Neural Collapse

Concept erasure methods that operate via linear projection assume that features occupy separable subspaces. We show this assumption fails under dense superposition: when two features are forced into an antipodal pair sharing a single subspace, state-of-the-art linear erasure destroys both, not just the target. Networks trained with gradient descent instead solve this problem non-linearly, but not uniformly: they converge to one of two distinct circuit-level solutions depending on initialization, which we call mirror and shadow solutions. We map this bifurcation as a function of feature entanglement, show it reflects a stable attractor structure rather than an artifact of our setup, and use targeted causal interventions to demonstrate that both solutions leave a substantial, measurable trace of the erased feature's representation intact, recoverable through a single scalar patch rather than requiring any further training. This mirrors a failure mode recently observed empirically in LLM unlearning, where suppression rather than deletion allows forgotten knowledge to resurface; our results offer a mechanistic, causally-validated account of why that failure mode occurs.

Recent years have witnessed growing interest in continual anomaly detection for industrial visual inspection. However, real-world manufacturing environments exhibit unpredictable shifts in data distributions, rendering task-dependent continual learning assumptions impractical. To address this limitation, we formulate industrial anomaly detection as a task-free continual learning problem and propose NC-TFAD, a neural-collapse-inspired, geometry-driven framework for learning from non-stationary data streams without task boundaries. NC-TFAD freezes a pretrained backbone and aligns streaming features to a simplex Equiangular Tight Frame (ETF) prototype space to stabilize representation geometry under non-stationary streams. To satisfy the NC-inspired geometric construction in the absence of real anomalies, we generate synthetic anomaly samples as auxiliary anchors during training. Building on this geometry, we further introduce inter- and intra-class regularization together with a Focal Neural Collapse Contrastive (FNCC) loss to suppress representation drift and improve normal-anomaly separability. Finally, a normal-patch-prototype-guided localization branch constructs calibrated patch-wise deviation maps from normal training samples and fuses them with a weak self-attention prior, producing anomaly heatmaps without pixel-level annotations. Extensive experiments on MVTec AD and VisA show that NC-TFAD consistently outperforms representative task-free continual learning methods adapted from general vision, as well as unified anomaly detection baselines, in both image-level detection and pixel-level localization under the task-free continual learning protocol. These results highlight that geometry-driven modeling offers an effective and robust solution for task-free continual anomaly detection in real-world industrial applications.

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

Neural spectral PDE solvers often learn an entire unresolved vector field even when an inexpensive approximate model can already capture most of the trajectory. Here we introduce Perturbative-NeuSA, a residual formulation that decomposes the target solution into a low-fidelity background and a high-resolution perturbation, so that only the unresolved dynamics is learned. Starting from the exact perturbation equation, the method combines a fixed spectral operator, a background-dependent correction, the background defect in the target PDE, and an optional neural closure. This construction makes the roles of physical structure and neural closure separately measurable. Across 2D Burgers, Klein-Gordon, and heterogeneous 2D wave equations, the deterministic structured solver outperforms the trained NeuSA baseline while requiring no neural-network training. The largest gains occur on Burgers, where the deterministic correction reduces training and extrapolation errors by factors of 24 and 44, respectively. In addition, a Klein-Gordon sweep over seven background resolutions shows that the effect of the closure is conditional: it improves a poor background by 3.6 times, becomes neutral at intermediate resolutions, and degrades a well-resolved background. For the wave equation, however, the closure provides an additional 18% reduction when the remaining residual is interface-localized. Multi-initial-condition diagnostics further show that the useful closure regime depends on the initial-condition spectrum and can disappear in extrapolation when structured correction already captures the dominant Burgers dynamics. Perturbative-NeuSA therefore reframes neural closure as a conditional, diagnosable correction governed by background fidelity, residual organization, and compatibility with the closure model.

