Curvature

Diffusion and flow-matching models are typically trained by corrupting data through independently sampled Gaussian noise. While simple and scalable, this forward process induces arbitrary data-noise couplings, forcing the network to learn high-curvature transports between unrelated endpoints. Existing optimal-transport methods reduce this burden by reassigning fixed noise samples to data, but the source noise distribution itself remains passive. To address this, we introduce Contrastive Noise Alignment (CNA), a training-time method that creates dynamic, contrastive couplings by optimizing the noise representations directly. By modeling the noise batch as an interacting particle system, CNA employs a cross-modal InfoNCE objective to align noise particles with their paired data targets. To prevent spatial collapse, this alignment is regularized using an angular entropy term and a radial norm penalty. We show theoretically that this equilibrium asymptotically preserves Gaussian structures, maintaining tractability during inference. Empirically, CNA improves the alignment between noise and data, reduces flow curvature, and provides better generation quality with fewer required sampling steps. For few-step, pixel-space generation (2-4 NFEs), CNA reduces FID by over 50% compared to standard rectified flow, and by at least 24% against Optimal Transport baselines.

Looped Transformers offer a parameter-efficient route to test-time scaling by reusing shared layers for iterative latent reasoning. However, additional iterations can reduce support for a reference answer, leaving unclear whether an update's direction is locally unhelpful or its full displacement moves too far. We study this distinction by analysing reference utility, which measures this support, along the model's own update direction, varying the fraction of the proposed displacement supplied to the readout. This reveals finite-step failures in which a locally improving direction produces a harmful full update. A pathwise curvature decomposition characterises how initial progress is lost, while a local quadratic model predicts full-step gains and useful step scales. Bounds based on accumulated curvature variation characterise the approximation error of these predictions. Experiments across two model families reveal this separation on mathematical and commonsense tasks. A fixed quarter step produces positive gains in reference utility for 72.2--83.2% of selected failures across four settings. These findings identify a mismatch between update direction and step scale as a mechanism of lost progress, explaining how some harmful updates retain useful computation.

Mixed-curvature representation learning seeks to capture rich geometric structures that cannot be adequately modeled by a single curvature regime. Existing approaches largely rely on product manifolds, which require manually specifying how different curvature spaces are combined and separate their curvature contributions across factors. We introduce the $\mathbb{SL}(n)$ space, a representation geometry defined by the simple $\det(A)=1$ constraint and a left invariant Schatten-$p$ Finsler structure. Despite this minimal construction, $\mathbb{SL}(n)$ exhibits pointwise negative, zero, and positive flag curvature around a common flagpole, while its mixed-curvature and curvature-coupling capacities are asymptotically maximal relative to the intrinsic geometric upper bound. Beyond geometry, its noncommutative group structure provides inherent order sensitivity, and its non-nilpotent Lie algebra admits nonzero nested Lie brackets at arbitrary depth, enabling deep order-aware composition. Empirically, $\mathbb{SL}(n)$ consistently outperforms a broad range of representation manifold baselines across graph benchmarks at different scales. It reduces average distortion over the strongest baselines by $44.3\%$ on KEGG and $40.5\%$ on HumanCyc, and improves Hits@20 by $42.8\%$ on OGBL-PPA. Experiments on Flickr30k-Order further support its ability to capture higher order dependencies from ordered composition. Together, these results show how a seemingly simple structural constraint can yield unexpectedly rich geometry, capacity, and composition within a unified representation space.

We study the classical Moreau--Yosida unadjusted Langevin algorithm (MYULA) for $\pi(\,\mathrm{d} x)\propto e^{-f(x)-g(x)}\,\mathrm{d} x$, where $f\in C^2(\mathbb{R}^d)$ is $m$-strongly convex with $L_f$-Lipschitz gradient and $g:\mathbb{R}^d\to\mathbb{R}$ is convex and globally $G$-Lipschitz. For the Moreau-smoothed target $\pi_\lambda$ and the MYULA invariant law $\widehat\pi_{\lambda,h}$, we prove $$\sqrt m\,W_2(\pi_\lambda,\widehat\pi_{\lambda,h}) =O(h)+\widetilde O(h^{3/4})$$ under $0<h(L_f+\lambda^{-1})\le c$, with only logarithmic dependence on $\lambda^{-1}$ in the error coefficients. Combining this estimate with the Moreau approximation bias yields $\widetilde O(\varepsilon^{-4/3})$ iterations to achieve $\sqrt m\,W_2(\mu_N,\pi)\le\varepsilon$, for fixed model parameters and initialization. The proof combines a discrete Poisson corrector with active-trace estimates and a shared-noise bound for the exact--Euler two-point curvature.

