation symmetries rather than be tied to a particular basis. We study this group action in the tractable exact setting of the final readout layer, where equivalent realizations induce affine changes of hidden coordinates. The resulting symmetry principle singles out a unique hierarchy of shallow coordinate-stable probes, with linear probes as its degree-1 member. We also show that a natural object for cross-model probe transfer is a shared probe-visible quotient--the representation modulo directions invisible to the probe family--rather than the full hidden state. Experiments on synthetic and real-world tasks support both predictions, showing where degree-2 probes help beyond linear ones and how quotient-based transfer enables coverage-aware monitor portability across model families. These results point toward a broader geometric representation theory of neural probing, with coverage-aware monitor transfer as a concrete operational consequence.

Neural Sheaf Diffusion (NSD) generalizes diffusion-based Graph Neural Networks by replacing scalar graph Laplacians with sheaf Laplacians whose learned restriction maps define a task-adapted geometry. While the diffusion limit of NSD is known to be the space of global sections, the representation-theoretic structure of this harmonic space remains largely implicit. We develop a quiver-theoretic interpretation of NSD by identifying cellular sheaves on graphs with representations of the associated incidence quiver. Under this correspondence, learned sheaf geometries become points in a finite-dimensional representation space. We show that direct-sum decompositions of the underlying incidence-quiver representation induce decompositions of the harmonic space reached in the diffusion limit. This gives an algebraic interpretation of oversmoothing as representation degeneration: learned sheaves may collapse toward low-complexity summands whose global sections fail to preserve discriminative information. Building on this viewpoint, we connect sheaf diffusion to stability and moment-map principles from Geometric Invariant Theory. We introduce moment-map-inspired regularizers that bias restriction maps toward balanced representation geometries, and identify a structural obstruction in equal-stalk architectures: when $d_v = d_e$, admissibility for learnable stability parameters forces the trivial all-object summand onto a stability wall. Non-uniform stalk dimensions remove this obstruction, making adaptive stability meaningful. Experiments on heterophilic benchmarks are consistent with this mechanism: breaking stalk symmetry can reduce variance or improve validation behavior, and adaptive stability becomes more effective in selected rectangular settings. Overall, our framework reframes oversmoothing as a degeneration phenomenon in the representation geometry underlying learned sheaf diffusion.

Understanding why independently trained neural networks from different modalities converge toward shared representations, and where this convergence leads, remains an open question in representation learning. All existing evidence relies on symmetric similarity measures, which can detect convergence but are structurally blind to its direction. We introduce directional convergence analysis using cycle-kNN, an asymmetric alignment measure, applied across dozens of independently trained unimodal models spanning point clouds, vision, and language. We uncover a consistent directional asymmetry: non-language modalities move toward the neighborhood structure of language significantly more than the reverse, and this pattern holds across all model families and scales--yet is entirely invisible to symmetric measures. Mechanistic analysis traces the directionality to feature density asymmetry, whereby language representations occupy the most compact regions of representational space. The Information Bottleneck framework provides a principled interpretation: optimization under compression drives representations toward discrete, compositional structures characteristic of language. We formalize this as the Wittgensteinian Representation Hypothesis: the semantic structure of language is the asymptotic attractor of multimodal representation convergence.

Scientific foundation models are expected to reuse representations under changes in dataset, acquisition protocol, and deployment domain, yet many sequence backbones treat scientific temporal structure as an unconstrained pattern to be fitted. We argue that this misses a central property of natural dynamical systems: neural and atmospheric time series are organized by interacting processes across multiple physical timescales, and failure to preserve this multiscale structure contributes to brittleness under distribution shift. We formalize this failure mode as temporal kernel mismatch, where a model fits in-distribution dynamics with an effective memory policy that is not anchored to the signal's physical timescales, leading to representation drift and degraded transfer. We propose Physics-Informed Multi-Scale Mamba (PIMSM), a state-space architecture that maps spectrum-estimated transition points between frequency regimes (knee frequencies) to scale-specific discretization parameters and anchors them to acquisition time units. On Human Connectome Project fMRI, PIMSM improves robustness and representation stability under severe temporal-context truncation, extreme low-resource transfer, and resting-state-to-task-state generalization. Without modality-specific adaptation, the same architecture also attains the lowest variable-wise MAE across all reported horizons and variables on Weather-5K held-out-station spatial out-of-distribution forecasting. These results support temporal-scale alignment as a practical inductive bias for scientific foundation models that must preserve structure, not only fit correlations, under deployment shift.