With the widespread deployment of deep neural networks (DNNs) in safety-critical domains, reducing the cost of model validation under limited testing budgets has become increasingly important. Existing test case prioritization techniques often rely on single-checkpoint confidence signals derived from output probabilities. However, DNNs can be confidently wrong, and the confidence margin between the predicted and competing classes is frequently small, which weakens early fault discovery. To address this limitation, we propose a Neural-Collapse-Inspired Prioritization (NCIP) framework that replaces absolute confidence with cross-checkpoint prediction variability in the terminal training regime, where model geometry becomes highly structured. NCIP introduces two key components. First, it selects an NC-guided representative subset of training checkpoints using an equiangularity score of classifier weights, quantified as the standard deviation of pairwise cosine similarities among class weight vectors. Second, it prioritizes test inputs by their prediction variability across the selected checkpoints, surfacing boundary-adjacent and failure-prone samples that are unstable under checkpoint-induced decision boundary shifts. Extensive experiments across multiple datasets and architectures show that NCIP achieves strong performance in early fault discovery compared with competitive baselines, with 1.5 to 16.6 percent RAUC-ALL gains and 4.9 to 20.6 percent RAUC-500 gains under the same testing budget. NCIP further attains the best average performance across all dataset-model pairs.

In this work, we introduce CoCo, a loss function aimed at learning normalized and well-structured representations. The proposed loss encourages intra-class collapse and inter-class contrast while preserving sufficient flexibility for neural networks to approximate geometrically optimal embeddings with large angular separation between classes. We provide a theoretical analysis positioning CoCo with respect to related objectives such as dot regression and cross-entropy, showing that the new proposed loss benefits from closer initialization to the optimal configuration, more informative gradients, and stronger incentives for class-wise representation collapse. Extensive experiments on diverse tabular datasets from the OpenML-CC18 benchmark show that CoCo achieves competitive performance with state-of-the-art methods, including kernel SVM, Random Forest, dot regression, and cross-entropy-based neural networks. In addition, both theoretical arguments and empirical analyses demonstrate that the proposal promotes tighter class clustering and faster convergence. These results highlight CoCo loss as an effective objective for learning discriminative representations while maintaining competitive predictive performance.

Independently trained neural networks have no shared neuron-index reference frame, so comparing them requires accounting for coordinate freedom. Neural Collapse sharpens this problem: networks converge toward a shared, low-dimensional geometry, raising the question of whether trajectory-specific functional variation remains distinguishable after convergence. We distinguish three claims - detectability, transplantability, and causal persistence - and address the first. Using five independently trained networks reconstructing Neural Collapse on MNIST, we apply a verified affine-correct alignment mapping donor heads into recipient coordinates. Donor-specific functional fingerprints remain distinguishable after recipient-level baseline correction: all 20 ordered donor-recipient pairs are correctly identified, with an exact permutation p=0.0083, robust to a leakage audit. These findings establish detectability under the test used here, but not transplantability or causal persistence. The study shows how alignment, ambiguity diagnostics, and leakage control combine to test cross-network variation in a controlled setting; whether this generalizes beyond it is open.

Within-class variance in language-model representations is commonly read as incomplete neural collapse. We argue it is allocated information storage, and that the allocation obeys a law. A one-line centering identity voids a family of simplex equiangular-tight-frame claims, including our own earlier ones; in dimensionless variance shares across 14 models, macro-category structure carries only 4-12% of representational variance and within-token context carries 79-91%, stable across a 100x parameter range. On the theory side, token-level weight decay penalizes a category in proportion to its type count, not its occurrence mass, reducing next-token prediction to an imbalanced K-class problem whose optimum orders category norms by type count. A converse floor, proved for binary categories, forces within-category dispersion to be at least proportional to the conditional mutual information I(token; context | category). The law holds: identity dispersion, not total variance, tracks this information across every tested model and partition, under a model-free estimate and even across models, where one model's information predicts another's dispersion; and over pretraining the category share overshoots, decays, and partially recovers, because the information it must carry never left.