Low-Rank Adaptation (LoRA) is an effective approach for adapting large pretrained models by learning low-rank weight updates. In practice, the LoRA rank is used to control an adapter's parameter budget and representational capacity. We show that this view is incomplete: while the nominal rank determines the representational capacity, the optimizer shapes how much of that capacity is used in the induced weight-space updates. In a case study of GPT-2 adaptation with LoRA, we observe a strong rank-dependent optimizer effect. Despite using the same nominal rank, AdamW often produces per-step updates with concentrated singular spectra and low effective rank, whereas Muon uses a richer set of directions and benefits more consistently from increasing LoRA rank. These observations motivate ISO-LoRA, an optimizer that couples the LoRA factor updates through spectral descent on the induced tangent perturbation in weight space. ISO-LoRA promotes updates that distribute energy more evenly across singular directions, improving rank utilization while preserving compatibility with the LoRA parameterization. We complement this design with theoretical guarantees showing that ISO-LoRA can achieve higher effective rank than standard factor-wise optimizers through a one-step analysis under a stylized spiked-gradient model. We validate this design on language-model adaptation across 0.1B-7B-parameter models, where ISO-LoRA improves effective rank and downstream performance, with the strongest gains at moderate-to-large LoRA ranks. Our results highlight rank utilization as a key factor in LoRA optimization and suggest that optimizer design offers an important path toward stronger parameter-efficient adaptation.

Recently, researchers have proposed a graph pooling operation, akin to the pooling process in conventional convolutional neural networks (CNN), aimed at reducing the computation cost of Graph convolutional neural networks (GCNNs). While most GCNN-based methods treat graph pooling as a node clustering problem and propose learning a cluster assignment matrix, existing clustering-based pooling methods tend to focus solely on the rough topology information of graphs, neglecting the exploitation of higher-order mutual connections among neighbors. In terms of message passing on graph, the ease of information passing on edges reflects the closeness between neighboring nodes, which significantly relies on the interconnectivity among neighbors. In this study, we address this gap by considering such local connection information and introducing a novel graph pooling method named RicciPool. We introduce discrete graph curvature, particularly Ollivier-Ricci curvature, as a measure of higher-order connectivity around an edge. Subsequently, we construct an Ollivier-Ricci flow formula to reweigh edge weights, leveraging the crucial information provided by Ricci curvature, particularly vital for extracting clusters in graphs. Building upon this foundation, we utilize the spectral clustering technique to learn a new cluster assignment matrix. Experimental results on multiple bioinformatics protein datasets and social networks underscore the effectiveness of our proposed method.

Divergence-based regularization and Sharpness-Aware Minimization (SAM) are two prominent approaches for improving generalization in deep learning, both motivated by robustness to perturbations. However, their relationship has remained largely unexplored. Building on classical second-order expansions of $f$-divergences, we show that the two methods are locally consistent under parameter-space perturbations: both induce curvature-sensitive penalties, with divergence regularization yielding a Fisher-weighted quadratic form and SAM penalizing sharpness through the dominant Hessian eigenvalue. For negative log-likelihood objectives with exponential-family output distributions, this correspondence becomes especially transparent, since the Fisher and Gauss-Newton matrices coincide. We further show that the same local geometric perspective extends to input-space perturbations, where divergence-based regularization is defined through transformations of the input. In this setting, the regularizer induces a pullback quadratic form on the input space, providing a more general perturbation framework than standard SAM while preserving the same local sensitivity interpretation. To validate the analysis empirically, we use the asymmetric $α$-skew Jensen-Shannon divergence (JSD) family as a controlled testbed. Its local curvature coefficient scales as $α(1-α)$ and is maximized at the symmetric point $α=\tfrac12$, which recovers the standard JSD. Loss-landscape visualizations in the input-perturbation regime show that stronger induced curvature penalization is associated with flatter local minima. Experiments on four benchmark datasets further demonstrate that both accuracy and negative log-likelihood are consistently best near this regime of maximal curvature penalization.