Multimodal self-supervised learning (MSSL) has emerged as a key paradigm for pretraining geospatial foundation models. However, existing geospatial MSSL methods are mainly designed for static pairs of modalities, such as satellite imagery, street-view imagery, and text, where learning is driven by aligning observations from the same or nearby locations. This assumption breaks down for human mobility trajectories, which represent continuous movement along paths rather than discrete observations at individual locations. Although trajectories are important for urban understanding through their ability to capture human activity across roads, neighborhoods, and places over time, they remain largely underexplored in current geospatial MSSL frameworks. We present TrajGANR, a novel trajectory-centric geospatial MSSL framework that aligns continuous movement patterns with static, location-based observations. TrajGANR learns a continuous neural representation of trajectories at arbitrary points along each path, which enables fine-grained alignment with nearby street-view images, even when they are not co-located with any trajectory waypoints. We leverage this capability to introduce an MSSL objective that jointly aligns three modalities: trajectories, street-view images, and their geographic locations. We evaluate TrajGANR on four urban mobility and road understanding tasks. Across these tasks, TrajGANR consistently outperforms existing geospatial MSSL frameworks and a trajectory-specific foundation model. Ablation studies further demonstrate that our proposed MSSL objective and the multimodal learning framework are the primary drivers of these improvements, highlighting the importance of fine-grained geospatial alignment over coarser aggregation, as well as geospatial multimodal learning.

We study the grokking phenomenon through the lens of topology. Using persistent homology on point clouds derived from the embedding matrices of a range of models trained on modular arithmetic with varying primes, we identify a clear and consistent topological signature of grokking: a sharp increase in both the maximum and total persistence of first homology ($H_1$). Persistence diagrams reveal the emergence of a dominant long-lived topological feature together with increasingly structured secondary features, reflecting the underlying cyclic structure of the task. Compared to existing spectral and geometric diagnostics -- specifically, Fourier analysis and local intrinsic dimension -- persistent homology provides a unified geometric and topological characterization of representation learning, capturing both local and global multi-scale structure. Ablations across data regimes and control settings show that these topological transitions are tied to generalization rather than memorization. Our results suggest that persistent homology offers a principled and interpretable framework for analyzing how neural networks internalize latent structure during training.

Artificial neural networks trained on visual tasks develop internal representations resembling those of the primate visual system, a discovery that has guided a decade of computational neuroscience. Research on building brain-aligned models has progressively embraced finer-grained learning ob- jectives, from object classification to contrastive self-supervised objectives that maximize distinc- tions among individual images. Yet the effect of learning-signal granularity on brain alignment remains largely unexamined. Here we systematically investigate how the granularity of a learning signal shapes representational alignment with human vision. We parametrically vary the number of training classes using a data-driven approach that partitions a set of training images into differ- ent numbers of categories via PCA-based splits of pretrained embeddings. We train hundreds of neural networks across convolutional and transformer architectures on these coarse classification tasks and compare their representations with human fMRI responses, macaque electrophysiology recordings, and human behavior. We find that networks trained to distinguish as few as eight broad categories learn representations that match or exceed the neural alignment of models distinguishing 1,000 classes. Even more strikingly, these coarsely trained networks align more closely with hu- man perceptual similarity judgments than all other models evaluated, including networks trained with fine-grained supervision or self-supervision as well as leading large-scale vision models. These results demonstrate that human-like visual representations can emerge from surprisingly simple learning objectives, reframing what learning signals vision may require and opening a path toward building AI systems that are more aligned with human perception.

