Jacobian

Fine-tuning foundation models on new tasks inevitably suffer from catastrophic forgetting. While existing works attempt to mitigate this on the basis of parameter-efficient fine-tuning methods, they adopted an overly restrictive Subspace Orthogonality condition. In this paper, we introduce a purely post-hoc and tuning-agnostic weight rectification framework that achieves Parameter Space Orthogonality, which is the necessary and sufficient condition for preserving historical performance to the first order. By projecting parameter updates into the JAcobian NUll Space (JANUS), our method significantly recovers compromised historical knowledge without interfering with the underlying fine-tuning process. To overcome the local validity of the Jacobian approximation, we further propose a Multi-step Adaptive Rectification mechanism that utilizes the JANUS shift to dynamically verify the valid trust region and adjust step sizes. Coupled with our proposed ghost projection, ghost orientation comparison, and sequence-level singular value decomposition compression techniques, JANUS also achieves great temporal and spatial efficiency. Experiments demonstrate that JANUS seamlessly integrates with various fine-tuning methods, significantly mitigating the stability-plasticity dilemma by recovering historical knowledge while preserving downstream task adaptation.

This paper provides a self-contained proof of the rank of the pathlifting Jacobian of a DAG ReLU network by performing an induction on the network's number of hidden nodes. In fact, the induction is elementary, and the key recipe is to consider the skeleton matrix of the network, a sparse matrix encoding the network paths, and transform the representation of one of its hidden neurons into an output node. The proof relies on intermediate propositions which link the pathlifting, its Jacobian, the network parameters, and its skeleton matrix, which, on top of permitting to conclude on the rank of the pathlifting Jacobian, also provide a way to compute it without backpropagation and whose computation cost is super efficient in practice compare to usual backpropagation. The paper is provided with a Python module that implements the different propositions of the paper for feed forward networks and is used to experimentally quantifies the computational gain of computing the pathlifting Jacobian with the proposed theory.

Dexterous in-hand manipulation of a grasped object with an anthropomorphic hand is an unsolved frontier for robot dexterity. The contact-richness and highly dynamic nature of object-hand interactions tend to require extensive modeling or data-collection efforts for learning-based approaches. Modern simulators used for reinforcement learning (RL) cannot fully replicate the required contact complexity, while collecting dexterous demonstrations for imitation learning (IL) remains an open problem. In this research, we present an embodied control approach based on real-time task Jacobian estimation of the combined hand and object system on the physical robot. Using only the CPU on a laptop, the proposed controller begins in-hand pen writing after approximately 18 s of initialization and continues to adapt online, without an analytic hand--object kinematic/contact model, simulation training, or precollected task demonstrations. We demonstrate that the same estimator/controller formulation works on three anthropomorphic robotic hand systems (one physical, two simulated) to show human-like, in-hand articulation of a grasped pen by an embodiment-independent formulation. Sub-millimeter in-plane precision (mean 0.6 mm across runs) is achieved across letters and shapes written in the air and on paper on a physical robot. To our knowledge, this is the first demonstration of an anthropomorphic hand writing arbitrary single-stroke trajectories with a grasped pen through purely in-hand motion, and it showcases an alternative to compute- and data-heavy approaches such as RL and IL for achieving dexterous manipulation through computationally simple and data-efficient algorithms.

This paper investigates the hypothesis that the first-order structure of physical interactions, i.e. gradients or Jacobians, characterizes the structure of phenomenal experience. It does so in an idealized world inhabited by neural networks, Gradland, where the physics are known and the functions are (mostly) differentiable. The paper introduces two measures of Jacobian structure: effective rank and cohesion, based on Kirchhoff complexity. Applying the measures to a series of worked examples shows the hypothesis accounts for: (1) the duration of experience, that it can prolong over hundreds of milliseconds; (2) the difference between what is experienced vividly and obscurely; (3) the experience of texture; (4) the blooming buzzing confusion presumably experienced by newborns; (5) the difference between ideas that are held distinctly in mind and ideas that are confused; (6) what learning is like; and finally (7) the paper explains the function of rich, dense experience.

Soft Actor-Critic (SAC) represents a continuous policy as an unbounded Gaussian that is squashed by tanh. The Jacobian of that map is $\partial a/\partial u = 1-a^2$, which vanishes as $|a|\to 1$. A natural concern is that this throttle starves the actor of critic signal exactly where extreme actions (full brake, full throttle) are optimal. We test a minimal intervention that restores the missing signal: one extra term in the actor loss whose gradient on the pre-tanh mean is the detached action-gradient of $Q$, with no gain parameter. On a minimum-time double integrator whose optimum is bang-bang at the action bounds, vanilla SAC already reaches near-optimal return ($-31.6$ vs. a calibrated optimum of $-30.3$) across ten paired seeds. An ungated bypass does saturate the policy (99% of eval steps with $|a|\ge 0.9$) and collapses return to $-195.5$. A gated bypass that fires only on the flat shoulder $|a|\in[0.9,0.999]$ also fails, and does so without leaving a saturated policy. Warm-started MetaDrive fine-tuning shows the same pattern: the bypass does not improve return, and where collision rate falls it is typically traded for out-of-road departures. Auto-tuned entropy coefficient rises against the bypass, which is a push toward the tails. The Jacobian effect is real. Treating it as a bug to be undone is not free, and on the tasks studied here it is not helpful. Saturating a bound is not the same as solving a problem whose optimum lives on that bound.

Key-value (KV) cache eviction is essential for scaling long-context inference in Large Language Models. However, existing policies predominantly rely on empirical heuristics, lacking a rigorous characterization of token utility under the inherently nonlinear softmax attention mechanism. In this work, we rethink KV cache eviction through the lens of local information geometry, modeling the attention process as a nonlinear Gaussian communication channel. By performing a first-order Taylor expansion of the attention mapping, we derive the Jacobian Information Capacity, a novel objective that explicitly captures query relevance, softmax sensitivity, and structural diversity. Guided by this theory, we introduce Jacap, a capacity-aware eviction method that utilizes softmax-aware importance weighting and statistical leverage scores for subset selection. Extensive experiments across diverse architectures and benchmarks demonstrate that \textsc{Jacap} delivers superior performance in most scenarios, particularly in high-compression regimes.

Constant optimization refines the numerical coefficients of candidate expressions in tree-based genetic programming for symbolic regression. But its per-generation cost has led modern GPU-accelerated frameworks to omit it or restrict it to lightweight forms. We present a GPU-resident, batched Levenberg--Marquardt solver that optimizes constants across a structurally heterogeneous population of expression trees using a fixed number of population-wide CUDA launches per iteration. Reverse-mode automatic differentiation assembles the per-tree Jacobian in one backward sweep, making the dominant per-iteration cost independent of the number of constants per tree, and a double-precision delivery guard guarantees that returned constants are never worse than their initial values. On early-generation populations, the solver sustains up to $5.1{\times}10^{5}$ trees per second on an NVIDIA A100; at a GPU-saturated benchmark configuration it delivers roughly $9.9{\times}$ the throughput of Operon running on a 64-core EPYC 7763, while matching fp64-reference quality. Integrated in-process into EvoGP, the solver enables end-to-end search to recover governing equations on $10$ of $18$ constructed problems versus 0 for stock EvoGP. Our code is at https://github.com/TensorConv/CuSR.

We revisit the Sinkhorn-Knopp (SK) algorithm for the matrix scaling problem. Despite extensive literature on the global convergence of SK and its variants, its local linear convergence behavior remains less understood. We address this gap by providing the first nonasymptotic local analysis of SK that matches the rate obtained from existing asymptotic Jacobian-based arguments. We show that under certain connectivity conditions, SK is a polynomial-time algorithm for doubly stochastic matrix scaling. With the developed tools, we showcase the local suboptimality of SK and provide accelerated variants. Finally, for dense matrices, we improve the complexity of existing first-order matrix scaling algorithms from $O(\tfrac{n^{7/3}}{\varepsilon^{2/3}})$ to $O(\tfrac{n^{9/4}}{\sqrt{\varepsilon}})$.

