Deep Network

Deep networks trained on structural MRI for Alzheimer's disease (AD) staging often reach reasonable accuracy while attending to anatomically irrelevant regions, and multimodal models that add clinical tables frequently rely on variables that were used to assign the diagnostic label in the first place. We study both issues with a deliberately lightweight slice-based encoder (ResNet18 with a one-layer Transformer over slices) on 1,075 baseline T1-weighted scans from ADNI-1. First, we use FastSurfer segmentations as an anatomical reference: YOLOv8 models trained on segmentation-derived labels localize Alzheimer-relevant structures with mAP_50 above 0.96, and a Grad-CAM comparison shows that the image-only classifier frequently attends to the skull, orbits and background. Second, we adapt a CLIP-style image - tabular contrastive framework and organize ADNIMERGE variables along a label-leakage spectrum. Fusion with cognitive scores yields 87.3% three-way accuracy, which we treat as a leakage-driven upper bound rather than an imaging result; fusion with regional volumes yields 73.0%. We observe that the choice of contrastive target changes what the image encoder learns: on MCI vs. CN, the image-only head reaches 52.4% when the encoder is aligned to cognitive scores and 73.8% when aligned to volumes, although no tabular input is used at inference. Third, restricting the input to a per-subject crop of the medial temporal lobe raises image-only three-way accuracy from 58.7% to 65.1%. All results come from single runs on a small balanced test set, and we report confidence intervals and the protocol differences that prevent direct comparison with published numbers.

We study compute reduction in neural networks through a unified partial versus full computation view, captured by one-shot magnitude pruning in the static regime and early exit in the adaptive regime. In an asymptotic single-neuron model, we prove a concentration theorem for one-shot magnitude pruning with explicit rates. We also introduce the conditional perceptron for early exit and show that its excess generalization error decays as a power of the compute gap, with an exponent that grows to infinity as the alignment between partial and full computations tends to one. We then extend the analysis to deep networks, characterizing how pruning-induced distortions accumulate with depth and deriving a corresponding compute-accuracy tradeoff for frozen-backbone early exit under a neural network Gaussian process model. Numerical simulations corroborate the predicted scaling laws.

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.

Floorplans are compact, appearance-invariant maps ideal for indoor localization, yet existing methods rely on depth networks that are brittle in cluttered scenes. We propose GALoc, a geometry-first framework that replaces depth prediction with gravity-aligned wireframes that satisfy verticality and coplanarity by construction. Given monocular RGB, camera intrinsics, relative poses, and IMU orientation, GALoc constructs a linear constraint matrix encoding verticality and coplanarity, and finds the camera gauge minimizing its smallest singular value via global search. The rectified wireframes are projected into bird's-eye-view layouts through a closed-form, FOV-consistent transformation and matched against the floorplan via metric-free SE(2) search. We evaluate end-to-end on Structured3D, with calibrated noise on Gibson, and on real-world author-collected sequences. When sufficient wall geometry is visible, GALoc matches or outperforms depth-based baselines -- achieving 88% sequential localization success at 0.1m over 100-step sequences on Gibson vs the baseline's 68% -- while abstaining in structure-blind scenes.

The learning-rate schedule is a consequential choice in training deep networks, yet the policies in common use are heuristic, and published comparisons are hard to read, because architecture, dataset, and budget tend to vary alongside the schedule. We study BrachistoneLR, a schedule built by mapping the vertical coordinate of the brachistochrone, the curve of fastest descent under gravity, onto the range between a peak and a floor rate. Expanding the definition shows it to be cosine annealing with the half-period set to E - 1 instead of E, the configuration a standard implementation gives when its period argument is one less than the number of epochs. The rate therefore reaches its floor at the last epoch trained rather than one epoch later, and we show this difference decays as E^-2, making it a short-horizon effect. We then benchmark six schedules over 72 runs on three image classification datasets (MNIST, Fashion-MNIST, CIFAR-10) and four architecture families (fully connected, convolutional, recurrent, residual), fixing the optimizer, data pipeline, and evaluation protocol so that only the schedule varies. Schedules that fall smoothly from peak to floor beat the constant rate and calendar-based decay by margins that grow with task difficulty, reaching 2.5 points of dataset mean on CIFAR-10. Within that leading group, BrachistoneLR, cosine annealing, and warmup-cosine lie within 0.06 accuracy points and 0.17 of a mean rank, which one seed per configuration cannot separate. BrachistoneLR is best on both residual networks and has the highest CIFAR-10 mean, and it sets no milestones, decay factor, warmup length, or restart period. We conclude that the shape of a schedule matters more than its parameterization, that the choice of whether to use a smooth schedule matters more than the choice among them, and that the terminal-rate distinction is worth attention only over short horizons.

Local learning methods offer an alternative to end-to-end backpropagation, but their unstructured local objectives can produce weak feature learning in deep networks. We study whether the order of hidden-state optimization can address this limitation. We propose a boundary-first inference schedule that partitions a model into chunks, first coordinates hidden states at chunk boundaries, and then refines representations within each chunk. We instantiate this schedule in predictive coding networks (PCNs), a local-learning framework in which hidden activities and prediction errors are explicitly exposed during inference. On CIFAR-10, the resulting boundary-first predictive-coding instantiation improves accuracy over standard predictive coding by $9.77\%$ under a standard parametrization and by $5.51\%$ under a $μ$-parametrization. Diagnostic analyses further show more non-trivial early-layer updates, lower initial-to-final CKA, and more diverse layerwise gradients, consistent with stronger feature learning. These results support boundary-first, chunk-based inference as a practical design principle for predictive-coding training and motivate its study in broader local-learning systems.

Representational similarity is foundational to analyses of deep networks, yet distances between point-valued representations are not intrinsically tied to downstream function: nearby states may produce different behaviors, while distant states may behave similarly. We instead give representations volume, turning similarity into statistical distinguishability. Overlapping stochastic representations necessarily induce overlapping downstream distributions, grounding latent comparison in model function and bringing it under information-theoretic tools such as the data-processing inequality. We realize this idea in pretrained transformers through a light-touch modification to LayerNorm: at each residual-stream read, we normalize the state, add isotropic Gaussian noise, and renormalize. During distillation fine-tuning, one learned allocation parameter per residual-stream read distributes a fixed global rate budget across the processing stack. The resulting model can be viewed as transformer blocks reading the residual stream with learned finite precision under a shared global rate budget. Using the Bhattacharyya coefficient, we trace which counterfactual distinctions are preserved through MLP blocks or selectively exposed to the query, key, and value computations of individual attention heads. Experiments on ViT-S and GPT-2 small reveal the depthwise propagation of continuous visual perturbations and head-specific sensitivity to token distinctions aligned with known attention motifs. These results establish distinguishability as a functionally grounded lens on transformer computation that complements existing interpretability approaches.

Most modern optimizers form their momentum as an exponential moving average (EMA) of past gradients, forgetting every direction at one fixed rate. However, the inputs a deep network sees during training can be highly anisotropic, with a few directions queried frequently while most are seen rarely. Preconditioning methods address this anisotropy by wrapping extra processing around this buffer and leave the momentum update itself unchanged. We propose Activation-Keyed Momentum (AK-Momentum), which builds direction-awareness into the momentum update rule. The gradient of a linear layer splits into an input activation that acts as a key and an output-side error that acts as a value. Keying on that activation, AK-Momentum updates the momentum buffer by the canonical delta rule, so each direction is forgotten at a rate set by how often it appears. We prove that it is a valid momentum, that it applies the input-side curvature correction without matrix inversion, and that it clears stale directions faster than EMA under both a fixed and a drifting optimum. It is a drop-in replacement for the momentum buffer of any optimizer, its coefficient transfers across widths under $μ$P, and its extra compute stays between $22.2\%$ and $25.0\%$ of a gated-MLP block's linear cost with no persistent memory. In FineWeb-Edu pretraining, AdamW with AK-Momentum (AK-AdamW) reaches AdamW's validation loss in up to $46.39 \pm 4.32\%$ fewer steps at 67M and $22.12 \pm 0.80\%$ at 370M over three seeds, and the gain persists at 1B on a Chinchilla-optimal budget. A Muon baseline tuned under the same protocol sits above AK-AdamW at both language-model scales, and the gain holds for SGD, ResNet-18, and ViT-Tiny on CIFAR-10. Training-time diagnostics confirm the predicted mechanism, better gradient tracking and healthier input directions.