Learning robust navigation policies directly from visual observations remains a fundamental challenge in vision-based robotic navigation. In end-to-end imitation learning approaches, the visual encoder and action decoder are jointly optimized using a single action loss, which provides only an indirect supervisory signal to the encoder. This indirect supervision frequently results in the encoder learning ambiguous, action-agnostic representations. The problem is further complicated by substantial variations in scene structure and appearance across diverse environments, as well as the prevalence of visual distractors inherent to real-world navigation settings. Such action-agnostic features cause the navigation policy to produce inconsistent actions at ambiguous decision points, leading to navigation failure. To overcome these limitations, we propose ORION (Ordinal Neural Collapse for Visual Navigation), a method that explicitly organizes the encoder's representation space according to the ordinal structure of navigation actions. In the context of goal-directed navigation, ego-centric control categories from Far Left to Far Right exhibit a natural ordinal relationship in which neighboring classes share similar visual contexts, while semantically opposing classes differ substantially in appearance. We encourage class representations to be arranged sequentially along a single discriminative axis, while suppressing off-axis variance within each class. The pretrained encoder is then integrated into a diffusion-based navigation framework, and the full pipeline is fine-tuned end-to-end. Extensive experiments in both simulation and real-world settings show that ORION consistently outperforms end-to-end and neural collapse baselines in navigation success rate and goal progress, with notable gains in visually challenging scenarios such as complex multi-way intersections.

Neural Collapse predicts that balanced one-hot classification pushes model representations to be equally far from each other; a symmetric configuration that depends only on the output label and ignores any semantic similarity in the inputs. This creates a puzzle: next-token prediction language models are trained predominantly (as context length increases) with one-hot labels: the same context is very unlikely to appear twice in training with different labels. However, they clearly learn latent structural features. That is, despite the one-hot training regime, a language model's contextual embeddings represent the fact that the next word in ''Mary broke the ___'' is likely to be filled by tokens in the latent classes of a) medium-sized, b) rigid, c) inanimate nouns. How does gradient descent find such categorical semantic structure when co-occurrence statistics collapse to one-hot sparsity, eliminating any shared next-tokens among different contexts? To investigate this tension we identify three synthetic controlled settings where inputs have latent semantic factors but are mapped to distinct one-hot labels. We find that semantic geometry emerges early in training, and that representations cluster by shared attributes despite receiving no explicit supervision to do so. This structure is transient: with sufficient capacity and time, the model eventually reaches the predicted symmetric state where all representations are equally separated. We study this phase transition through Gram matrix analysis and propose a preliminary modification to the commonly used unconstrained features model to capture the emergent semantic geometry.

Large Transformer models function as Dense Associative Memories (DAMs), retrieving knowledge via high-dimensional attractor dynamics driven by the self-attention mechanism \citep{ramsauer2020hopfield, wu2024attention}. However, adapting these frozen memory systems to new tasks presents a fundamental ``Plasticity-Stability'' dilemma. Current methods either risk catastrophic interference by modifying synaptic weights directly (e.g., LoRA) \citep{hu2021lora} or degrade associative capacity by clogging the retrieval buffer with static prompt tokens (e.g., VPT) \citep{jia2022vpt}. In this work, we propose \textbf{H-Res} (Hierarchical Residual Steering), a mechanism that modulates the effective energy landscape of the Transformer without altering its global equilibrium or expanding its sequence length. By formulating adaptation as a control problem on the activation manifold \citep{chen2018neuralode}, H-Res learns a state-dependent vector field that steers token trajectories into task-specific basins of attraction. We formally prove that H-Res preserves the attention entropy of the foundation model and facilitates Neural Collapse \citep{papyan2020prevalence}. Empirically, Manifold Steering outperforms global weight modification by 26% on associative retrieval tasks and eliminates the computational overhead of prompt-based methods, scaling effectively to structured domains \citep{zha2023vtab}.

Self-organized criticality (SOC), a dynamical regime associated with maximal information processing, offers a promising foundation for reservoir computing. Recent work has shown that neural cellular automata (NCA) can be evolved toward critical avalanche dynamics and employed as effective reservoirs for memory and classification tasks. Here, we investigate whether mass conservation -- a local redistribution rule that preserves total lattice mass -- serves as an inductive bias toward SOC in evolved NCA reservoirs. We compare mass-conserving and standard NCA across multiple independent runs and evaluate both on three downstream benchmarks: 5-bit sequential memory, MNIST digit classification, and CartPole-v1 temporal control. Mass-conserving NCA consistently exhibit stronger criticality, with more runs achieving perfect power-law fits across avalanche distributions, while also being 1.27$\times$ faster during evolution. Importantly, conservation does not impair downstream utility: both variants achieve comparable performance across all three tasks. Furthermore, the reservoir with perfect criticality achieves the highest temporal control score, suggesting a positive link between SOC quality and sequential computation. Our results demonstrate that mass conservation is a simple, effective mechanism for promoting robust criticality in evolved NCA reservoirs without sacrificing downstream performance.