Deep neural network (DNN) training with stochastic gradient descent (SGD) and its variants achieves strong empirical performance, yet classical optimization theory does not fully explain this success. This limitation arises because conventional analyses rely on assumptions such as differentiability, convexity, or smoothness, which are often violated by DNN objectives. In this paper, we establish a unified optimization framework for DNN training by generalizing classical convexity and smoothness through Legendre functions and convex conjugation. Specifically, we introduce $\mathcal{H}(ψ)$-convexity and $\mathcal{H}(Ψ)$-smoothness, which unify convex and non-convex as well as smooth and non-smooth objectives within a single formalism and reveal a natural duality between generalized smoothness and convexity. Building on these generalized properties, we introduce generalized gradient descent (GD) and generalized SGD through convex conjugation. We theoretically prove that generalized GD admits an optimal learning rate of exactly $1$, and derive rigorous gradient-energy-based convergence rates for both proposed optimizers. We further reformulate DNN training as a composite optimization problem, demonstrating that its convergence relies on jointly reducing the gradient energy and controlling the induced norm of the network Jacobian. To characterize the practical influences of network architectures and training configurations, we introduce the gradient correlation factor and model capacity risk, and quantitatively analyze how architectural designs, batch size, and model capacity shape training convergence. Extensive experiments across diverse network architectures, datasets, optimizers, and loss functions validate our theoretical bounds and demonstrate precise alignment between our theoretical predictions and empirical training dynamics.

We introduce the neural echo as a tool for understanding the behavior of neural networks. It generalizes the model-based concepts of impulse responses, diffusion echoes, and filter echoes to learning-based methods. It provides local, space-adaptive impulse responses and filter kernels for a neural network, its so-called echoes. These echoes depend on the input image and can be visualized to understand the learned dynamics of the network via an affine mapping. Neural echoes build a bridge from classical signal processing to modern explainable AI. They are very general and can be applied to both image-to-image and classification networks, with convolutional or fully connected structure, of feedforward or recurrent type, including modern transformer networks. Network differentiability is not required. In the differentiable case, neural echoes comprise concepts based on the network Jacobian, such as saliency maps and the analysis of adversarial perturbations, as special instances. As a simple blueprint to explain our framework, we derive neural echoes for the denoising convolutional neural network (DnCNN). Our experiments suggest that this network weights pixels based on their spatial and gray value distances. This not only clarifies its behavior, but also shows that it can reproduce key concepts of classical model-based denoisers such as bilateral filtering.

Echo State Networks (ESNs) offer an efficient framework for temporal prediction, but their randomly initialized reservoirs are often over-parameterized and dynamically redundant. Existing pruning methods largely rely on static connectivity or activation statistics, which may overlook neurons that shape input-driven state transitions. We propose Dynamical Mode Pruning (DMP), a reservoir pruning method that ranks neurons by their contribution to dominant transition modes obtained from a trajectory-averaged Jacobian Gramian. DMP removes low-impact units and retrains only the readout. Experiments on chaotic and real-world time-series benchmarks show that DMP improves or preserves forecasting accuracy while reducing redundant reservoir components. Our results suggest that dynamical influence is a useful criterion for reservoir refinement beyond static structural importance alone.

A companion paper studies when activation patching and weight-space ablation agree, inside an idealized model where a conditional computation is carried additively through a residual stream. For the one composition in that model where two carriers are architecturally dependent, an attention head and its own layer's normalization-MLP composition, it derives an exact first-order interaction formula, zero when only the MLP is ablated and second-order bounded when the head is also ablated. That result is confined to a single residual block and checked only on small transformers on a synthetic task. This paper extends the result past both limits. First, the interaction from ablating carriers spanning several layers decomposes exactly into same-block terms, one per touched layer, plus a cross-layer remainder on which the decomposition makes no claim of smallness. Second, we isolate that remainder exactly, for two layers, as a double integral of a mixed second derivative, and name the missing ingredient needed to bound it: a Jacobian bound for the attention sub-block. We derive this bound in closed form and verify it, without a single violation, against Qwen2.5-1.5B-Instruct's real weights, though we do not yet chain it across layers. We also give, in closed form, the curvature constant the companion paper's bound leaves unexhibited. Third, on that same model, we search for and find an emergent circuit for indirect object identification, never designed into it, using the original activation-patching method for this task, and test collapse, dissociation, and interaction on it. The result is mixed: a shared carrier emerges across all five tested instances, collapse and dissociation hold on most but not all, and a nonzero interaction is measurable on three of five, at layer pairs outside the same-block case the companion theorem covers.

Mixed complementarity problems (MCPs) arise as the first-order optimality conditions of nonlinear programs and noncooperative games, and provide a natural formulation for multi-agent trajectory optimization problems that appear throughout robotics. The dominant solver for problems of this form is PATH, which offers strong performance on robotics problems but remains closed-source. We present MixedComplementarityProblems.jl, an open-source, pure Julia implementation of an interior point method for parametric MCPs that: (i) matches PATH's reliability on standard benchmarks, (ii) natively supports batched, parallel processing of many parameter instances, either across CPU threads or on an NVIDIA GPU, and (iii) supports efficient automatic differentiation of solutions with respect to problem parameters. On a multi-agent lane-change trajectory game representative of robotics planning problems, our CPU-multithreaded batched solver clears a batch of parametric instances ~100x faster than sequential calls to PATH. A GPU backend, running the same solver implementation unmodified, also clears these batches far faster than PATH, but does not outperform the multithreaded CPU on this problem; the GPU pulls ahead only once each per-instance KKT system grows large, and we characterize this regime dependence. We describe the solver's interior point formulation, the abstraction that lets a single solver implementation run unmodified across dense, batched-sparse, and single-large linear-algebra backends, and report benchmarks against PATH on both randomly generated quadratic programs and trajectory games.

Large language models can answer scientific questions, yet a correct output does not reveal whether the model represents or uses the governing physics. Here, using three open-weight Gemma 4 models (google/gemma-4-E4B-it, google/gemma-4-12B-it, google/gemma-4-31B-it) we identify three experimentally separable signatures of materials-science mechanism information: selective concept readability, relational encoding of qualitative constitutive orientation, and causal, context-dependent control of constrained engineering answers. We combine matched direct and Jacobian vocabulary readouts, option-free state geometry, a 60-law counterfactual benchmark and causal interventions. In 50 held-out materials descriptions, three independently fitted Jacobian lenses reproduced concept ranks, and target-free word sets from both readouts enabled blinded identification of 9 of 10 mechanism families. A separate 72-prompt benchmark produced mechanism-specific hidden-state neighborhoods, but an exact graph audit showed that this apparent physical organization was equally explained by numerical comparison. We therefore compared otherwise identical prompts in which only the direction of the physical input was reversed, asking whether the resulting hidden-state movement followed the supplied constitutive law. These state transformations ordered direct, physically neutral and inverse laws across 60 frozen relations and correctly oriented 39 of 40 directional laws, whereas lexical controls were near chance. Bidirectional interventions shifted answer probabilities toward or away from the physically appropriate outcome across all 12 matched cases, while counterfactual state patches transferred opposing decision signals across mechanisms and answer formats. Physical relationships were therefore more visible in controlled state changes than in absolute states alone.

Contrastive learning is increasingly moving toward settings with three or more modalities instead of image-text pairs. Yet, extending models from pairwise to higher-order multimodal alignment can introduce optimization and representation challenges. We identify encoder Jacobian conditioning as a key factor in trimodal contrastive learning: poorly conditioned encoders exhibit collapsing or amplified singular-value spectra, leading to exploding Jacobian condition numbers and degraded multimodal alignment. We introduce geometry-preserving encoders (GPEs) by directly conditioning the Jacobian through regularization and demonstrating that simple modifications like LeakyReLU activations and residual paths recover these geometric benefits. Across a synthetic benchmark and four real-world datasets including missing modalities, improving Jacobian conditioning boosts retrieval and linear probe performance across multiple contrastive objectives, whereas expressive objectives yield little benefit in linear probes. More broadly, our results show that multimodal contrastive learning depends not only on objective expressivity, but also on the geometric and optimization properties of the underlying encoders.