Human-scene interaction is an active research topic with several industrial applications in virtual reality, gaming, robotics, and surveillance. Despite significant progress in network architectures to improve the results or optimize models' parameters for fast inference speed, the efficient representation of contact between humans and their environments remains an open challenge. In this paper, we propose a new efficient human-contact representation for human-scene interaction. Our primary contribution is the introduction of sparse contact masks that strategically select essential contact information, significantly reducing redundant data in high-dimensional inputs. Leveraging this efficient contact representation, we propose a suite of sparse operators to replace traditional dense operators within deep network layers for faster computation. Our approach not only enhances computational speed but also filters out non-essential contact data, thereby improving the precision of human-scene interaction models. To validate the effectiveness of our method, we conduct intensive experiments across three public benchmark datasets, focusing on two critical tasks for human-scene interaction: contact prediction and scene synthesis. The experimental results show that our approach outperforms state-of-the-art models in reconstruction accuracy and achieves a computation speed-up of at least 12 times over recent baselines.

Biologically plausible learning models aim to explain how neural circuits can implement effective learning under the constraints of real neurons. Although significant progress has been made, a major remaining challenge is that existing models often allow neurons or synapses to represent mixed-sign values, both positive and negative, in violation of a basic aspect of cortical circuitry -- Dale's constraint: biological neurons are either excitatory or inhibitory, but not both, and synapses cannot change sign. In this work, we address this discrepancy by introducing a biologically motivated neural architecture in which both neural activations and learning signals are represented by non-negative activity, and synapses have fixed sign, while still supporting backpropagation-like learning. Our approach uses two complementary interacting non-negative channels to represent positive and negative contributions, inspired by evidence of on-off representations in the brain. These channels are implemented through a simple neural circuit motif, which is repeated throughout the network in both bottom-up and top-down pathways. Combined with a local Hebbian learning rule, the resulting model propagates learning signals and updates weights using only local interactions between neurons. We show theoretically that our learning scheme can exactly recover the backpropagation update despite relying solely on non-negative error signals. Empirically, beyond satisfying stronger biological constraints, the on-off architecture learns efficient representations, yielding substantial gains over comparable vanilla networks on the Tiny ImageNet benchmark. These results demonstrate that effective learning can emerge from biologically plausible mechanisms without requiring mixed-sign signals, providing a step toward more realistic models of neural computation.

Humanoid robots, with their human-like shape and multi-tasking capabilities, are well-aligned with human-dominated workplaces, like those in civil and construction engineering, where they could collaborate with human workers or autonomously perform physically demanding and hazardous tasks. Despite this promise, limited research has explored how to endow these robots with the practical capabilities needed to perform construction tasks. To this end, this study proposes a novel perception-and-action system that enables humanoid robots to learn and perform construction tasks from worker demonstrations. This system contains two deep networks: Humanoid-PoseNet, which extracts human postures and translates them into mechanically feasible poses for a humanoid robot; and Humanoid-ActionNet, which learns robot-executable actions based on these translated poses. Experimental results demonstrate that the humanoid robot reliably executed eight construction-related actions, achieving an average motion-tracking error of 82.45 mm MPJPE (Mean Per Joint Position Error). This work provides an early step toward deploying humanoid collaborators in construction.

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.

Despite the empirical advantages of deep networks over shallow ones, theoretical depth separations largely concern approximation power, while algorithmic results are mostly limited to comparisons between two- and three-layer networks. In this work, we prove the first algorithmic separation between constant-depth and logarithmic-depth networks. Specifically, we identify a class of Boolean functions with hierarchically structured Fourier spectra that logarithmic-depth networks can learn efficiently using layerwise coordinate descent by reconstructing the spectra hierarchically and adaptively. We also exhibit a subclass for which every constant-depth, polynomial-width network with sufficiently regular activations and controlled spectral norms must incur constant $L^2$ approximation error under the uniform distribution over the hypercube.

Deep neural networks encode complex representations, but deconstructing this internal knowledge remains a challenge. Given the link between learning and compression, network compression offers a promising lens to analyze this knowledge. However, standard compression heuristics often suffer from scale symmetries and architectural biases. To resolve these, we introduce Hilbert Operator for Progressive Encoding (HOPE), a mathematical framework to gradually deconstruct the representations in trained network weights. HOPE shifts network compression from the discrete domain into a Hilbert space of continuous functions. By modeling individual neurons as rank-1 Hilbert-Schmidt operators, HOPE unifies pruning and neuron merging as low-rank subspace projection. Extending this formulation, HOPE introduces macro block eviction to encompass multi-layer structures like entire residual pathways under the same unified metric. This unified approach enables unbiased architectural decisions across layers with different types and sizes. HOPE is a data-free and hyperparameter-free framework. We present proof-of-concept experiments in model compression and fine-tuning to highlight the practical potential of our theory.

Backpropagation makes training deep networks memory intensive because it must store intermediate activations. Forward-mode methods avoid this cost, but their gradient estimates become increasingly noisy as the number of trained parameters grows. We introduce Split Forward Gradient (Split-FG), which splits a network at an intermediate representation: it computes the output head gradient exactly and estimates only the trunk gradient with a Jacobian--vector product. This reduces estimator variance and requires no backward pass through the trunk, while retaining an Adam-style convergence guarantee. Our experiments reveal an important practical failure mode. On WikiText-103, naive forward-gradient training of the trunk performs worse than leaving a randomly initialized trunk frozen, likely because Adam updates every noisy, under-determined trunk coordinate too aggressively. Simply using a much smaller learning rate for the trunk reverses this result: a $16$M-parameter GPT-2-style model reaches validation perplexity $387$, compared with $668$ for the frozen-trunk control and $2{,}885$ for a matched pure forward-gradient baseline (backpropagation reaches $150$). Split-FG also produces the strongest backprop-free results on our tabular benchmarks and reaches $60.5\%$ on CIFAR-10 and $35.2\%$ on CIFAR-100 with a heavy-head design. It reduces peak memory by up to $35\%$ relative to matched backpropagation, although the performance gap widens as the forward-mode trunk grows.

We investigate how each component of the Transformer feedforward block architecture design determines how much rank survives across depth at initialization. We reinterpret skip connections and normalization, long understood as controlling magnitude, as mechanisms for preserving gradient rank across depth, since the very matrix multiplications and nonlinear activations that make the network expressive also reduce the rank. We show that skip connections trade off rank collapse against ensemble-like behavior, controlled by the relative scales of the branch and the skip: skip connections route the gradient around the residual branch, where rank is lost, rather than along the long gradient paths that encourage the layers to compose. The placement of the normalization layer controls this same tradeoff by setting the branch-to-skip ratio across depth, unifying much of the normalization placement and depth scaling literature, in particular why rank collapses for Post-Norm but plateaus for Pre-Norm. Other aspects of the architecture, like the two-matrix structure that expands and contracts the width, use additional parameters to preserve the representation or branch Jacobian rank. The second matrix decorrelates a coherent mean spike that would grow across blocks with a single matrix and uncentered activation, preventing the residual representation from collapsing. The width expansion between the two matrices keeps the branch Jacobian full rank: applying the rank-reducing activation in this expanded space leaves enough directions to span the original, at a width that follows a Marchenko--Pastur law. The initialization rank of the input--output Jacobian predicts which networks train on CIFAR-10. Taken together, we recast architecture design for deep networks as navigating an intrinsic tradeoff among rank collapse, ensemble-like behavior, and parameter count.