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.

Few-shot class-incremental learning (FSCIL) in synthetic aperture radar imagery presents unique challenges due to severe data scarcity and SAR-specific variability. In particular, strong azimuth sensitivity in SAR induces large intra-class variation and inter-class confusion, and FSCIL sequential updates further lead to catastrophic forgetting of previously learned classes. Inspired by neural collapse, we propose an optical-guided SAR FSCIL framework, which derives orthogonal feature subspaces from a data-rich optical ATR dataset and uses them as geometric priors to guide SAR feature learning. SAR features are projected onto these orthogonal subspaces via principal angle constraints, effectively transferring discriminative structure from the optical to the SAR domain. Specifically, our projection loss and the classifier loss optimized with a frozen simplex-ETF geometry jointly induce neural collapse by concentrating features around class means while maintaining large inter-class angles. We evaluate the approach on a benchmark comprising an optical ATR dataset and a SAR ATR dataset with 24 target classes, organized into a base training session and seven incremental sessions. Compared with recent FSCIL methods including NCFSCIL and so on, our method achieves the highest final accuracy and a favorable trade-off between final performance and performance degradation. Moreover, neural collapse metrics show improved intra-class compactness and inter-class separability, indicating that the learned features more closely approximate the ideal simplex-ETF geometry.

Neural collapse is a structural property of the last-hidden-layer activations in neural network classification models, when trained beyond a zero classification error. In this work, we explore the role of label encoding in neural collapse by relying on the unrestricted feature model with mean squared error training loss. We demonstrate that, for one-hot encoded labels and balanced data, the uncentered mean features associated with each class transition from a simplex equiangular tight frame to an orthogonal frame when increasing the bias regularization coefficient associated with the final classifier. These structures are reminiscent of the orthogonal frame structure of one-hot encoded labels. For any arbitrary encoding, we also show that the final classifier's bias aims at centering the labels, compensating for the discrepancy between the global mean of the labels and the origin. We further discuss the role of the encoding in other neural collapse properties.

Neural collapse (NC) describes the structured geometry that emerges in the features and weights of trained classifiers. Recent theory suggests NC can be suboptimal in deep architectures, attributing this to an explicit low-rank bias from L2 regularization. We study the deep unconstrained feature model (UFM)-equivalent to a deep linear network with orthogonal inputs-trained without regularization, to isolate how gradient descent and depth alone shape NC. We show that depth induces an implicit low-rank bias: low-rank matrices propagate norm more efficiently through successive multiplications, promoting low-rank alternatives to NC. These alternatives, we argue, correspond to softmax codes: max-margin solutions previously found in width-bottlenecked networks. Analyzing training dynamics under spectral initialization, we identify an early-time repulsion among singular values that drives low-rank emergence, and characterize how depth shrinks NC's basin of attraction. Finally, we show that some effects act in the opposite direction: for randomly initialized networks, increasing width biases training toward higher-rank solutions. Our results provide the first asymptotic and dynamic characterization of implicit bias in deep UFMs trained with unregularized multiclass cross-entropy.

Supervised classification has a theoretical optimum, Neural Collapse (NC), yet neither of its two dominant paradigms reaches it in practice. Cross entropy (CE) leaves radial degrees of freedom unconstrained and converges to a degenerate geometry, while supervised contrastive learning (SCL) drives features toward NC during pretraining but discards this structure in a post hoc linear probing phase. We show that both paradigms are different appearances of the same method that contrasts prototypes on the unit hypersphere, and that closing the gap requires fixing each at its point of failure. From the CE side, we propose NTCE and NONL, two normalized losses that import contrastive optimization's missing ingredients into classifier learning: a large effective negative set and decoupled alignment and uniformity terms. From the SCL side, we prove that SCL's objective already optimizes throughout training for a principled classifier whose weights are the class mean embeddings, making linear probing both redundant and harmful. Empirically, on four benchmarks including ImageNet-1K, NTCE and NONL surpass CE accuracy, closely approximate NC ($\geq 95\%$), and match CE's converged NC on 4/5 metrics in under $7.5\%$ of its iterations, while SCL with fixed prototypes matches linear probing without the hours-long classifier training phase. The learned geometry yields $+5.5\%$ mean relative improvement in transfer learning, up to $+8.7\%$ under severe class imbalance, and improved robustness to corruptions on ImageNet-C. Our work recasts supervised learning as prototype learning on the hypersphere, with NC reached by design.