We introduce \emph{gate-zero growth}, a function-preserving (FP) operator for continual learning that adds new residual blocks through a zero-initialised gate. Under a transversality condition, gate-zero growth induces \emph{rank separation} in the functional Jacobian: old directions are unchanged, new-weight directions are exactly flat at the growth point, and new gate directions are the only first-order source of new functional variation. As gates open during continual learning, function drift is $O(\|\boldsymbolα\|^2)$ and Jacobian leakage $O(\|\boldsymbolα\|_\infty)$, giving a controlled departure from the FP locus. On a $300\mathrm{M}\to857\mathrm{M}$ Transformer adapted from WikiText-103 to BookCorpus, gate-zero growth reaches near-zero old-domain forgetting ($Δ_A < 0.1$) under both exact-preservation (Isolation) and joint-frontier (Freeze-Nothing) operating points, while a non-FP control ($G_{\text{stack}}$) suffers an order-of-magnitude larger forgetting under the same recipe. The same geometric analysis covers LoRA, ReZero, and zero-init adapter constructions, establishing gate-zero growth as the canonical instance of a shared local geometry that governs safe capacity activation in CL.

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.

Predictive Coding (PC) offers a biologically motivated alternative to backpropagation via local weight updates, yet routing error between layers still relies on an autograd Jacobian-transpose ($J^\top$) product - the last non-local operation in PC. We show that this dependency is largely avoidable. For any layer $f(x)=\mathrm{Act}(\mathrm{Norm}(L(x)))$ with frozen normalization statistics, the exact $J^\top$ factors into three locally available terms, $J^\top v = L^\top(s \odot σ'(z) \odot v)$, where $σ'$ is the activation derivative, $z$ is the pre-activation, and $s=γ/σ_{\mathrm{run}}$ is the normalization gain. Prior weight-feedback methods omitted both corrections; restoring them closes the transport gap for this layer class. Locality here holds up to three assumptions, which we state upfront: weight symmetry ($L^\top$ mirrors the forward operator, as assumed by all PC), a soft spectral-norm control that is not synapse-local, and a nearest-neighbour approximation for MaxPool. Substituting the identity into PC yields WF-Act-PC, which removes the autograd backward pass from error transport. On CIFAR-10/100 (50 epochs, 5 seeds), WF-Act-PC is the only PC method whose accuracy improves with depth, surpassing iPC - the strongest classical PC baseline - by 2.7-22.3 pp on CIFAR-10. With both methods tuned per architecture, it matches or exceeds a comparably-tuned backpropagation baseline on the deeper CIFAR-10 architectures (VGG-9: 93.57% vs. 92.43%; ResNet-18: 92.76% vs. 91.54%) and on the harder Tiny-ImageNet benchmark, while trailing tuned BP on the deeper CIFAR-100 VGG cells. Our WF-Act-PC implementation is publicly available at https://github.com/jlshen025/pcax

Jailbreak-robustness research typically evaluates safety through generated responses using an LLM-as-judge approach. Such evaluations, however, are sensitive to the benchmark's grading procedure and capture only observed behavior on a given set of attacks, without directly revealing the hidden fragility of the underlying safety mechanisms. This work proposes JADR (Jacobian Assessment of Danger Recognition), a protocol that measures a model's internal representation through Jacobian space (J-space, a recently proposed workspace of verbalizable concepts) before the first response token is generated. For every prompt and layer we record the top-k J-space tokens; these are grouped into six behavioral scenario axes and compared between a danger sample based on StrongREJECT and a safe control drawn from XSTest and OKTest. The method does not call on an external judge model: the computation runs entirely locally, on the activations of the model under evaluation, which lets us compare both different models against each other and modifications of a single model -- quantization and fine-tuning in particular -- on the same terms. The final comparison rests on the proposed SafetyAUC metric, complemented with bootstrap confidence intervals. The protocol is applied to six models (Qwen3-1.7B, Qwen3-4B, Qwen3-8B, Qwen3-Uncensored-4B, Qwen3-SafeRL-4B, Gemma 2 9B) across three weight-representation regimes -- BF16, INT8, and INT4 -- and checked against an independent behavioral evaluation with the StrongREJECT grader. The metric separates models with a strong versus a weak internal safety mechanism with statistical significance and captures substantively different effects across quantization regimes.

In radiation transfer simulations, an M1 method achieves substantial computational savings by replacing the full angular transport equation with a low-order moment system. Because this reduced system is not closed, a closure model is required to represent the unknown higher-order moments using lower-order moments. While machine learning (ML)-based closures can improve accuracy beyond classical analytic closures, unconstrained learned closures may produce non-real characteristic speeds and consequently cause numerical solver breakdown. To guarantee real eigenvalues of the Jacobian associated with ML closures, we propose a hyperbolic neural closure for the M1 radiative transfer system. Rather than directly predicting closure terms, we parameterize the Jacobian through two neural networks: (i) a symmetric matrix network and (ii) a strictly convex entropy network whose Hessian defines a positive definite symmetrizer. These components are combined to yield a Jacobian that is similar to a symmetric matrix, thereby ensuring real eigenvalues. The closure is then reconstructed by numerical integration of the learned Jacobian field along a prescribed integration path. Numerical experiments show that the proposed closure not only achieves higher closure accuracy than classical analytic closures, but also improves solution accuracy and remains stable in discontinuous Galerkin simulations for radiative transfer problems.

We study the small-noise asymptotics of Euclidean heat regularizations of probability measures supported on manifolds with corners. Near a boundary or corner stratum, the relevant regime is a conical boundary layer in which the observation point approaches the stratum at the same scale as the Gaussian smoothing parameter. After rescaling this layer, the support is replaced to leading order by its inward tangent cone. We prove a two-term expansion for the heat-regularized density in this regime. The leading coefficient is the Gaussian mass of the linearized cone, weighted by the density on the support and by the adapted corner Jacobian; the first correction records the variation of the density, the Jacobian, and the quadratic geometry of the embedding. A localization argument then yields the corresponding expansion for the full heat regularization, with the nonlocal contribution exponentially small. From this density expansion we derive logarithmic asymptotics and uniform expansions for the score, the log-Hessian, and the scale derivative of the score. These formulas show how lower-dimensional support, boundary faces, corners, and curvature are encoded in the singular differential structure of small-noise Gaussian regularizations.

Generating realistic human avatars in complex motions--such as clothing dynamics--requires modeling of global and local deformations which remains challenging in monocular settings. We address this problem by leveraging neural Jacobian fields (NJFs) for representing semi-rigid deformations. We train self-supervised neural networks for predicting Jacobian matrices that give the pose-dependent deformations, by solving a Poisson equation. However, monocular input presents several difficulties such as self-occluded regions and invisible surfaces. To address these issues, we introduce three key components: a constrained Poisson solver, signed distance-based Jacobian regularization, and a deformation-guided residual flow loss, which together suppress boundary artifacts, recover frequently occluded regions such as armpits and thighs, and enforce temporal consistency during motion. Experiments on benchmark and in-the-wild videos demonstrate that our method generates temporally stable and geometrically coherent avatars, outperforming state-of-the-art approaches.

Dual-energy CT (DECT) exploits attenuation differences across different X-ray spectra to provide richer material information and has been widely used in medical imaging. While sparse-view acquisition can lower radiation exposure, it makes DECT material decomposition even more challenging, as the problem is nonlinear and ill-posed. Existing deep unrolling approaches generally do not explicitly incorporate the Jacobian operator induced by the nonlinear forward model, and their sparsity priors are still mainly built on conventional convolutions, which are insufficient for modeling global structural information. This study addresses the challenge of DECT multi-material decomposition in sparse-view settings by representing it as a sparse-regularized nonlinear least-squares problem. To solve it, we propose an iterative dual-domain refinement network (DECT-DRNet). In each iteration, the filtered back-projection (FBP)-based Jacobian approximation module is used first to generate an intermediate material decomposition result. Here, we characterize the forward process of material decomposition using a nonlinear operator, and then construct a theoretically grounded learnable approximation of the adjoint Jacobian operator by integrating the FBP algorithm with a U-Net into the backward process. In addition, to address the limitation of existing deep learning-based decomposition methods in globally suppressing noise and artifacts, we introduce a learnable sparse dual domain regularization term that incorporates Fourier convolutional residual blocks. This refinement block combines geometric feature extraction in the image domain with noise suppression in the frequency domain, allowing the model to capture both global and local features while maintaining structural details. DECT-DRNet demonstrates its ability to achieve more accurate material decomposition under sparse-view conditions.