Deep networks trained with label noise often learn clean structure before memorizing corrupted labels. We show that this transition leaves a spectral signature in the centered scatter of per-example last-layer gradients. Its effective rank transiently expands during memorization and contracts after corrupted labels are fit. We call this phenomenon Fisher Rank Inflation. Corrupted labels increase effective rank by injecting spectral mass into low-energy or previously unused eigendirections, increasing the entropy of the gradient spectrum. We derive a first-order leave-one-out attribution formula, identify conditions under which corrupted examples contribute more strongly than clean examples, and explain why attribution signals weaken once the normalized Fisher-gradient spectrum stabilizes. We test these predictions on CIFAR-10, CIFAR-100, and CIFAR-10N using SmallCNN, ResNet18, and Vision Transformers. Across settings, Fisher effective rank exhibits a consistent inflation--collapse trajectory aligned with memorization. At peak-rank checkpoints, corrupted examples are enriched among the highest rank-contributing samples, with top-100 noisy fractions from $69.2\%$ to $96.2\%$ across five-seed synthetic-corruption experiments and $94.4\%\pm1.9\%$ on CIFAR-10N. First-order spectral attribution closely matches exact leave-one-out contributions in convolutional models and remains enriched in the Vision Transformer. Peak effective rank increases monotonically with corruption severity, from $28.88\pm1.95$ under clean training to $97.09\pm1.78$ at $60\%$ corruption. In several settings, the retrospectively identified onset of rank inflation precedes observable test degradation. These results establish Fisher Rank Inflation as a spectral signature connecting corrupted-example enrichment, corruption severity, and the transition from structure learning to memorization.

The Hessian of the training loss governs the local geometry of the loss landscape, yet despite existing explanations for its largest eigenvalues, the origin of the vast multitude of vanishingly small eigenvalues remains elusive. We argue that the bulk consists of the weakly lifted pseudo-Goldstone modes of the continuous symmetries of the network parametrization. In deep linear networks these symmetries are exact: they generate flat directions and hence exact zero modes, whose eigenvectors we construct explicitly. Introducing a ReLU nonlinearity as a perturbation, we show that it breaks these symmetries weakly and explicitly. Resolving the spectrum at the level of eigenvectors, we find that the high-curvature directions are orthogonal to the symmetry subspace, while the bulk lies almost entirely within it. We demonstrate the mechanism in a two-layer ReLU student--teacher model and in a network trained on CIFAR-10. A convolutional example demonstrates that the same diagnostic extends beyond fully connected layers. Together, these results link the Hessian bulk to weakly broken symmetries and clarify the origin of near-zero modes.

Deformable shape representations have proven to be robust complements to texture features in cardiac image classification, offering geometric priors that are invariant to imaging artifacts and intensity variations. However, existing deep networks perform simple concatenation to combine these distinct feature representations, which neither fully exploits their complementary nature nor learns cross-modal feature dependencies. Furthermore, this results in uniform attention across all timepoints; hence ignoring the varying diagnostic importance across the cardiac phases. In this paper, we propose a novel cardiac video classification model that, for the first time, learns temporal features in an integrated space of deformable shape and image texture representations. In particular, we design a bi-directional cross-attention in the latent space to fuse latent deformable shape and image features, allowing each modality to adaptively weight the other based on spatio-temporal correspondence. In contrast to current methods that apply uniform weighting across all the cardiac phases, our approach learns to dynamically adjust the contributions of shape and texture representations, derived from images, over time. We demonstrate state-of-the-art classification performance on a cine cardiac magnetic resonance (CMR) video dataset, achieving improved interpretability from attention mechanisms that identify diagnostically critical cardiac phases and modality contributions.

We develop a unified function space theory of deep fully connected neural networks. Functions in our spaces are defined recursively as $\ell^1$-bounded linear combinations of activated functions from preceding layers, with a dictionary of affine functions at the first layer. Unlike existing theories that are largely specialized to homogeneous activations such as the ReLU, our framework provides a meaningful notion of functional complexity for deep networks with a broad range of homogeneous and non-homogeneous activation functions commonly used in practice. This simple construction unites several seemingly disparate ideas from the literature, including norm-based complexity bounds and variational characterizations of depth, and facilitates novel analyses of what kinds of functions deep norm-constrained networks can represent. To this end, we prove a novel representer theorem for our spaces and establish novel function-space complexity bounds showing that the associated function classes remain qualitatively small at arbitrary depth. In the univariate ReLU case, we prove a "depth saturation" result: depth in this setting yields only a small constant rescaling of the function class, with no added functional diversity. As a consequence, we show that deep norm-controlled ReLU functions in any dimension cannot exhibit high frequencies along any direction. This finding reveals that some commonly cited expressivity benefits of depth disappear once network complexity is controlled by an appropriate function space norm, rather than parameter count or other representational costs that permit compounded rescaling across layers. Overall, our results illustrate how a function space perspective yields new structural insights into the relationship between depth and complexity.

3D Gaussian Splatting (3DGS) has emerged as a powerful representation for high-fidelity rendering. However, existing assets often suffer from quality bottlenecks such as missing details and texture noise. Prior attempts to enhance these assets via 2D image processing introduce multi-view inconsistencies and high computational costs. In this paper, we propose a novel 3D-native refinement paradigm named AnchorSplat. AnchorSplat is an end-to-end deep network operating directly on 3D structures, avoiding the expensive optimization overhead of traditional 3D-2D-3D pipelines. Crucially, AnchorSplat is a strictly source-free solution requiring no original multi-view images. Central to the proposed method is the Point Anchor Mechanism, which enforces geometric consistency via local offset constraints, mitigating ill-posed mapping and gradient confounding. Furthermore, AnchorSplat replaces iterative densification with a single-pass multiplication mechanism. To facilitate research, we construct 3DGS-SR, the first large-scale benchmark for this task. Experiments demonstrate state-of-the-art results on the 3DGS-SR dataset, with throughput up to $10^5$ times faster than optimization methods. Notably, AnchorSplat exhibits robust zero-shot generalization across diverse data distributions, including generative model outputs and real-world scans.

A deep network's loss is invariant to continuous symmetries of its parameters: the logit shift, the ReLU rescaling, the LayerNorm scale, the per-head attention rotation. Adam's per-coordinate preconditioner drifts along each symmetry orbit, which pulls the trajectory off the symmetry quotient where the optimization lives and blurs the singular-learning rate the quotient makes readable. We build DDC, a Dead-Direction Conditioner that lifts a base optimizer into a $G$-equivariant one: it conditions the optimizer's state in the orbit decomposition of a $G$-invariant metric, so the trajectory stays a preconditioned gradient flow on the quotient $\barΘ= Θ/G$. The construction carries four architectural gauges (cross-entropy shift, ReLU and SwiGLU rescaling, LayerNorm and RMSNorm scale, and a per-head $O(d_{\rm head})$ attention rotation matched to RoPE), proves exactly equivariant on an Adam base, and composes with a Muon base through a gauge-equivariant orthogonaliser. Respecting the symmetry changes both the minimum the optimizer reaches and what it leaves measurable there. On a language model trained past the point of fit, DDCAdam resists the over-training collapse AdamW falls into, holding a validation-train loss gap of 0.67 against 5.88, and reads the dead-direction rate in 32 of 65 layer-by-observable cells where AdamW reads it in 7. A vision transformer trained from scratch reaches lower validation loss (1.71 against 2.12) while compressing spare feed-forward capacity a matched AdamW leaves intact. On a Muon base, where the rotation gauge composes exactly, DDCMuon groks ten of eleven seeds at depth 24 that a plain Muon never reaches. Built into the optimizer, a network's gauge symmetry sharpens the minimum it finds and turns that minimum's geometry into something the trajectory can measure.