All 26 neural network merge strategies we tested including weight averaging, SLERP, TIES, DARE, Fisher merging, and evolutionary approaches -- fail the algebraic properties (commutativity, associativity, idempotency) required for conflict-free distributed operation. We prove that this failure is structural: normalisation-based merges cannot simultaneously satisfy all three properties. To resolve this, we present a two-layer architecture -- CRDTMergeState -- that wraps any merge strategy in a CRDT-compliant (Conflict-Free Replicated Data Type) layer. Layer 1 manages contributions via OR-Set CRDT semantics, where the merge operation is set union -- trivially commutative, associative, and idempotent. Layer 2 applies merge strategies as deterministic pure functions over a canonically-ordered contribution set, with randomness seeded from the Merkle root. We prove that this separation guarantees Strong Eventual Consistency: all replicas receiving the same contributions compute identical merged models, regardless of message ordering. Empirical validation spans three tiers: controlled 4x4 tensors (104/104 tests pass), production-scale models up to 7.24B parameters (208 strategy-level tests, 43,368 layer-level property checks at capped tensor resolution), and multi-node convergence under gossip and partition healing (100 nodes, 20 orderings), with CRDT overhead below 0.5 ms. Because the wrapper is transparent, downstream performance is identical by construction, confirmed via byte-identical output verification. The reference implementation is available as crdt-merge v0.9.4.

Out-of-distribution (OOD) detection is a critical component for ensuring the reliability of deep neural networks in safety-critical applications. In this work, we present a key empirical observation: for in-distribution (ID) samples, class-wise Mahalanobis distances exhibit a pronounced sharp minimum structure, where the distance to the nearest class is small while distances to all other classes remain large, resulting in high variance across classes. In contrast, OOD samples tend to exhibit a less pronounced sharp minimum structure, producing comparatively lower variance across classes. We further provide a theoretical analysis grounding this observation in Neural Collapse geometry: under relaxed Neural Collapse assumptions on within-class compactness and inter-class separation, ID samples are shown to structurally exhibit high class-wise distance variance, offering a theoretical basis for its use as an OOD score. Motivated by this observation and its theoretical backing, we propose MahaVar, a simple and effective post-hoc OOD detector that augments the Mahalanobis distance with a class-wise distance variance term. Following the OpenOOD v1.5 benchmark protocol, MahaVar achieves state-of-the-art performance on CIFAR-100 and ImageNet, with consistent improvements in both AUROC and FPR@95 over existing Mahalanobis-based methods across all benchmarks.

Prompt learning has emerged as an efficient alternative to fine-tuning pre-trained vision-language models (VLMs). Despite its promise, current methods still struggle to maintain tail-class discriminability when adapting to class-imbalanced datasets. In this work, we propose cluster-aware neural collapse prompt tuning (CPT), which enhances the discriminability of tail classes in prompt-tuned VLMs without sacrificing their overall generalization. First, we design a cluster-invariant space by mining semantic assignments from the pre-trained VLM and mapping them to prompt-tuned features. This computes cluster-level boundaries and restricts the constraints to local neighborhoods, which reduces interference with the global semantic structure of the pre-trained VLM. Second, we introduce neural-collapse-driven discriminability optimization with three losses: textual Equiangular Tight Frame (ETF) separation loss, class-wise convergence loss, and rotation stabilization loss. These losses work together to shape intra-cluster geometry for better inter-class separation and intra-class alignment. Extensive experiments on 11 diverse datasets demonstrate that CPT outperforms SOTA methods, with stronger performance on long-tail classes and good generalization to unseen classes.