Predicting a patient's physiological trajectory under a planned treatment sequence is a prospective interventional problem, not standard time-series extrapolation. We study this problem in glucose management, where insulin and carbohydrate records are policy-dependent: future drivers are coupled to patient state, behavior, and clinical decision rules, so observational forecasting accuracy alone does not guarantee correct responses to planned interventions. We introduce Interventional Flow Matching (IFM), a continuous-time generative framework for physiologically constrained prospective forecasting. IFM conditions a flow-matching velocity field on patient history and planned future drivers in a bounded latent glucose space. Rather than embedding strict mechanistic glucose--insulin ODE equations or enforcing causality through rollout-based simulations, IFM uses a solver-free regularization: it penalizes the Jacobian of the instantaneous velocity field with respect to smoothed treatment drivers. This imposes signed, dose-bounded local sensitivities directly on the learned dynamics: insulin lowers glucose, carbohydrates raise it, and both responses remain within plausible ranges. On a simulated UVA/Padova type 1 diabetes cohort, IFM achieves the strongest balance between observed-driver RMSE and interventional response metrics. Across experiments, it consistently produces physiologically correct responses to both insulin and carbohydrate drivers while maintaining high directional, and ranking consistency.

Automated diabetic retinopathy (DR) grading from colour fundus photographs can achieve strong predictive performance, but clinical interpretation requires more than an image-level label. It requires understanding how lesion evidence is distributed around retinal vessels and how this evidence relates to quantitative vascular biomarkers. We present a dual-edge spatial-Jacobian image graph for interpretable DR grading. Each fundus image is represented as a graph node with four aligned evidence streams: AutoMorph vessel information ($X_1$), DR-XAI-style lesion evidence maps ($X_2$), a 128-dimensional lesion-based contrastive image embedding ($X_3$), and AutoMorph morphometric biomarkers ($X_4$). The spatial edge branch ($X_{12}$) encodes vessel-lesion geometry, while the Jacobian branch ($X_{34}$) models embedding-biomarker sensitivity. Lightweight two-token attention fuses both edge families into a final image graph. On 2,910 matched non-augmented APTOS images, the full graph achieves 0.8076 accuracy, 0.8312 quadratic weighted kappa, 0.5915 macro-F1, and 0.9330 adjacent-grade accuracy; referable DR reaches 0.9055 accuracy and 0.9711 AUROC. The framework is positioned as an explainable representation-learning tool for lesion-biomarker hypothesis generation, rather than as a deployment-ready clinical classifier. The code is available at https://github.com/Inamullah-Colab/dual-edge-dr-graph-xai.

Predictive coding networks (PCNs) offer a biologically-plausible, local-learning alternative to back-propagation of errors (backprop). Nevertheless, they have remained largely confined to shallow architectures and evaluated on simple machine intelligence benchmarks. A central obstacle to scaling PCNs is that the learning signal decays rapidly as it propagates away from the clamped boundaries, leaving interior layers effectively unchanged. To directly counter this problem, we propose highway error propagation (HEP), a scheme that augments the free energy function underlying predictive coding (PC) by altering its neural structure with feedback matrices $V_{L\to i}$ that couple selected hidden states directly to the clamped output error. Since this coupling is linear in the hidden state, the highway pathway delivers a correction at every inference step whose magnitude is independent of depth, in contrast to vanilla PC where the output error reaches the $i$-th hidden layer with attenuation that decays exponentially in depth. This bypasses the Jacobian chain while preserving the local PC synaptic update rule. On MNIST and Fashion-MNIST, we show that HEP effectively trains MLPs of up to 128 layers with accuracy that is robust with respect to depth.

Many central machine learning tasks, from entropy tuning in reinforcement learning to equilibrating generative adversarial networks, are fundamentally stochastic root-finding problems rather than loss minimization. Yet, they are frequently forced into a minimization framework via squared residuals, introducing a critical flaw we identify as the Variance Trap. Standard bilevel minimization algorithms require estimating hypergradients involving implicit Jacobians; in stochastic settings, these terms act as noise amplifiers, destabilizing convergence. We formalize Root-Finding Bilevel Optimization (RF-BO) as a distinct problem class that bypasses this pathology. We propose a Jacobian-free solution using Two-Time-Scale Stochastic Approximation (TTSA) that updates directly along the root error, structurally avoiding variance amplification. We provide the first non-asymptotic convergence guarantees for TTSA in this setting under Markovian noise. Extensive experiments demonstrate the decisive advantage of this paradigm: compared to squared-residual and implicit-gradient baselines, our framework achieves a 2.6% top-1 accuracy gain in SimCLR, 17$\times$ faster convergence in non-linear ODE control where baselines fail, significantly improved entropy stability in reinforcement learning, and an 11.1% quality improvement in generative modeling.

The Categorical Jacobian of Zhang et al. (2024) reads protein contacts from a language model by perturbing every residue with every alternative amino acid, about $19L$ forward passes. We show the signal it reconstructs is already concentrated in a small subset of attention heads: averaging the top-$K$ contact-relevant heads -- selected on as few as 10 labeled proteins, with no fitted per-pair or per-head weights -- recovers contacts in a single forward pass and matches or beats the Categorical Jacobian for every bidirectional model where it is defined (bar the smallest, 8M). Our primary test is leakage-clean: on a CAMEO split where neither selection nor evaluation touches data the models have plausibly memorized, the head readout beats the Categorical Jacobian on ESM-2-650M by +9pp ($N = 29$, $p < 0.001$), with the within-model margin reproducing across architectures. Ablations localize the gain to labeled head selection, not to averaging: at a matched label budget the unweighted mean ties a supervised $L_1$ logistic regression on the same heads. Both methods fall 30-36pp from their in-distribution Zhang numbers to the leakage-clean split, which we read as an upper bound on how much prior numbers reflect pretraining overlap. We additionally introduce representation-CJ, a hidden-state generalization of the Jacobian to architectures without a masked-LM head (the output-head-independent analogue of logit-CJ), agreeing with the Categorical Jacobian where both are defined (per-protein Pearson $r \approx 0.95$); show that the optimal $K$ tracks how diffusely a model spreads its contact heads; and find both methods lose the signal on the two causal LMs we test, suggesting attention-encoded pair structure may depend on bidirectional pretraining.

Neural operators are increasingly used to warm-start Newton solvers for nonlinear PDEs, on the premise that a low test error places the initial guess inside the basin of attraction. We show that this premise is unreliable. An operator trained to the relative $L^2$ error $O(10^{-3})$ can still produce an initial state in which the discrete Jacobian is indefinite, because the mean-squared training controls error on average while leaving localized pointwise violations of the underlying physics. For a nearly incompressible hyperelasticity problem, we trace this to the predicted volume change: the operator disperses $\mathrm{det} F$ well away from one, and the resulting Jacobian acquires negative eigenvalues even when the predicted field is visually indistinguishable from the reference. At a small scale, this is a nuisance; at a multi-million degree-of-freedom scale, it is disqualifying, since the conjugate gradient and other Krylov solvers needed for memory-feasible Newton steps assume a definite spectrum. We then show that a short, label-free fine-tuning phase -- penalizing the operator against the discrete energy, with no additional solution data -- shifts the Jacobian spectrum back to positive definite. Combined with an inexact outer loop, this gives a warm-started Newton method that converges across the full loading range where the unregularized operator fails, reaching up to 5.4$\times$ wall-clock speedup over incremental continuation on a 3D problem with 6.4 million degrees of freedom.

Fine-tuning a vision-language-action model (VLA-JEPA) on a single GPU should be simple: load a pretrained checkpoint, run training, deploy. There is a hidden danger. Run the same fine-tuning code thirteen times -- same data, same architecture, different random seed -- and twelve runs produce a robot succeeding 91--94% of the time, while one run silently degrades to 65.2%: a 29 pp gap with no error message, no warning, and no way to predict which seed will fail. We call this the seed lottery. We trace the cause to output collapse: the action predictor quietly learns to produce nearly identical outputs regardless of what the robot sees. Existing weight-level methods (L2, EWC) are structurally blind to this collapse -- they penalize weight changes, but collapse occurs in directions weights can move freely without affecting outputs, a gap we formalize via the Jacobian null-space. Across 7 methods x up to 13 seeds x 3 LIBERO benchmarks, three output-level regularizers -- VICReg (n=12 seeds), Dropout (n=4), and a halved learning rate (n=5) -- each eliminate every catastrophic seed (0/21 combined collapses vs. 1/13 Baseline; F(12,11)=28.7, p<0.001), while weight-level methods (L2, EWC) preserve the lottery. The simplest fix is changing one number in your optimizer config.