Neural representations carry rich geometric structure; but does that structure causally shape behavior? To address this question, we intervene along paths through activation space defined by different geometries, and measure the behavioral trajectories they induce. In particular, we test whether interventions that respect the geometry of activation space will yield behaviors close to those the model exhibits naturally. Concretely, we first fit an activation manifold $M_h$ to representations and a behavior manifold $M_y$ to output probability distributions. We then test the link $M_h \leftrightarrow M_y$ via interventions: we find that steering along $M_h$, which we term manifold steering, yields behavioral trajectories that follow $M_y$, while linear steering -- which assumes a Euclidean geometry -- cuts through off-manifold regions and hence produces unnatural outputs. Moreover, optimizing interventions in activation space to produce paths along $M_y$ recovers activation trajectories that trace the curvature of $M_h$. We demonstrate this bidirectional relationship between the geometry of representation and behavior across tasks and modalities. In language models, we use reasoning tasks with cyclic and sequential geometries as well as in-context learning tasks with more complex graph geometries. In a video world model, we use a task with geometry corresponding to physical dynamics. Overall, our work shows that geometry in neural representation is not merely incidental, but is in fact the proper object for enabling principled control via intervention on internals. This recasts the core problem of steering from finding the right direction to finding the right geometry.

Activation-alignment measures such as Representational Similarity Analysis (RSA), Canonical Correlation Analysis (CCA), and Centered Kernel Alignment (CKA) are widely used to compare biological and artificial neural representations. Recent theoretical work interprets many of these methods as assessing agreement between optimal linear readouts over broad families of global tasks. However, agreement at the level of global readouts does not determine how a system uses local stimulus evidence. Specifically, representations may align in activation space yet differ in their sensitivity to small perturbations. To address this challenge, we introduce a complementary framework based on local decodable information, which focuses on a representation's ability, under noise, to discriminate small perturbations within a specified stimulus-coordinate subspace. Building on Fisher information and local representation geometry, we summarize each representation using the expected projected pullback/Fisher metric over that subspace. This formulation induces a second-moment family of local discrimination tasks, for which the resulting operator provides a minimal, complete dataset-level summary of expected discriminability. We compare these regularized signatures using a log-spectral distance on the manifold of symmetric positive definite (SPD) matrices, yielding the Spectral Riemannian Alignment Score (S-RAS) and a uniform multiplicative certificate over the corresponding family of lifted task values. Empirically, this framework enables the recovery of corresponding layers across independently trained artificial neural networks, supports transferable class-conditional probes, reveals controlled dissociations between standard and robust training, and uncovers stimulus-coordinate family effects across mouse visual cortex using the Allen Brain Observatory static gratings dataset.

Understanding how neural models represent human-interpretable concepts is challenging. Prior work has explored linear concept subspaces from diverse perspectives, such as probing and concept erasure. We introduce a unified framework to study these subspaces along two axes: \textit{containment}, which tests if a concept is fully represented in a subspace but not outside it, and \textit{disentanglement}, which tests for isolation from other concepts. In experiments on both text and speech models, we first highlight that concept subspaces may not be uniquely determined, and discuss the implications for concept subspace analysis. Then, we compare properties of concept subspaces estimated using five estimators, proposed in different communities. We find that (1) the choice of estimator impacts the containment and disentanglement properties; (2) the state-of-the-art concept erasure method, LEACE, performs well on both testing axes, but still struggles to generalize to unseen data; and (3) in HuBERT speech representations, phone information is both contained and disentangled from speaker information, while speaker information is hard to contain in a compact subspace, despite being disentangled from phones.

Neural representations (NRs), such as neural fields and 3D Gaussians, effectively model volumetric data in computed tomography (CT) but suffer from severe artifacts under sparse-view settings. To address this, we propose DiffNR, a novel framework that enhances NR optimization with diffusion priors. At its core is SliceFixer, a single-step diffusion model designed to correct artifacts in degraded slices. We integrate specialized conditioning layers into the network and develop tailored data curation strategies to support model finetuning. During reconstruction, SliceFixer periodically generates pseudo-reference volumes, providing auxiliary 3D perceptual supervision to fix underconstrained regions. Compared to prior methods that embed CT solvers into time-consuming iterative denoising, our repair-and-augment strategy avoids frequent diffusion model queries, leading to better runtime performance. Extensive experiments show that DiffNR improves PSNR by 3.99 dB on average, generalizes well across domains, and maintains efficient optimization.