We study the local errors of classical machine-learning surrogate models, which approximate the time evolution of the one-dimensional viscous Burgers equation. Four models are compared on the same prediction task, using the spatial grid values directly: radial basis function (RBF) kernel ridge regression (KRR), linear Ridge, ExtraTrees, and Random Forests. Across all four models, the one-step residual, defined here as the true value minus the predicted value at each grid point, forms clear curved branches near predicted maxima and minima. A more detailed analysis of KRR shows that these errors are much more strongly related to the second spatial derivative, which measures local curvature, than to the first spatial derivative. Near a smooth extremum, predicted value and curvature form a local two-branch fold. Under our local curvature-based model of the residual, this fold predicts a leading-order near-parabolic relation between predicted value and residual. This geometric result motivates a direct test of the Burgers advection (transport) and diffusion (smoothing) terms. For KRR and Ridge, regression tests on held-out trajectories, a control that breaks the spatial alignment of the diffusion term, and a spectral test of high-frequency content are consistent with insufficient viscous smoothing at moderate and high viscosity. In this case, the surrogate retains more small-scale structure than the true future state. The same physical explanation is much weaker for the tree models. Finally, a correction that uses only predicted quantities reduces both one-step error and error during recursive rollout, where each prediction is used as the next input.

Muon can be interpreted as optimizing a linear local objective over a spectral-norm ball. This gives a matrix-sign update that preserves the singular directions of the gradient and assigns the same magnitude to all active singular modes. We ask whether these two properties remain optimal when local curvature is taken into account. To answer this question, we keep Muon's spectral-norm constraint unchanged and replace the linear local model with a quadratic one. We call the resulting method \emph{Quadratic Spectral Descent} (QSD). We show that curvature can change both the singular values and the singular directions of the optimal update. To make QSD practical, we approximate curvature with Kronecker-factored statistics and solve the constrained quadratic with a small number of Frank--Wolfe steps, each of which has a closed-form matrix-sign subproblem. We further provide an optimality certificate, a comparison with Muon under the same quadratic surrogate, and an $O(1/K)$ convergence rate for the inner solver. Experiments on GPT pre-training show that QSD consistently improves validation loss over Muon and recent Muon variants, and reduces wall-clock training time by up to $8.49\%$ at matched validation loss.

At a junction, a score field can reveal weighted tangent rays, yet these first-order quantities do not determine how individual branches bend or how their densities change away from the center. Recovering this missing information is necessary for describing local continuation beyond a single point, but finite observations must separate branchwise second-order effects while allowing error in the estimated center. We address this inverse problem using matched score queries at noise scales $σ$ and $λσ$. For a finite union of $C^{2,α}$ half-branches in $\mathbb{R}^D$, the normalized score has the expansion $F_σ=F_0+σG+O(σ^{1+α})$. Matched subtraction cancels the tangent contribution and exposes $G$, which depends linearly on branchwise curvature and log-density slope. Given tangent directions and weights on distinct rays, $G$ uniquely identifies all $sD$ branch parameters, and $sD$ scalar component observations are necessary. An $O(σ^2)$ center error introduces $D$ translation modes, leading to $(s+1)D$ observations under full-rank calibration, except for a translation-invariant full line. We also establish a perturbation bound and a conditional kernel-density-estimation rate. Experiments reproduce the predicted population and $N^{-1/5}$ trends and remain full rank up to $D=20$ with 16 supplied branches. In end-to-end tests for $D=3$--$5$, a known-count first-order frontend yields full rank in all 135 population systems and a median relative jet error of 0.132. With strong first-order error, matched responses reduce median parameter error by a factor of 49.4 relative to naive tangent subtraction.