Moonshine is an autonomous agent whose central objective is to generate mathematical conjectures. Its core capability is to extract structure from classical problems, distill new concepts, and formulate conjectures of mathematical significance. Rather than treating the solution of a single proposition as its endpoint, Moonshine builds an extensible theoretical framework through conjecture generation, bridge building, and obstacle identification. This article uses Moonshine's exploration of the Jacobian conjecture as an example. It shows how the central logic of whether local nondegeneracy can force global injectivity is transferred to one-hidden-layer affine-ridge sigmoid networks. This leads to the formulation of the \emph{Neural Jacobian Conjecture} (NJC): if such a network has strictly positive Jacobian determinant on the whole space, then it must be globally injective. By invoking GPT-5.5-pro and DeepSeek-V4-pro separately, Moonshine obtained independent complete proofs for the case $N=n+1$. In addition, with the assistance of ChatGPT through interactive use of its web interface with GPT-5.5-pro, a geometric-topological proof was developed. These results provide preliminary evidence for the plausibility of the conjecture. The general higher-width case $N\ge n+2$, however, remains unresolved and is left for further investigation. This work illustrates Moonshine's ability to autonomously generate meaningful mathematical problems and make rigorous progress on them.

The NISQ era places stringent constraints on quantum computation, where noise and decoherence fundamentally limit performance. In classical deep learning, model robustness and resilience to perturbations are well studied: deep neural networks (DNNs) maintain high performance despite pruning, noise injection, and structural perturbations due to inherent redundancy in their representations. A central challenge in quantum machine learning is to transfer this notion of robustness to quantum neural networks (QNNs) under realistic NISQ noise. While classical deep learning exhibits robustness through structural redundancy, analogous principles for QNNs remain underdeveloped. We propose JGRA: a framework for assessing robustness in noise-aware QNNs via Jacobian geometry, capturing model sensitivity to parameter perturbations induced by noise. Our method includes entropy-matched noise calibration, noise-aware training, and noise-conditioned Jacobian extraction, yielding geometric descriptors that link clean-regime structure to noisy inference behaviour. We also empirically demonstrate that these descriptors encode predictive information about robustness under unseen noise.

We study the generalization of ridge-regularized nonlinear least-squares models via on-average algorithmic stability, deriving error bounds for local minimizers in terms of a data-dependent effective dimension that reflects the geometry of the gradient model at the trained parameters, through the empirical Jacobian Gram matrix and a residual-curvature term. In the linear case, where the curvature term vanishes, this recovers the classical effective dimension of the Jacobian kernel covariance, but evaluated at the trained model rather than at initialization as is typical in neural tangent kernel analyses. We further bound this effective dimension via covering complexity of the gradient features, leading to guarantees that depend on learned geometry rather than parameter count. In particular, for manifold-supported data and piecewise Lipschitz Jacobians, the bounds scale with intrinsic dimension, while for one-hidden-layer ReLU networks, the mechanism can be made explicit through counts of activation-stable regions. Experiments on synthetic manifolds, clustered distributions, and benchmark datasets illustrate trained-Jacobian compression, the tightness of the residual-curvature linearization, and agreement between the stability bound and observed generalization gaps. A key feature of our bounds is the simplicity of their derivation, which follows from first principles using the Brascamp-Lieb inequality under strongly log-concave noise.

Many important outcomes unfold as dynamic cascades, including product adoption, disease spread, financial distress, and information diffusion. A central challenge is to recover the hidden influence network behind these cascades. Existing methods typically assume a specific diffusion model, and their performance degrades substantially when that assumption is misspecified. We propose CascadeNet, a Jacobian-based machine learning framework for network recovery that does not require specifying a diffusion mechanism. The key idea is that the underlying influence structure can be characterized by the Jacobian of the one-step transition function. CascadeNet first constructs a flexible estimator of the transition function, and further applies Neyman-orthogonal debiasing via the Riesz representer, so that the debiased Jacobian is $\sqrt{n}$-consistent and asymptotically normal, enabling formal inference on the network structure. We validate CascadeNet in both a simulation exercise and a real-world empirical application. In simulations, where the data-generating process is known, CascadeNet achieves the highest network recovery accuracy across nine common data-generating processes. In an empirical application to COVID-19 transmission across Spain's 52 provinces, CascadeNet recovers transmission networks that are significantly correlated with the true inter-province mobility network, whereas networks recovered by baseline methods show no significant alignment with the ground truth.

Controllable generation with discrete diffusion models is often hindered by high computational overhead or the need for retraining. In this paper, we present \underline{\textbf{G}}radient-\underline{\textbf{I}}nformed \underline{\textbf{L}}ogit \underline{\textbf{C}}orrection (\textbf{GILC}), a plug-and-play framework that efficiently estimates guidance signals by repurposing the pretrained denoising network as a variational proxy. To circumvent the gradient instability inherent in high-dimensional discrete spaces, we introduce a Jacobian-free mechanism that directly corrects the clean prediction logits, facilitating stable and effective guidance. Our method accommodates both differentiable and non-differentiable reward functions. Extensive experiments across DNA, protein sequence, and molecular generation tasks demonstrate that GILC achieves state-of-the-art performance without additional training, frequently outperforming fine-tuning approaches.

Whilst the vulnerability of graph neural networks (GNNs) to adversarial attacks poses a critical threat to graph representation learning, the understanding of the robust generalization behavior remains a fundamental challenge in the adversarial setting. Recently, PAC-Bayesian margin-based generalization analysis substantially advances this line of research by providing a flexible and data-dependent analytical framework. However, existing robust analyses often rely on isotropic Gaussian posteriors and control weight perturbations in the full parameter space, which limits the ability to capture heterogeneous parameter sensitivity yet hinges on hidden-width-dependent complexity terms, resulting in not-tight-enough generalization bounds. In this paper, we extend a recently proposed sensitivity-aware PAC-Bayesian framework from deep neural networks to message passing GNNs (MPGNNs) and derive a tighter robust generalization bound in the adversarial setting. Specifically, we first quantify how sensitive the perturbations across different parameter blocks are to the network outputs by deriving the output Jacobians with respect to the weight parameters. Exploiting the fact that these Jacobian matrices have rank at most $K$ in $K$-class graph classification, we then construct Jacobian-aligned sensitivity matrices and use anisotropic Gaussian posteriors with optimized covariances to upper bound the KL divergence in a tight way. Notably, by refining the spectral-norm dependence on the learned weights and reducing the leading dimension factor from hidden-width-dependent terms to the number of classes $K$, our analysis yields much tighter robust generalization guarantees for MPGNNs, thereby guiding their designs to enhance adversarial robustness.

Coupled gradient descent - where the update of one parameter depends on another - arises naturally in bilevel optimization, two-time-scale stochastic approximation, and generative adversarial networks. When the coupled Jacobian is block-triangular, asymptotic stability is determined by the spectral radii of the diagonal blocks, yet transient amplification before convergence can be arbitrarily large due to non-normality. We develop a sharp pseudospectral theory for block-triangular Jacobians J = [[A, 0], [C, D]], proving Kreiss-constant bounds of the form K(J) <= 2/(1-γ) + ||C||/(4(1-γ)) when ρ(A), ρ(D) <= γ< 1 and A, D are symmetric, and establishing matching minimax lower bounds. We characterize the critical coupling threshold for spectral instability and extend the theory to nearly self-referential systems via a Neumann-series perturbation framework. As a consequence, we obtain a finite-horizon O(K(J)^2 log(1/δ)) iteration complexity bound. Framed as scaling laws for stochastic two-time-scale optimization, our results expose a non-asymptotic, instance-dependent regime of high-dimensional learning dynamics that is invisible to spectral-radius analysis. Experiments on linear-quadratic problems, IQC-based comparisons, and neural-network training confirm the theory.