Deep Learning (DL) has advanced various fields by extracting complex patterns from large datasets. However, the computational demands of DL models pose environmental and resource challenges. Deep Shift Neural Networks (DSNNs) present a solution by leveraging shift operations to reduce computational complexity at inference. Compared to common DNNs, DSNNs are still less well understood and less well optimized. By leveraging AutoML techniques, we provide valuable insights into the potential of DSNNs and how to design them in a better way. We focus on image classification, a core task in computer vision, especially in low-resource environments. Since we consider complementary objectives such as accuracy and energy consumption, we combine state-of-the-art multi-fidelity (MF) hyperparameter optimization (HPO) with multi-objective optimization to find a set of Pareto optimal trade-offs on how to design DSNNs. Our approach led to significantly better configurations of DSNNs regarding loss and emissions compared to default DSNNs. This includes simultaneously increasing performance by about 20% and reducing emissions, in some cases by more than 60%. Investigating the behavior of quantized networks in terms of both emissions and accuracy, our experiments reveal surprising model-specific trade-offs, yielding the greatest energy savings. For example, in contrast to common expectations, quantizing smaller portions of the network with low precision can be optimal with respect to energy consumption while retaining or improving performance. We corroborated these findings across multiple backbone architectures, highlighting important nuances in quantization strategies and offering an automated approach to balancing energy efficiency and model performance.

Despite the success of deep learning, training deep networks in biologically plausible and hardware-efficient ways remains an open challenge. Feedback alignment (FA) methods address this by replacing backpropagation's symmetric backward weights with fixed random matrices, but their effectiveness depends critically on whether they can be accurately evaluated. The standard evaluation relies on two quantities: task accuracy and cosine similarity between the method's credit signal and the backpropagation gradient. We show that this reporting pair is insufficient by identifying two independent failure modes, both silent under current reporting: (1) measurement degeneracy, where the BP reference gradient collapses to the numerical floor in terminal-LayerNorm residual architectures, rendering cosine uninterpretable; and (2) aggregation collapse, where the aggregate cosine masks layerwise heterogeneity that concentrates credit at one end of the network. To address these limitations, we propose a diagnostic evaluation protocol based on three checks -- scale stability, reference validity, and depth utility -- together with per-layer rather than aggregate cosine reporting. Across multiple architectures and methods, the standard reporting pair gives no signal of failure in any audited case, while our protocol identifies all failures with wide calibration margins. The two failure modes are causally independent: a per-block scale penalty alleviates Mode 1 (residual scale explosion driving reference collapse) without affecting Mode 2 (cosine ranking that contradicts every functional metric we measured). Identifying these silent failures prevents researchers from building on non-functional credit assignment and provides actionable guidance for developing FA methods that genuinely train deep layers.

Compositionality is believed to be the foundation for generalization, enabling models to reuse meaningful primitives in novel combinations. Yet, models trained with standard gradient-based optimization rarely, and often only weakly, exhibit compositional internal structure, and it remains unclear how or why such compositionality forms. In this work, we show that compositionality emerges in a narrow connectivity-depth sweet spot. Along the connectivity axis, compositionality only appears in some specifically sparse networks, heavily depends on which connections remain rather than on weights' sparsity alone. Along the depth axis, compositionality emerges within a narrow, target-dependent regime, peaking at specific depths, while both shallower and deeper networks fail. When either the depth or connectivity condition is violated, gradient descent silently converges to fractured solutions rather than compositional ones. To discover and exploit this emergence, we introduce (i) similarity-based pruning (SP) to recover compositional connectivity and (ii) a heuristic depth predictor to estimate where compositionality is most likely to appear. Finally, we support these empirical findings with a theoretical framework based on compositional sparsity, volume-ratio arguments, and feature-interference bounds, explaining why compositional solutions are reachable only in a narrow depth-connectivity regime.

On biomedical tabular data, flexible models such as deep networks, gradient-boosted trees, and kernel methods are repeatedly matched or beaten by linear and logistic regression given the same features. The usual reaction is to treat this as a model-side shortfall, to be fixed with more data, a better architecture, or tuning, on the assumption that the nonlinear structure is there and the model has failed to capture it. We argue that these fixes cannot help when the binding limit is the measurement rather than the model, as it frequently is in biomedicine. Additive noise blurs the population-optimal predictor, and because blurring removes a function's fine, rapidly varying detail before its broad shape, it erases nonlinear structure faster than linear structure. A degree-$k$ interaction is attenuated by the $k$-th power of feature reliability, while the linear part is attenuated only once. At the reliabilities typical of biomedical measurement, the nonlinear advantage can vanish even when the underlying biology is strongly nonlinear, and what the noise removes cannot be recovered by a larger cohort or a more flexible model, only by better measurement. The nonlinearity is hidden, not absent, and a tie between linear and flexible models is not by itself a verdict on the biology. These pieces are classical, drawn from measurement-error statistics, psychometrics, and Gaussian analysis, and we assemble them into an exact excess-risk identity. Measurement reliability is one of three conditions, alongside sample size and feature representation, that must align for a flexible model to help, and together they leave only a narrow window that most biomedical tasks fall outside. Across 140 UK Biobank tasks, the gap between flexible and linear models, where it exists, carries the predicted noise signature, and the three conditions can be separated by intervention but not by a benchmark alone.

We extend the entropy formula of Menon and Yu for the real Deep Linear Network (DLN) to its complex and quaternionic analogues, obtaining a unified formula for DLNs over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$.

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.

In computer vision, the basic pipeline of most convolutional neural networks consists of multiple feature extraction layers, where the input signal is downsampled to a lower resolution in each subsequent layer. This downsampling process is commonly referred to as pooling, which is an essential operation in CNNs. Pooling improves robustness against transformations, reduces the number of trainable parameters, increases the receptive field, and lowers computation time. Since pooling is a lossy process but remains important for extracting high-level information from low-level representations, it is important to preserve the most prominent information from previous activations to improve network discriminability. Standard pooling is usually performed using dense pooling methods, such as max pooling or average pooling, or through strided convolutional kernels. In this paper, we propose a simple yet effective adaptive pooling method, called FlexPooling, which generalizes average pooling by learning a weighted average over activations jointly with the rest of the network. We further show that attaching Simple Auxiliary Classifiers (SAC) to the CNN improves performance and demonstrates the effectiveness of the proposed method compared with standard pooling methods. Experiments on multiple popular image classification datasets show that FlexPooling consistently outperforms baseline networks, achieving approximately 1 to 3 percent improvement in accuracy.