The Nearest Class Mean (NCM) classifier is widely favored in Class-Incremental Learning (CIL) for its superior resistance to catastrophic forgetting compared to Fully Connected layers. While Neural Collapse (NC) theory supports NCM's optimality by assuming features collapse into single points, non-linear feature drift and insufficient training in CIL often prevent this ideal state. Consequently, classes manifest as complex manifolds rather than collapsed points, rendering the single-point NCM suboptimal. To address this, we propose Hierarchical-Cluster SOINN (HC-SOINN), a novel classifier that captures the topological structure of these manifolds via a ``local-to-global'' representation. Furthermore, we introduce Structure-Topology Alignment via Residuals (STAR) method, which employs a fine-grained pointwise trajectory tracking mechanism to actively deform the learned topology, allowing it to adapt precisely to complex non-linear feature drift. Theoretical analysis and Procrustes distance experiments validate our framework's resilience to manifold deformations. We integrated HC-SOINN into seven state-of-the-art methods by replacing their original classifiers, achieving consistent improvements that highlight the effectiveness and robustness of our approach. Code is available at https://github.com/yhyet/HC_SOINN.

In this paper, we provide a computable characterization of the geometry of optimal representations in Contrastive Learning (CL) when the classes are imbalanced. When classes are balanced and the representation dimension is greater than the number of classes, it is well-known that the optimal representations exhibit Neural Collapse (NC), i.e., representations from the same class collapse to their class means and the class means form an Equiangular Tight Frame (ETF). For imbalanced classes and a large, generalized family of CL losses, we prove that the optimal representations of all samples from the same class collapse to their class means and their geometry exhibits an angular symmetry structure that is determined by the relative class proportions. In general, we show that the geometry can be determined by solving a convex optimization problem. Exploiting this symmetry structure, we analytically investigate a special case where class imbalance is extreme and prove that CL exhibits a phenomenon called Minority Collapse (MC) where all samples from the minority classes (classes with small probabilities) collapse into a single vector, whenever the class imbalance exceeds a threshold, which in turn depends on the regularity properties of the CL loss used and on the number of negative samples. Numerical results are provided to illustrate these phenomena and corroborate the theoretical results. We conclude by identifying a number of open problems.

Loss reweighting is a widely used strategy for long-tailed classification, but existing reweighting strategies often rely on heuristics and rarely define a well-specified target. Inspired by Neural Collapse (NC), the ideal simplex Equiangular Tight Frame (ETF) terminal geometry suggests equal per-class average loss as a reasonable target for reweighting. Based on the ideal equal loss objective, we consider loss reweighting as an inverse problem and propose an inverse-view reweighting strategy that infers class weights dynamically to match this ideal objective. Empirically, NC metrics suggest our method can effectively reduce the loss imbalance coefficient and closer alignment with NC geometry while consistently outperforming strong long-tailed baselines on different datasets.

Alignment, the tendency of adjacent weight matrices in deep networks to develop compatible subspace orientations, underlies gradient flow, Neural Collapse, and representation similarity across architectures. Despite extensive empirical documentation, these phenomena have resisted unified theoretical treatment: existing explanations are post-hoc, each fitted to a specific observation with whatever mathematics is at hand. We reverse this direction by deriving the mathematical structure that layerwise alignment inherently demands. Using geometric invariant theory, we prove that alignment geometry has a canonical closed, polystable stratum given by a flag variety, and that subspace intersection dimension is its unique reparameterization-invariant observable, establishing that subspace metrics are not empirical conventions but mathematical necessities. This unified framework yields two dynamical consequences: ridge regularization drives subspace alignment at an exponential rate set by weight decay, whereas nonlinear activations induce a commutator obstruction to exact basis alignment, generically present in nonlinear networks and absent in linear ones. Together these give a geometric explanation of the Level-2/3 hierarchy in Neural Collapse from first principles rather than post-hoc analysis. The commutator magnitude and head subspace overlap further serve as weight-space windows into internal alignment structure, requiring no forward passes. Experiments on multilayer perceptrons, residual networks, and pretrained language models support the proposed diagnostics and delineate their scope.