Existing evaluations of neural operators and in-context operator learning rely primarily on prediction error, but accurate output prediction does not guarantee the correct local dynamical structure. A model may match solutions while exhibiting incorrect sensitivities, distorted frequency response, spurious mode coupling, or unstable tangent behavior. We introduce a Jacobian-based spectral audit for in-context operator learning. For a fixed prompt, we differentiate the network output with respect to the query function and view the resulting Jacobian as a learned tangent operator. Projecting it onto Fourier modes, we obtain a local spectral characterization of the inferred operator, including frequency-dependent gains, phase structure, and cross-mode coupling. The audit complements standard prediction metrics by testing whether the model reproduces local mechanisms of the underlying PDE operator rather than only outputs. Across benchmarks, the audit reveals distinct operator-level phenomena, including phase transport, viscosity-dependent damping, nonlinear mode coupling, and reaction--diffusion stability structure. It also detects failures partially hidden by prediction-error metrics, including high-frequency degradation, incorrect phase recovery, and prompt--operator inconsistencies. Corrupted or internally inconsistent prompts lead to degraded tangent-operator structure even when pointwise predictions remain partially accurate. Our results suggest that prediction accuracy and local operator fidelity are distinct properties of learned neural operators. Our framework also provides a diagnostic for stability, sensitivity, and operator consistency.

Learning chaotic dynamical systems from data requires more than short-term predictive accuracy: the learned model must preserve the attractor geometry and its invariant statistics. Trajectory (zero-order) and Jacobian (first-order) matching supervise the values and tangent structure of the vector field, but neither constrains how the field bends away from its tangent plane. A model can thus match values and tangents at the supervised states yet curve differently from the truth, remaining locally accurate while drifting toward spurious attractors and distorting long-time statistics. We show that enforcing second-order consistency mitigates these failures, but forming the full Hessian is prohibitive in high dimensions. We propose model-constrained randomized Jacobian matching, which compares the Jacobians of the true and learned vector fields at randomly perturbed inputs. A Taylor expansion shows that the expected randomized Jacobian loss decomposes into the nominal Jacobian mismatch plus a Hessian mismatch scaled by the noise variance, implicitly enforcing second-order consistency at $\mathcal{O}(d^2)$ cost without forming the $\mathcal{O}(d^3)$ Hessian tensor. Using only Jacobian evaluations, the method scales to high dimensions where explicit Hessian matching does not. Numerical experiments confirm that second-order methods are robust. For Lorenz63, first-order methods produce catastrophic Lyapunov-exponent outliers under minimal temporal supervision, which second-order methods eliminate while recovering the correct attractor. For coupled Lorenz96, an out-of-distribution forcing sweep separates the methods: all agree up to $F=16$, but beyond $F=18$ only second-order methods preserve the invariant measure and Lyapunov spectrum. On both systems, randomized Jacobian matching performs comparably to explicit Hessian matching at much lower cost.

Dynamical systems reconstruction (DSR) aims to learn surrogate models that capture the dynamics underlying time-series data. Reliably deploying these surrogates requires uncertainty estimates consistent with the learned dynamics. We expose a dynamic-probabilistic consistency (DPC) gap: the pursuit of finite-horizon probabilistic objectives can degrade dynamics or decouple predictive uncertainty from the local tangent dynamics it ought to reflect. We isolate three mechanisms behind this gap: core collapse, noise masking, and blind uncertainty. Specifically, we show that open-loop Gaussian rollout objectives can penalize Jacobian-generated covariance growth in chaotic systems, encouraging optimization shortcuts that weaken physical expansion or decouple uncertainty from it. To mitigate this gap, we propose KAFFEE (Kalman-Aware Framework For Ergodic Emulation), a differentiable extended Kalman filter-based training framework that evaluates likelihood on local predictive residuals (innovations) while transporting covariance through learned local Jacobians. On stochastic hyperchaotic Lorenz-96, KAFFEE reduces the identified failure modes, improves reconstruction of dynamical invariants relative to open-loop objectives, and maintains competitive predictive scores. We further show that the DPC gap appears when probabilistically adapting a DSR foundation model across 13 chaotic systems, where KAFFEE enables in-context Bayesian filtering while largely preserving zero-shot dynamics.

Inverse Kinematics (IK) plays a critical role in robotic motion planning and control. The IK solutions of a robot manipulator could be done by conventional ways such as geometric, algebraic, or Jacobian methods, which have drawbacks. The Artificial Neural Networks (ANNs) have become a promising alternative for approximating IK solutions due to their generalization ability and computational efficiency. This approach basically trains only a few samples of the end effector that are recorded for the solution of the IK problem. However, a fundamental question remains: how many training samples are sufficient to achieve reliable and accurate IK predictions? This study investigates the mathematical framework of relating the size of training datasets and the accuracy of ANN-based IK solvers. Using an articulated robotic manipulator, we generate varying amounts of joint-position pairs to train feedforward neural networks and assess their accuracy, convergence, and generalization capability. The results reveal more training samples than 125 did not contribute to the improvement of the model efficiency that the comparable measure dealing with the approximation accuracy over the sampling size, offering valuable insight into data efficiency. This work provides practical guidance for optimizing the data sizing of ANN solutions, balancing computational cost and model accuracy for real-world robotic applications.

Message-passing neural networks (MPNNs) often suffer from an information bottleneck when capturing long-range dependencies, leading to the oversquashing (OSQ) phenomenon. Alongside spatial connectivity enrichment (e.g., rewiring), recent studies have shown that spectral filtering can yield strong long-range learning outcomes, as spectral operators enable global information mixing that alleviates OSQ. These approaches achieve this either by stabilizing the Jacobian energies in deep propagation or by guaranteeing OSQ mitigation under strong theoretical assumptions. We revisit these conclusions and show that the associated Jacobian sensitivity lower bound is generally difficult to achieve in practice. We then propose S$^3$GNN, which mitigates OSQ without such restrictive assumptions by lightweightly reintroducing omitted components with substantially lower computational complexity, while standard stability constraints on feature transformations remain effective under our new dynamics. Extensive experiments across diverse domains (e.g., long-range benchmarks, KGQA, and mesh-based fluid dynamics) demonstrate that S$^3$GNN achieves up to an order-of-magnitude error reduction with up to 50% fewer parameters. Our code can be found in https://github.com/EEthanShi/S3-GNN.git.

JEPAs often regularize one-view embeddings toward an isotropic Gaussian, implicitly baking Euclidean symmetry into the representation. We show that this is not merely a benign default. For a known structured downstream geometry $H\succ0$, the minimax and maximum-entropy covariance under a Hamiltonian energy budget is $(c/d)H^{-1}$, and Euclidean isotropy incurs a closed-form price of isotropy. More importantly, when the downstream geometry is unknown, no geometry-independent fixed marginal target is canonical: every fixed covariance shape can be maximally misaligned for some structured geometry. We further show that even oracle one-view marginals do not identify the JEPA view-to-view predictive coupling. These results suggest that the structural bias in JEPAs should enter the cross-view coupling rather than a fixed encoder marginal. We instantiate this principle with \textbf{HamJEPA}, which encodes each view as a phase-space state $(q,p)$ and predicts view-to-view transitions with a learned Hamiltonian leapfrog map, while non-isotropic scale and spectral floors prevent collapse. In a deliberately headless token protocol, HamJEPA improves over SIGReg on CIFAR-100 by $+4.89$ kNN@20 and $+3.52$ linear-probe points at 30 epochs, and by $+6.45$ kNN@20 and $+10.64$ linear-probe points at 80 epochs, while a matched MLP predictor ablation shows that the symplectic coupling is the ingredient driving the neighborhood-geometry gain. On ImageNet-100, HamJEPA-$q$ improves by $+4.82$ kNN@20 and $+7.52$ linear-probe points at 45 epochs.

Generative analysis often models multi-domain observations as nonlinear mixtures of domain-invariant content variables and domain-specific style variables. Identifying both factors from unpaired domains enables tasks such as domain transfer and counterfactual data generation. Prior work establishes identifiability under (block-wise) statistical independence between content and style, or via sparse Jacobian assumptions on the nonlinear mixing function, but such conditions can be restrictive in practice. In this work, we introduce content-style differential independence (CSDI), an alternative structural condition requiring that infinitesimal variations in content and style induce orthogonal directions on the data manifold, thereby enabling identifiability even when content and style are dependent and the Jacobian is dense. We operationalize this condition through a blockwise orthogonality constraint on the Jacobian subspaces associated with content and style. To support high-dimensional generative models, we design a stochastic regularizer based on numerical Jacobian approximation, enabling scalable training in settings such as high-resolution image generation. Experiments across multiple datasets corroborate the identifiability analysis and demonstrate practical benefits on counterfactual generation and domain translation.