Individual animal recognition can be useful in the search for lost or stolen pets, the tracking of individuals of endangered species, and the recognition of animals in crowded farms. Present recognition techniques mostly use physical devices, e.g., microchips, often impractical and difficult to apply. These could be replaced by remote recognition via the animal's face; if accurate enough, it provides several advantages: it is non-invasive, can work at a distance, and is difficult to counterfeit, as, for instance, in the case of substituting sick animals for healthy ones in the food industry. The few existing datasets with sufficient per-subject images annotated with a single animal identity are not large enough to train current deep learning architectures. We rather investigate the possibility of transfer learning, exploiting pre-trained network models as backbones. Our experiments compared FaceNet, which is specifically trained on large databases of human faces, with the Vision Transformer (ViT) pre-trained on ImageNet, i.e., on object categories. We used three face datasets of very different animals: dogs, primates (lemurs, golden monkeys, and chimpanzees), and cattle. We report the results and, for each dataset, compare them with the state of the art (SOTA) ad hoc-trained deep networks. The capture conditions differ among the three datasets. Image quality (resolution, motion blur, diverse poses, etc.) decreases from dogs to cattle to primates. The best performance was achieved with dogs, where ViT reached a mean verification accuracy of 96.85% and a Rank-1 Identification Rate of 84.34%. The results for endangered primates are still encouraging, but performance varies across animal classes and tasks (verification or identification), and does not always outperform SOTA. For cattle, the ViT results outperform SOTA, while FaceNet is still competitive.

Singular learning theory and information geometry study the same spaces: the former in resolved coordinates, the latter in original coordinates under a non-degeneracy assumption that overparameterised models violate. This paper carries one direction of the bridge between them, from Watanabe's invariants to Fisher geometry, through one primitive, the dead direction: a unit vector along which the Fisher metric degenerates, equivalently a direction crossing the analytic singular set along which the KL divergence keeps a zero of high order, its KL order set by how fast that divergence vanishes. Our central result recovers the KL order as the decay rate of the directional Fisher quadratic form approaching the singularity, in original coordinates, without a Hironaka resolution. A selection rule on smooth fibres translates this rate into Watanabe's single-direction contribution to the real log canonical threshold, and the recovery extends to multi-component crossings, multiplicity $m$, the singular fluctuation $ν$, prior-RLCT shifts, and tempered posteriors. We then carry the rate into a deep network: a multi-layer K-FAC factorisation writes each Fisher block as a product of activation- and gradient-side rates with a duality between them, instantiated at residual streams, layer normalisation, and attention. A quotient theorem carries the rate to the gauge quotient for optimizers whose update commutes with the group action; Adam's per-coordinate preconditioner fails that condition, so we construct DDCAdam, an equivariant Adam-family preconditioner, and prove the quotient rate along its trajectory. The result is a trajectory-rate readout of Watanabe's triple $(λ, m, ν)$ from one checkpoint's forward and backward passes, without posterior sampling.

Grokking suggests that fitting the training data and learning a simple underlying rule may occur on different time scales. We formalize this phenomenon by separating the fast decay of the classification loss from the slower simplification of the learned representation, and we call the resulting pair of stopping times two training clocks. For deep linear networks, we show that a post-margin gap-growth or one-step tail-contraction condition reduces the cross-entropy loss to level epsilon on a logarithmic time scale. In contrast, when layerwise weight decay is present, the induced regularization on the end-to-end map can be expressed as a Schatten-type penalty; under a sharp late-time Kurdyka-Lojasiewicz tail, this structural energy closes on a polynomial time scale. The two clocks, therefore, separate fitting from representation simplification. We then explain how the same mechanism can appear in ReLU MLPs. In regions where the activation patterns on the training set remain fixed, the network reduces to a linear model in the active coordinates. In a two-layer ReLU embedding model, chain-rule estimates further show that the classifier head can receive larger effective gradients than the embedding block under controlled downstream norms. This supports a two-stage mechanism in which the classifier fits first, while the representation continues to simplify later. We use modular addition as the main experimental setting. The deep linear theory provides the rigorous core of the analysis. But the ReLU results are formulated as conditional reductions that account for empirical behavior without claiming a global proof for nonlinear training dynamics.

Deep learning proxies for Alternating Current Optimal Power Flow (ACOPF) lack systematic methods for determining architectural size. This paper conducts a constructive thought experiment to answer a fundamental inquiry: how wide must a neural network be to almost accurately approximate the ACOPF manifold? We introduce a Loss-Guided Neural Densification (LG-ND) algorithm that incrementally discovers necessary capacity by expanding only when the current deep neural network topology fails to improve further. Empirical results across various IEEE systems show that LG-ND achieves performance parity with literature baselines using up to ten times fewer neurons per layer. Such architectural minimalism is critical for the formal verification required in safety-critical grid operations.

Normalization layers such as BatchNorm and LayerNorm have long been considered essential for stable training in deep networks. This work demonstrates that they can be fully replaced by a single learnable activation mechanism. We identify a plasticity suppression effect induced by standard normalization: learnable activation parameters rapidly lose adaptability when paired with normalization layers. Motivated by this observation, we introduce SALU (Saturated Adaptive Linear Unit), $$\operatorname{SALU}(x;a,b) = \frac{a x}{\sqrt{1 + a b x^2}},\quad a>0,\; b>0$$ a bounded, learnable activation that provides intrinsic signal stabilization without relying on batch statistics or external affine parameters. Building on SALU, we propose SaluNet, a paradigm grounded in total plasticity: SALU replaces normalization layers, while SWALU and GALU replace standard activations. With ResNet-18, SaluNet-C-18 achieves 97.35% on CIFAR-10 and 83.25% on CIFAR-100 without normalization, maintaining 93.44% and 76.23% at batch size 1 where normalized architectures fail. For transformers, SaluNet-T improves over LayerNorm-GELU from 90.92% to 91.01% on CIFAR-10 and from 66.54% to 68.10% on CIFAR-100. SaluNet-C-50 reaches 78.67% Top-1 on ImageNet-1K at $224\times224$, and $79.23\%$ at $288\times288$. These results suggest normalization layers suppress total plasticity, a property biological neurons inherently possess, enabling deep networks to learn effectively.

Recent analyses of multi-pathway Deep Linear Networks use Gradient Flow to predict a "winner-takes-all" specialization in which path symmetry breaks and each feature concentrates in a single pathway. In this work, we show that discrete Gradient Descent (GD) with a large step size tells a different story. We prove that single-path solutions are sharp minima, whereas distributing signals across pathways reduces sharpness by a factor that decreases with both the number of pathways and depth. Consequently, while early training reproduces the depth-driven symmetry breaking predicted by GF, oscillations at the Edge of Stability subsequently override this tendency and drive the network into a re-balancing phase, where signals redistribute across pathways. Together, these results clarify how depth shapes pathway competition and explain why large-step GD favors shared representations rather than persistent single-pathway dominance.

Mislabeled samples in training datasets severely degrade the performance of deep networks, as overparameterized models tend to memorize erroneous labels. We address this challenge by proposing a novel approach for mislabeled data detection that leverages training dynamics. Our method is grounded in the key observation that correctly labeled samples exhibit consistent entropy decrease during training, while mislabeled samples maintain relatively high entropy throughout the training process. Building on this insight, we introduce a signed entropy integral (SEI) statistic that captures both the magnitude and temporal trend of prediction entropy across training epochs. SEI is broadly applicable to classification networks and demonstrates particular effectiveness when integrated with contrastive language-image pretraining (CLIP) architectures. Through extensive experiments on four medical imaging datasets -- a domain particularly susceptible to labeling errors due to diagnostic complexity -- spanning diverse modalities and pathologies, we demonstrate that SEI achieves state-of-the-art performance in mislabeled data identification, outperforming existing methods while maintaining computational efficiency and implementation simplicity. Our code is available at https://github.com/MedAITech/SEI.