Understanding how deep neural networks learn representations remains a central challenge in machine learning theory. In this work, we propose a feature-centric framework for analyzing neural network training by relating weight updates to feature evolution. We introduce a simple identity, the Feature Learning Equation, which identifies the weight Gram matrix as the key object capturing feature dynamics. This enables us to interpret gradient descent as implicitly inducing a hypothetical evolution of features, whose covariance structure - termed the Virtual Covariance - characterizes how representations evolve during training. Building on this perspective, we introduce Target Linearity, a measure quantifying the linear alignment between features and targets. By analyzing the training and layer-wise dynamics, we show that deep networks learn to sequentially transform representations toward target-linear structure. This linearization perspective provides a unified interpretation of several empirical phenomena, including Neural Collapse and linear interpolation in generative models.

This work investigates the phenomenon of Neural Collapse (NC) in multi-label classification, extending its conceptual framework from multi-class learning to general correlated and imbalanced multi-label settings. Although recent studies have identified a ''tag-wise averaging'' structure for multi-label features, this view relies on implicit assumptions of label balance and combinatorial symmetry. Consequently, it fails to account for the geometrical distortions caused by intrinsic label correlations and data imbalance, which are common in practice. We resolve the multiplicity-one imbalance conjecture raised by Li et al. (2024), showing that higher-multiplicity prototypes obey a class-frequency-weighted synthesis rule rather than uniform averaging. To address this, we propose a rigorous spectral-control framework to analyze the terminal phase of multi-label learning under general imbalanced conditions. We introduce the label covariance spectrum $κ_m$, a scalar controlling the distribution-dependent lower-bound geometry, derived from the second-order moment matrix of the label distribution. Contrary to the averaging perspective, our analysis reveals that the centered label covariance spectrum controls the stability of terminal geometry by quantifying the weakest centered inter-class contrast directions. We prove that the classical Tag-wise Averaging emerges only as a special case under perfect orthogonality. Numerical experiments on synthetic distributions validate our theoretical bounds. This work resolves the scaled-average aspect of the imbalance conjecture and establishes a unifying theoretical framework that extends Neural Collapse to complex, imbalanced multi-label settings.

Representational collapse, where embeddings become anisotropic and lose multi-scale structure, can erode downstream performance long before performance metrics react. We propose an online, topology-aware monitor for evolving neural representations that couples Modular Morse Homology Maintenance (MMHM) with a composite Collapse Index (CI). Instead of rebuilding complexes each epoch, we apply sparse edits at a fixed scale and maintain a discrete Morse matching, yielding fast, incremental updates. Across LLM fine-tuning and temporal KGE training, CI provides a low-latency early-warning signal suitable for in-training interventions. Code and experimental scripts will be released publicly

Chain-of-thought reasoning has driven striking advances in language model capability, yet every reasoning step grows the KV cache, creating a bottleneck to scaling this paradigm further. Current approaches manage these constraints on the model's behalf using hand-designed criteria. A more scalable approach would let end-to-end learning subsume this design choice entirely, following a broader pattern in deep learning. After all, if a model can learn to reason, why can't it learn to forget? We introduce Neural Garbage Collection (NGC), in which a language model learns to forget while learning to reason, trained end-to-end from outcome-based task reward alone. As the model reasons, it periodically pauses, decides which KV cache entries to evict, and continues to reason conditioned on the remaining cache. By treating tokens in a chain-of-thought and cache-eviction decisions as discrete actions sampled from the language model, we can use reinforcement learning to jointly optimize how the model reasons and how it manages its own memory: what the model evicts shapes what it remembers, what it remembers shapes its reasoning, and the correctness of that reasoning determines its reward. Crucially, the model learns this behavior entirely from a single learning signal - the outcome-based task reward - without supervised fine-tuning or proxy objectives. On Countdown, AMC, and AIME tasks, NGC maintains strong accuracy relative to the full-cache upper bound at 2-3x peak KV cache size compression and substantially outperforms eviction baselines. Our results are a first step towards a broader vision where end-to-end optimization drives both capability and efficiency in language models.