This paper presents a novel framework for Jacobian computation in motion optimization problems involving multi-link systems, where physical quantities are represented using higher-order time derivatives. In motion optimization of robots and humans, cost functions may incorporate higher-order time derivatives, such as jerk or the time variation of forces, to capture smoothness and perceptual characteristics, particularly in motion skill analysis and expressive behaviors, thereby necessitating Jacobian computations involving these quantities. However, such Jacobians are typically computed using numerical or automatic differentiation without explicitly exploiting the underlying multi-link structure, which can lead to increased computational cost and numerical instability. To address this limitation, we propose a structured Jacobian formulation for motion optimization, based on the comprehensive motion computation framework, in which physical quantities and their higher-order time derivatives are systematically represented along the multi-link structure. The proposed method systematically derives analytical expressions for Jacobians of kinematic and dynamic quantities, including momentum, forces, and joint torques, with respect to generalized coordinates and their higher-order derivatives. The resulting framework is applicable to both direct and inverse optimization. Through numerical experiments, we demonstrate that the proposed method improves computational efficiency compared to numerical and automatic differentiation, while achieving comparable accuracy. Furthermore, we demonstrate its effectiveness in inverse optimization by recovering cost function weights from motion data. Together, these results indicate that the proposed formulation provides a scalable and structured computational foundation for motion optimization involving higher-order time derivatives in multi-link systems.

Chaotic systems pose fundamental challenges for data-driven dynamics discovery, as small modeling errors lead to exponentially growing trajectory discrepancies. Since exact long-term prediction is unattainable, it is natural to ask what a good surrogate model for chaotic dynamics is. Prior work has largely focused either on reproducing the Jacobian of the underlying dynamics, which governs local expansion and contraction rates, or on training surrogate models that reproduce the ground-truth dynamics' long-term statistical behavior. In this work, we propose a new framework that aims to bridge these two paradigms by training surrogate dynamics models with accurate Jacobians and long-term statistical properties. Our method constructs a local covering of a chaotic attractor in phase space and analyzes the expansion and contraction of these coverings under the dynamics. The surrogate model is trained by minimizing the maximum mean discrepancy between the pushforward distributions of the coverings under the surrogate and ground-truth dynamics. Experiments show that our method significantly improves Jacobian accuracy while remaining competitive with state-of-the-art statistically accurate dynamics learning methods. Our code is fully available at https://anonymous.4open.science/r/neighborwatch.

bde is a user-friendly Python package for Bayesian Deep Ensembles with a particular focus on tabular data. Built on an efficient JAX implementation of the sampling-based inference method Microcanonical Langevin Ensembles (MILE), it provides scikit-learn compatible estimators for fast training, efficient Markov Chain Monte Carlo sampling, and uncertainty quantification in both regression and classification tasks.

Nonlinear inverse problems often trade inexpensive but fragile first-order updates against curvature-aware methods such as Gauss-Newton and Levenberg-Marquardt, which obtain stronger directions by repeatedly solving Jacobian-based linearized systems. We propose a learned alternative: amortize local inverse geometry into a reusable reverse operator. Our framework learns a bidirectional surrogate, Deceptron, and deploys it through D-IPG (Deceptron Inverse-Preconditioned Gradient), an iterative solver that pulls residual-corrected measurement-space proposals back to latent space. The key mechanism is a Jacobian Composition Penalty (JCP), which trains the reverse Jacobian to act as a local left inverse of the forward Jacobian; its runtime counterpart, RJCP, measures the same inverse-consistency error along optimization trajectories. We prove that D-IPG is first-order equivalent to damped Gauss-Newton under local pseudoinverse consistency, with deviation controlled by composition error and conditioning. Across seven PDE inverse-problem benchmarks, D-IPG outperforms standard baselines, achieves 94.8% mean success across the six-problem reliability suite, and reaches comparable or better recovery quality at up to 77x lower inference-time solve cost on the main benchmarks.

Training a deep neural network with the outputs of selected layers satisfying linear constraints is required in many contemporary data-driven applications. While this can be achieved by incorporating projection layers into the neural network, its end-to-end training remains challenging due to the lack of rigorous theory and efficient algorithms for backpropagation. A key difficulty in developing the theory and efficient algorithms for backpropagation arose from the nonsmoothness of the solution mapping of the projection layer. To address this bottleneck, we introduce an efficiently computable HS-Jacobian to the projection layer. Importantly, we prove that the HS-Jacobian is a conservative mapping for the projection operator onto the polyhedral set, enabling its seamless integration into the nonsmooth automatic differentiation framework for backpropagation. Therefore, many efficient algorithms, such as Adam, can be applied for end-to-end training of deep neural networks with linear constraints. Particularly, we establish convergence guarantees of the HS-Jacobian based Adam algorithm for training linearly constrained deep neural networks. Extensive experiment results on several important applications, including finance, computer vision, and network architecture design, demonstrate the superior performance of our method compared to other existing popular methods.

Linear attention reduces the quadratic cost of softmax attention to $\mathcal{O}(T)$, but its memory state grows as $\mathcal{O}(T)$ in Frobenius norm, causing progressive interference between stored associations. We introduce \textbf{Variational Linear Attention} (VLA), which reframes the memory update as an online regularised least-squares problem with an adaptive penalty matrix maintained via the Sherman-Morrison rank-1 formula. We prove that normalising the write direction to unit length gives the recurrence Jacobian spectral norm exactly $1$ for all sequence lengths and head dimensions (Proposition 2), and that the state norm is self-limiting under bounded inputs (Proposition 1). Empirically, VLA reduces $\|S_t\|_F$ by $109\times$ relative to standard linear attention at $T{=}1{,}000$, achieves near-perfect exact-match accuracy on multi-query associative recall within the effective per-head memory regime ($n_\text{pairs} < d_h$), maintaining substantially higher retrieval performance than DeltaNet and standard linear attention under increasing memory load, and maintains 62% accuracy at the per-head capacity boundary. A Triton-fused kernel achieves $14\times$ speedup over sequential Python and $\mathcal{O}(T)$ scaling, crossing below softmax attention latency at approximately 43,000 tokens.

Neural warm starts can sharply reduce the number of Newton-Raphson iterations required to solve the AC power flow problem, but existing supervised approaches generalize poorly on heavily loaded instances near voltage collapse. We prove a lower bound on the Newton-Raphson iteration count that depends on the direction of the warm start error rather than on its magnitude, and show as a corollary that the bound becomes vacuous as the smallest singular value of the power-flow Jacobian shrinks, identifying the failure mode of supervised regression near the saddle-node bifurcation. Motivated by this analysis, we introduce Newton's Lantern, a finetuning pipeline that combines group relative policy optimization with a learned reward model trained on perturbations of the base model's predictions, using the iteration count itself as the supervisory signal. Across IEEE 118-bus, GOC 500-bus, and GOC 2000-bus benchmarks, Newton's Lantern is the only method that converges on every test snapshot while attaining the smallest mean iteration count.

jNO (jax Neural Operators) is a JAX-native library for neural operators and foundation models with unified support for both data-driven and physics-informed training. Its core design is a tracing system in which domains, model calls, residuals, supervised losses, and diagnostics are written in one symbolic language and compiled into one optimization pipeline. This allows users to move between operator regression, mesh-aware residual evaluation, and PDE-constrained training without restructuring the surrounding code. jNO also supports multi-model compositions, fine-grained control at parameter level (model, optimizer, and learning rate), hyperparameter tuning, and JAX-native workflows for translated PDE foundation-model families. The source repository is available at https://github.com/FhG-IISB/jNO.

Autoregressive neural simulators now match classical solvers on short-horizon prediction of physical systems, yet their accuracy degrades rapidly when rolled out over long horizons. In this work, we identify transient amplification of perturbations around rollout trajectories as a structural mechanism driving rollout error. Using a linearization analysis we show that when the Jacobians along an autoregressive trajectory are non-normal and non-commuting, the model amplifies errors transiently, resulting in model rollout drift even when the overall system is asymptotically stable. Building on the analysis, we propose commutativity regularization: a combination of two penalties designed to reduce the normality defect of individual Jacobians and the commutator norm of Jacobians across steps. The penalties are estimated with Jacobian-vector products and have no inference-time cost. We show a propagator bound that quantifies rollout error under approximate commutativity and normality. We evaluate UNet and FNO variants with commutativity regularization on 1D and 2D spatio-temporal data in synthetic and real settings, showing successful long-horizon rollouts over thousands of steps. Further, we show that the method improves FourCastNet climate forecasts on ERA5 without using any new data. The gain is most pronounced out-of-distribution: trained on trajectories of a few hundred steps, regularized models remain in-distribution for thousands of rollout steps on initial conditions where baselines diverge.