Predictive coding (PC) is a local-learning alternative to backpropagation (BP), training deep networks via local energy-minimization dynamics rather than a global backward pass. We introduce Augmented Lagrangian Predictive Coding (PC-ALM), which maintains PC's inference budget but aligns each weight update toward BP by accumulating per-layer constraint errors into a layer-local Lagrange multiplier. In linear PC networks, PC-ALM converges to an equilibrium with exact BP gradients distributed across the network via only layer-local updates. We analyze PC-ALM in nonlinear PC networks up to depth 128 and show that it matches BP performance across all width-depth regimes, notably in deep narrow networks where PC underperforms. PC-ALM introduces recurrent dynamics in each layer's activations. Compared to PC's heat flow on a scalar energy, PC-ALM dynamics are driven by dual ascent on the augmented Lagrangian. We observe "ballistic" credit propagation across very deep networks, with credit signals evenly distributed across layers, compared to PC's slow, diffusive credit propagation. Beyond the algorithm itself, the augmented Lagrangian framework offers a generalization of PC, and may yield insights into how distributed systems could compute and propagate BP-like credit signals through purely local dynamics.

Deep networks often exhibit a preference for "simple" solutions, and such a simplicity bias is widely believed to play a key role in generalization. Yet a broadly applicable, quantitative measure of simplicity remains elusive. We introduce polynomial representations as a distribution-aware, low-dimensional surrogate for neural functions: we approximate a network's predictive behavior along data-dependent interpolation paths using orthogonal polynomial bases, yielding a compact functional representation. We show that the effective degree of this representation serves as a practical simplicity metric that is predictive of generalization across tasks and architectures, and consistently outperforms existing generalization proxies such as sharpness. Finally, polynomial representations naturally yield a differentiable simplicity regularizer, which consistently improves generalization in image and text classification, fine-tuning contrastive vision-language models, and reinforcement learning.

Understanding the relationship between generalization and privacy remains a central challenge in modern machine learning theory, particularly for deep networks trained by variants of differentially private stochastic gradient descent (DP-SGD). In this work we make progress on this persistent open problem by proving a finite-sample bound on the approximate max-information of DP-SGD that exhibits scaling properties comparable with (Dwork et al, 2015)'s classic result for $ε$-differentially private algorithms, namely at most linear in the dataset size. From our result we obtain a general-purpose PAC-Bayes generalization bound in which the necessary prior distribution can be learned by DP-SGD, as well as a generalization bound for DP-SGD-trained models themselves, with a complexity term that is fully explicit and controlled by the optimization hyperparameters.

Training instabilities in deep networks - loss spikes, oscillatory convergence, and gradient pathologies - are empirically prevalent but lack a rigorous operator-theoretic explanation. We show that the linearized update operators for practically used optimizers are generically non-normal: for Adam, non-normality is controlled by the commutator [H, M] between the Hessian and the diagonal adaptive preconditioner, while for SGD with momentum it arises from the augmented state-space structure of the update map. Applying non-normal stability theory to these operators, we derive a conservative pseudospectral precursor bound in which κ(V) serves as an early-warning indicator of transient amplification even when the spectral radius remains below one, and we establish that exceptional points of the update operator appear as the κ(V) -> \infty limiting case of this framework. Numerical experiments on two-layer networks confirm that the spectral radius ρ(J) provides no separation between stable and unstable training phases while κ(V) separates them by approximately one order of magnitude, complementing the classical sharpness criterion with a continuous severity measure of non-normal amplification. These results establish non-Hermitian operator theory as a useful and underexplored framework for neural network optimization stability, offering a diagnostic language and proof-of-concept benchmark for understanding adaptive optimization stability.

We address the fundamental question of why deep neural networks generalize by establishing a pointwise generalization theory for fully connected networks. This framework resolves long-standing barriers to characterizing the rich nonlinear feature-learning regime and builds a new statistical foundation for representation learning. For each trained model, we characterize the hypothesis via a pointwise Riemannian Dimension, derived from the eigenvalues of the learned feature representations across layers. This establishes a principled framework for deriving hypothesis-dependent, representation-aware generalization bounds. These bounds offer a systematic upgrade over approaches based on model size, products of norms, and infinite-width linearizations, yielding guarantees that are orders of magnitude tighter in both theory and experiment. Analytically, we identify the structural properties and mathematical principles that explain the tractability of deep networks. Empirically, the pointwise Riemannian Dimension exhibits substantial feature compression, decreases with increased over-parameterization, and captures the implicit bias of optimizers. Taken together, our results indicate that deep networks are mathematically tractable in practical regimes and that their generalization is sharply explained by pointwise, feature-spectrum-aware complexity.

Quantization is essential for reducing the computational cost and memory usage of deep neural networks, enabling efficient inference on low-precision hardware. Despite the growing adoption of uniform and floating-point quantization schemes, selecting optimal quantization parameters remains a key challenge, particularly for diverse data distributions encountered during training and inference. This work presents a novel statistical error analysis framework for uniform and floating-point quantization, providing theoretical insight into error behavior across quantization configurations. Building on this analysis, we propose iterative quantizers designed for arbitrary data distributions and analytic quantizers tailored for Gaussian-like weight distributions. These methods enable efficient, low-error quantization suitable for both activations and weights. We incorporate our quantizers into quantization-aware training and evaluate them across integer and floating-point formats. Experiments demonstrate improved accuracy and stability, highlighting the effectiveness of our approach for training low-precision neural networks.

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.

Activation functions are central to deep networks, influencing non-linearity, feature learning, convergence, and robustness. This paper proposes the Adaptive Arctangent Gated Activation (ArcGate) function, a flexible formulation that generates a broad spectrum of activation shapes via a three-stage non-linear transformation. Unlike conventional fixed-shape activations such as ReLU, GELU, or SiLU, ArcGate uses seven learnable parameters per layer, allowing the neural network to autonomously optimize its non-linearity to the specific requirements of the feature hierarchy and data distribution. We evaluate ArcGate using ResNet-50 and Vision Transformer (ViT-B/16) architectures on three widely used remote sensing benchmarks: PatternNet, UC Merced Land Use, and the 13-band EuroSAT MSI multispectral dataset. Experimental results show that ArcGate consistently outperforms standard baselines, achieving a peak overall accuracy of 99.67% on PatternNet. Most notably, ArcGate exhibits superior structural resilience in noisy environments, maintaining a 26.65% performance lead over ReLU under moderate Gaussian noise (standard deviation 0.1). Analysis of the learned parameters reveals a depth-dependent functional evolution, where the model increases gating strength in deeper layers to enhance signal propagation. These findings suggest that ArcGate is a robust and adaptive general node activation function for high-resolution earth observation tasks.

Predictive Coding (PC) is an influential account of cortical learning. Much of recent work has focused on comparing PC to Backpropagation (BP) to find whether PC offers any advantages. Small scale experiments show that PC enables learning that is more sample efficient and effective in many contexts, though a thorough theoretical understanding of the phenomena remains elusive. To address this, we quantify the efficiency of learning in BP and PC through a metric called ``target alignment'', which measures how closely the change in the output of the network is aligned to the output prediction error. We then derive and empirically validate analytical expressions for target alignment in Deep Linear Networks. We show that learning in PC is more efficient than BP, which is especially pronounced in deep, narrow and pre-trained networks. We also derive exact conditions for guaranteed optimal target alignment in PC and validate our findings through experiments. We study full training trajectories of linear and non-linear models, and find the predicted benefits of PC persist in practice even when some assumptions are violated. Overall, this work provides a mechanistic understanding of the higher learning efficiency observed for PC over BP in previous works, and can guide how PC should be parametrised to learn most effectively.