Low-Rank Adaptation (LoRA) reparameterizes a weight update as a product of two low-rank factors, but the Jacobian $J_{G}$ of the generator mapping the factors to the weight matrix is rank-deficient, so the factor-space preconditioner $J_{G}^* {F}_t J_{G}$ induced by any ${W}$-space preconditioner ${F}_t$ is singular, and consequently the standard chain rule cannot be uniquely inverted to map a preconditioned ${W}$-space direction back to a factor-space update. We cast existing LoRA optimizers in a unified framework parameterized by two choices: (i) which invertible surrogate for $J_{G}^* {F}_t J_{G}$ to use, and (ii) which ${F}_t$ on ${W}$ to use. Existing methods occupy four families along these axes: factor-space adaptive updates, block-diagonal surrogates for $J_{G}^* J_{G}$, Frobenius-residual pseudoinverse methods, and Riemannian manifold constraint. Within this design space, a gradient-statistics-aware ${F}_t$ paired with a closed-form factor-space solve at ${O}((m+n)r)$ memory remains underexplored. We propose \textbf{AdaPreLoRA}, which fills this gap by adopting the Adafactor diagonal Kronecker preconditioner ${H}_t$ on ${W}$ and selecting from the resulting factor-space solution family the element minimizing an ${H}_t$-weighted imbalance between the two factor contributions; by construction, the resulting factor update is the closest LoRA approximation to the preconditioned ${W}$-space direction under the ${H}_t$-weighted norm. Across GPT-2 (E2E), Mistral-7B and Qwen2-7B (GLUE, ARC, GSM8K), and diffusion-model personalization, AdaPreLoRA is competitive with or improves over a representative set of LoRA optimizers while keeping peak GPU memory at the LoRA optimizer level.

Modern deep learning treats neural networks primarily as endpoint functions from inputs to outputs. Inspired by the shift from force to geometry in physics, we ask whether a network should instead be understood through the geometry of its internal propagation. We define a neural propagation field as the collection of hidden-state trajectories and local Jacobian operators across depth. Endpoint losses constrain only the boundary behavior of this field, leaving its interior geometry underdetermined. We show that endpoint-equivalent models can differ by orders of magnitude in trajectory and Jacobian structure, and introduce observable field metrics such as path sensitivity, solver consistency, and trajectory/Jacobian retention. In controlled teacher-flow and PDE systems, endpoint fitting fails to recover the underlying propagation law. In real multi-path tasks, field-aware objectives improve unseen-path generalization, OOD robustness, and calibration when aligned with the observation structure, but can collapse when over-constrained. In continual learning, field-preservation regularization complements replay and distillation: on Split CIFAR-100, DER++ with field preservation improves average accuracy, backward transfer, and field-retention metrics. These results identify propagation-field quality as a measurable and trainable property of neural networks beyond endpoint performance.

Formal verification of neural control barrier functions (NCBFs) remains challenging, especially for neural networks with nonlinear activations like $\tanh$. Existing CROWN-based methods rely on conservative linear relaxations for Jacobian bounds, limiting scalability. We propose LightCROWN, which computes tighter Jacobian bounds by exploiting the analytical properties of activation functions. Experiments on nonlinear control systems including the inverted pendulum, Dubins car, and planar quadrotor demonstrate that LightCROWN improves verification success rates up to 100%, while enhancing speed and scalability. Our approach provides a generalizable improvement for CROWN-based frameworks, enabling more efficient verification of complex NCBFs. The code can be found at github.com/Autonomous-Systems-and-Control-Lab/verify-neural-CBF.

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

Cocoa (Theobroma cacao) is a critical cash crop for millions of smallholder farmers in West Africa, where Cocoa Swollen Shoot Virus Disease (CSSVD) and anthracnose cause devastating yield losses. Automated disease detection from leaf images is essential for early intervention, yet deploying such systems in resource-constrained settings demands models that are small, fast, and require no internet connectivity. Existing edge-deployable plant disease systems rely on end-to-end deep learning without uncertainty quantification, while Bayesian methods for edge devices focus on hardware-level inference architectures rather than agricultural applications. We bridge this gap with TinyBayes, the first framework to combine a closed-form Bayesian classifier with a mobile-grade computer vision pipeline for crop disease detection. Our pipeline uses YOLOv8-Nano (5.9 MB) for lesion localisation, MobileNetV3-Small (3.5 MB) for feature extraction, and the Jacobi prior; a Bayesian method that provides a closed form non-iterative estimators via projection, for the classification. The Jacobi-DMR (Distributed Multinomial Regression) classifier adds only 13.5 KB to the pipeline, bringing the total model size within 9.5 MB, while achieving 78.7% accuracy on the Amini Cocoa Contamination Challenge dataset and enabling end-to-end CPU inference under 150 ms per image. We benchmark against seven classifiers including Random Forest, SVM, Ridge, Lasso, Elastic Net, XGBoost, and Jacobi-GP, and demonstrate that the Jacobi-DMR offers the best trade-off between accuracy, model size, and inference speed for edge deployment. We have proved the asymptotic equivalence and consistency, asymptotic normality and the bias correction of Jacobi-DMR. All data and codes are available here: https://github.com/shouvik-sardar/TinyBayes

Augmented Lagrangian and optimistic primal--dual methods stabilize equality-constrained optimization through seemingly different mechanisms: the former adds constraint-dependent primal curvature, while the latter adds dual memory. Recent work has shown that these mechanisms are equivalent for scalar parameters. We extend this equivalence to matrix-valued correction. We prove an additivity principle: for symmetric matrix parameters, the ideal primal trajectory depends only on the summed correction matrix, not on how it is split between augmented and optimistic channels. This exposes a design freedom: algebraically equivalent decompositions can have different finite-step feasibility because augmented correction affects primal curvature, whereas optimistic correction affects the scale of the dual memory correction. We formulate the resulting step-size-limited design problem and derive a closed-form hybrid rule that selects a matrix correction, splits it between the two channels, and chooses primal and dual steps using local spectral weights. Experiments on nonlinear equality-constrained problems with controlled constraint-Jacobian conditioning show that the hybrid design improves over pure augmented and pure optimistic endpoints, closely tracks a grid-search hybrid oracle, and is competitive with first-order primal--dual baselines under mild-to-moderate ill-conditioning. The experiments also identify the expected limitation: exact cancellation requires increasingly large matrix corrections as the constraint Jacobian becomes ill-conditioned.

Variational quantum circuits with angle encoding implement truncated Fourier series, and architectures arranging $N$ qubits with $L$ encoding layers each -- sharing encoding budget $E = NL$ -- generate identical frequency spectra, identical frequency redundancy, and require the same minimum parameter count for coefficient control. Despite this equivalence, trainability varies substantially with architecture shape $(N,L)$ at fixed $E$. We identify structural rank deficiency of the coefficient matching Jacobian $J$ as the mechanism responsible. For serial single-qubit architectures, we prove $\mathrm{rank}(J) \leq 2L+1$ regardless of parameter count $P$, with $\dim(\ker J) \geq P-(2L+1)$ growing without bound -- a phenomenon we term \emph{structural gradient starvation}: a growing fraction of parameters become structurally decoupled from the loss as $P$ increases at fixed $L$. Parallel architectures avoid this via independent phase trajectories, ensuring $σ_{\min}(J^{(\mathrm{par})}) > 0$ generically for $P \leq 2E+1$, so no parameter lies in $\ker J$. For practitioners, we further show that the two natural routes to increasing parameter count have fundamentally different effects: adding feature map (FM) layers monotonically strengthens the Jacobian QFIM eigenvalue spectrum and achieves $R^2 \geq 0.95$ with $1.6$--$2.2\times$ fewer parameters than adding trainable blocks across all tested architectures, while trainable blocks improve training only through the classical interpolation mechanism with no quantum-specific benefit.