Activation functions are what make deep networks expressive: without them, the model collapses to a linear map. Yet we still evaluate training mostly from the outside, through loss, accuracy, return, or final calibration, while the internal structural evolution of the network remains largely unobserved. In this paper, we argue that the Overfitting--Underfitting Indicator (OUI) should be understood as a first practical observable of that internal structure. Across our recent results, OUI consistently appears as an early, label-free, activation-based signal that reveals whether a network is entering a poor or promising training regime before convergence. In supervised learning, it anticipates weight decay regimes; in reinforcement learning, it discriminates learning-rate regimes early in PPO actor--critic; and in online control, it can drive layer-wise weight decay adaptation. Read together with recent evidence that activation patterns tend to stabilize earlier than parameters, these results suggest a broader research direction: an activation-centric theory of training dynamics. OUI is becoming an empirical foothold toward this theory.

Modern object detectors are static, fixed-depth networks optimized for a single operating point, requiring separate models for different deployment scenarios. We present an any-depth detection framework that enables a single network to span a continuous range of accuracy--efficiency trade-offs by controlling depth at inference time without retraining. Each backbone and neck stage is divided into an essential path, which always executes, and a skippable refinement path; this decomposition preserves the full multi-scale feature hierarchy at every depth configuration, unlike conventional early exiting that discards entire stages. To train such a network, jointly optimizing many sub-networks of varying depth introduces conflicting gradient signals. We address this via self-distillation between only the two extremes, with prediction-level and feature-level alignment losses that enforce stage-wise modularity, ensuring the outputs of each stage remain compatible regardless of the paths taken. Instantiated on RT-DETR and YOLOv12, our full-depth configurations match or surpass their respective SOTA baselines with negligible parameter overhead, while the most efficient configurations achieve up to $1.82\times$ speedup at a cost of only 2.0 AP, all from a single set of weights.

The manifold hypothesis (MH) is often used to explain how machine learning can overcome the curse of dimensionality. However, the MH is only applicable in regimes where the training data provides a sufficiently dense sample of the underlying low-dimensional data manifold, or where such a low-dimensional manifold is conceivably present. We describe the regimes where the MH is not applicable as sparse. In this paper, we demonstrate that models succeed in the sparse regime by exploiting a highly structured local geometry, a property we formalize as normal alignment. We prove that normal-aligned classifiers -- whose input-output Jacobians are rank-one and align perfectly with the training data -- minimize the training objective under norm constraints and achieve maximal local robustness under a non-zero Jacobian constraint. For continuous piecewise-affine deep networks, normal alignment manifests geometrically as centroid alignment within the network's induced power diagram partition and results from the feature-learning regime. Motivated by these theoretical insights, we introduce GrokAlign, a regularization strategy that actively induces normal alignment. We demonstrate that GrokAlign significantly accelerates the training dynamics of deep networks relevant to the grokking phenomenon. Furthermore, we apply the principle of normal alignment to Recursive Feature Machines (RFMs) to introduce Recursive Feature Alignment Machines (RFAMs). We show that RFAMs exhibit greater adversarial robustness compared to RFMs when trained on tabular data.

We present CoopNet, an approach that improves the cooperation of co-trained networks by dynamically adapting the apportionment of gradient, to ensure equitable learning progress. It is applied to motion-aware self-supervised prediction of depth maps, by introducing a new hybrid loss, based on a distribution model of photo-metric reconstruction errors made by, on the one hand the depth + odometry paired networks, and on the other hand the optical flow network. This model essentially assumes that the pixels from moving objects (that must be discarded for training depth and odometry), correspond to those where the two reconstructions strongly disagree. We justify this model by theoretical considerations and experimental evidences. A comparative evaluation on KITTI and CityScapes datasets shows that CoopNet improves or is comparable to the state-of-the-art in depth, odometry and optical flow predictions.

We study the evolution of hidden-weight spectra in wide neural networks trained by (stochastic) gradient descent. We develop a two-level dynamical mean-field theory (DMFT) that jointly tracks bulk and outlier spectral dynamics for spiked ensembles whose spike directions remain statistically dependent on the random bulk. We apply this framework to two settings: (1) infinite-width nonlinear networks in mean-field/$μ$P scaling and (2) deep linear networks in the proportional high-dimensional limit, where width, input dimension, and sample size diverge with fixed ratios. Our theory predicts how outliers evolve with training time, width, output scale, and initialization variance. In deep linear networks, $μ$P yields width-consistent outlier dynamics and hyperparameter transfer, including width-stable growth of the leading NTK mode toward the edge of stability (EoS). In contrast, NTK parameterization exhibits strongly width-dependent outlier dynamics, despite converging to a stable large-width limit. We show that this bulk+outlier picture is descriptive of simple tasks with small output channels, but that tasks involving large numbers of outputs (ImageNet classification or GPT language modeling) are better described by a restructuring of the spectral bulk. We develop a toy model with extensive output channels that recapitulates this phenomenon and show that edge of the spectrum still converges for sufficiently wide networks.

Continual learning research attempts to conserve two fundamental capabilities: new knowledge acquisition and the preservation of previously acquired knowledge. While knowledge in this case can be measured through performance over an implicit or explicit task space, model plasticity generally concerns adaptability as data distributions evolve. Though much of the literature has focused on catastrophic forgetting, deep networks can also suffer from loss of plasticity, becoming progressively harder to update under continued training. Recent research has identified multiple mechanisms underlying this phenomenon, including neuron saturation, parameter norm growth, and loss of useful curvature directions. Adaptive reset-based interventions, which selectively reinitialize low-utility network parameters, have emerged as practical solutions to restore trainability. Existing utility measures used to guide resets, such as activation magnitude, contribution utility, or gradient-based activity, rely on proxy signals that can become misaligned with the intervention they are meant to guide. In this paper, we introduce gradient times difference from reference (GXD), a theoretically motivated utility measure based on reference-based gradient attribution that estimates the first-order functional cost of replacing a unit. Our results show that utility measures aligned with the functional cost of the reset can make interventions more reliable in settings where existing reset criteria degrade. GXD reframes adaptive resetting as an intervention cost estimation problem, providing a practical path toward more robust continual learning systems.

Deep networks with continuous piecewise affine activations induce polyhedral partitions of the input space, making the number of realized affine regions a natural measure of expressive capacity and a key determinant of how well the model can approximate nonlinear target functions. In practice, standard training realizes far fewer region refinements in data-visited neighborhoods than the architecture could in principle support, while existing region-count theory is primarily architectural and offers little guidance on how optimization shapes the realized partition near the data. Our theory provides a sufficient condition under which bringing neuron switching surfaces sufficiently close to data points ensures their intersection with local neighborhoods, which in turn implies a strict increase in the local affine-region count, yielding a principled training-time handle for seeding data-relevant partitions early in optimization. Guided by these results, we propose a plug-and-play region-seeding regularizer that encourages early partitioning while allowing task-driven refinement to dominate later in training. Experiments show that the regularizer increases the number of realized affine regions via exact enumeration and improves overall performance on toy datasets, while also improving early-stage accuracy and achieving comparable (or slightly improved) final accuracy on ImageNet-1k for classical models.

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.

Geometry-aware optimizers such as Newton and natural gradient can improve conditioning in deep learning, but scalable variants such as K-FAC, Shampoo, and related preconditioners usually impose structural approximations early, often discarding cross-layer interactions induced by the network computation. We introduce Layerwise LQR (LLQR), a framework for learning structured inverse preconditioners under a global layerwise optimal-control objective. The starting point is an exact equivalence: the steepest-descent step under a broad class of divergence-induced quadratic models--including Newton, Gauss-Newton, Fisher/natural-gradient, and intermediate-layer metrics--can be written as a finite-horizon Linear Quadratic Regulator (LQR) problem. This formulation serves as a reference that exposes the layerwise dynamics and cost matrices encoding the original dense geometry. We then derive a scalable relaxation that learns diagonal, (E-)Kronecker-factored, or other structured inverse preconditioners by minimizing the LQR objective and reusing them across iterations. The resulting optimizer wraps standard methods while retaining a principled connection to second-order geometry, without forming or inverting the global curvature matrix. Experiments on ResNets and Transformers show that LLQR improves optimization dynamics and often translates these gains into improved final test performance, while adding only modest wall-clock overhead. It establishes LLQR as a practical framework for geometry-aware second-order methods and a reference for evaluating scalable approximations.

In deep networks with small initialization, training exhibits long plateaus separated by sharp feature-acquisition transitions. Whereas shallow nonlinear networks and deep linear networks are well studied, extending these analyses to deep nonlinear networks remains challenging. We derive an exact identity for the imbalance of Frobenius norms of layer weight matrices that holds for any smooth activation and any differentiable loss and use this to classify activation functions into four universality classes. On the permutation-symmetric submanifold, the identity combines with an approximate balance law to reduce the full matrix flow to a scalar ODE, giving a critical-depth escape time law $τ_\star = Θ(\varepsilon^{-(r-2)})$ governed by the number $r$ of layers at the bottleneck scale rather than the total depth $L$. We find that this same $r-2$ exponent is recovered under He-normal initialization with $r$ bottleneck layers rescaled by $\varepsilon$, where the symmetry manifold is preserved by the flow but not attracting. We find close agreement between our theory and numerical simulations.

Object recognition (OR) in humans relies heavily on shape cues and the ability to recognize objects across varying 3D viewpoints. Unlike humans, deep networks often rely on non-shape cues such as texture and background, leading to vulnerabilities in generalization and robustness. To address this gap, we introduce ShapeY, a novel and principled benchmarking framework designed to evaluate shape-based recognition capability in OR systems. ShapeY comprises 68,200 grayscale images of 200 3D objects rendered from multiple viewpoints and optionally subjected to non-shape ``appearance'' changes. Using a nearest-neighbor matching task, ShapeY specifically probes the fine-grained structure of an OR system's embedding space by evaluating whether object views are clustered by 3D shape similarity across varying 3D viewpoints and other non-shape changes. ShapeY provides a suite of quantitative and qualitative performance readouts, including error rate graphs, viewpoint tuning curves, histograms of positive and negative matching scores, and grids showing ordered best matches, which together offer a comprehensive evaluation of an OR system's shape understanding capability. Testing of 321 pre-trained networks with diverse architectures reveals significant challenges in achieving robust shape-based recognition: even state-of-the-art models struggle to generalize consistently across 3D viewpoint and appearance changes, and are prone to infrequent but egregious matches of objects of obviously completely different shape. ShapeY establishes a principled framework for advancing artificial vision systems toward human-like shape recognition capabilities, emphasizing the importance of disentangled and invariant object encodings.

Block-sequential continual learning demands that a single model both protect prior solutions from catastrophic forgetting and efficiently infer at inference time which prior solution matches the current input without task labels. We present Functional Task Networks (FTN), a parameter-isolation method inspired by structural and dynamical motifs found in the mammalian neocortex. Similar to mixture-of-experts, this method uses a high dimensional, self-organizing binary mask over a large population of small but deep networks, inspired by dendritic models of pyramidal neurons. The mask is produced by a three-stage procedure: (1) gradient descent on a continuous mask identifies task-relevant neurons, (2) a smoothing kernel biases the result toward spatial contiguity, (3) and k-winner-take-all binarizes the resulting group at a fixed capacity budget. Like mixture-of-experts, each neuron is an independent deep network, so disjoint masks give exactly disjoint gradient updates, providing structural guarantees against catastrophic forgetting. This three-stage procedure recovers the sub-network of a previously-trained task in a single gradient step, providing unsupervised task segmentation at inference time. We test it on three continual-learning benchmarks: (1) a synthetic multi-task classification/regression generator, (2) MNIST with shuffled class labels (pure concept shift), and (3) Permuted MNIST (domain shift). On all three, FTN with fine grained smoothing (FTN-Slow) results in nearly zero forgetting. FTN with a large kernel and only 2 iterations of smoothing (FTN-Fast) trades off some retention for increased speed. We show that the spatial organization mechanism reduces the effective mask search from the combinatorial top-k subset problem in O(C(H,K)) to the complexity of a near-linear scan in O(H) over compact cortical neighborhoods, which is parallelized by the gradient-based update.

We study whether deep networks for medical imaging learn useful nonrobust features - predictive input patterns that are not human interpretable and highly susceptible to small adversarial perturbations - and how these features impact test performance. We show that models trained only on nonrobust features achieve well above chance accuracy across five MedMNIST classification tasks, confirming their predictive value in-distribution. Conversely, adversarially trained models that primarily rely on robust features sacrifice in-distribution accuracy but yield markedly better performance under controlled distribution shifts (MedMNIST-C). Overall, nonrobust features boost standard accuracy yet degrade out-of-distribution performance, revealing a practical robustness-accuracy trade-off in medical imaging classification tasks that should be tailored to the requirements of the deployment setting.

Raman spectra obtained in real world applications are often a noisy combination of several spectra of various substances in a tested sample. Unmixing such spectra into individual components corresponding to each of the substances is of great value and has been a longstanding challenge in Raman spectroscopy. Existing unmixing methods are predominantly designed to invert an overdetermined mixed model and therefore require multiple mixed spectra as input. However, open domain and/or non-cooperative detection applications in Raman spectroscopy such as controlled substance detection, call for single-channel solutions which can identify individual components from thousands of candidates by analyzing only a single noisy mixed spectrum. To our knowledge, sparse regression is the only existing solution which can cope with this scenario, yet it has very low tolerance to noises and can hardly be applicable in practice. To address these limitations, we introduce a novel neural approach for single-channel Raman spectrum unmixing inspired by speech separation. It aims at solving underdetermined systems and can decompose a noisy mixed spectrum from a library of thousands of components (substances). The core of our method is a deep separation neural network (RSSNet) which takes a mixed spectrum as input and outputs spectra of pure components. We created two synthetic datasets of single-channel Raman spectra unmixing and demonstrated feasibility and superiority of RSSNet on these datasets (outperform competing methods by >4dB). Furthermore, we verified that RSSNet, trained solely on synthetic data, can successfully unmix real-world mixed spectra of mixtures of mineral powders, exhibiting strong generalization. Our approach represents a new paradigm for Raman unmixing and enables new possibilities for fast detection of Raman mixtures.

Modern neural networks can achieve high accuracy while remaining poorly calibrated, producing confidence estimates that do not match empirical correctness. Yet calibration is often treated as a post-hoc attribute. We take a different perspective: we study calibration as a training-time phenomenon on small vision tasks, and ask whether calibrated solutions can be obtained reliably by intervening on the training procedure. We identify a tight coupling between calibration, curvature, and margins during training of deep networks under multiple gradient-based methods. Empirically, Expected Calibration Error (ECE) closely tracks curvature-based sharpness throughout optimization. Mathematically, we show that both ECE and Gauss--Newton curvature are controlled, up to problem-specific constants, by the same margin-dependent exponential tail functional along the trajectory. Guided by this mechanism, we introduce a margin-aware training objective that explicitly targets robust-margin tails and local smoothness, yielding improved out-of-sample calibration across optimizers without sacrificing accuracy.