Gradient flow is widely used as a continuous-time surrogate for gradient descent, but state convergence does not imply convergence of differentiated training maps in nonsmooth networks. We characterize the fixed-horizon, vanishing-step limit of exact automatic differentiation through hard-ReLU gradient descent. Under a stable finite itinerary of separated, same-direction transverse activation events, gradient-descent states converge at first order to the corresponding piecewise-smooth gradient flow, while the exact derivative of every nonresonant discrete program converges to an event-free regional propagator. The true flow derivative instead interleaves classical saltation matrices that encode event-time sensitivity. For globally convex objectives, any strict activation event prevents complete cancellation of these missing transfers. Moreover, minimal globally 1-strongly convex residual-ReLU risks can realize arbitrarily large reciprocal sensitivity gaps, subject to an explicit transversality-scale tradeoff, and a coupled strongly convex construction yields an open set on which the largest initialization-gradient coordinate is reversed. In a controlled 17-parameter ReLU MLP, state and regional-AD errors vanish under mesh refinement while AD-to-flow errors remain between 0.18 and 0.39; an event-aware corrected product restores convergence. Resolved smoothing likewise recovers the flow sensitivity when the transition layer is sufficiently resolved. These results show that the gradient-flow limit of hard-ReLU training need not remain valid after differentiation.

LLM agents need to sustain goal-consistent reasoning across long multi-turn interactions under strict resource constraints. However, as the multi-turn context accumulates, it can destabilize the underlying LLM's internal representation of task-relevant information from earlier turns, blurring the boundary between constructive reasoning and representation drift. We formulate multi-turn reasoning as a hidden-state trajectory of the underlying LLM that is characterized via two complementary signals: temporal curvature that captures the directional consistency of turn-to-turn updates, and variance slope which measures the expansion or contraction of the exploration space. Across four tasks and three underlying LLMs, we observed that these geometric signals distinguish between correct and incorrect episodes prior to completion. We further decompose each episode into three-action chains formed from four actions (Read, Write, Respond, Transfer) and show that separability is action-dependent, with different signals distinguishing various chain patterns. Our experiments demonstrate that trajectory geometry can identify critical turns in the reasoning process, increasing task success rates on $τ$-Bench from 24.1% to 39.6% while reducing token cost by 11.2%.

Probabilistic Circuits (PCs) are generative models that support exact inference and, unlike deep neural networks, admit an exact and tractable measure of loss-surface curvature: the trace of the Hessian of the log-likelihood. Recent work regularizes this trace globally to bias learning toward flatter, better generalizing optima. We show that treating sharpness as a global regularizer can be misspecified for PCs, whose curvature is inherently compositional. We prove that each sum node's contribution to the Hessian trace factorizes exactly into its circuit flow, which measures how heavily the node is used, and a local sharpness term determined by its output distribution. This decomposition provides insights into why global sharpness regularization is depth biased and can lead to underfitting. Building on it, we introduce an adaptive sharpness aware regularizer that penalizes nodes based on intrinsic local curvature and preserves closed form EM updates. We also show that empirically, this targeted regularization recovers the generalization that global regularization sacrifices while retaining the robustness and benefits of sharpness aware learning.

When a reinforcement learning agent cannot observe the full state, we usually blame its policies: it cannot see enough to represent a good one. We show that in a solvable case the bigger problem lies elsewhere. Even when a good policy is available and the agent's value function is expressive enough to describe it exactly, learning still ends up somewhere far worse. We study a partially observed linear-quadratic problem in which a standard actor-critic learner can be solved in closed form. At our default setting the best policy the agent can represent is already close to optimal, costing 10.4% more than the ideal controller that observes everything. Learning does not find it. The algorithm instead comes to rest at a policy that is 35% worse than the best one available to it, and we can say exactly where and why. The cause is a bias in what the critic learns rather than a limit on what the actor can express. Because the agent cannot attribute what it sees to the part of the state it cannot observe, the critic misreads that unexplained variation as sharp curvature in its own value estimates, and the actor follows that error away from the optimum. We derive closed-form expressions for the resulting policy, for its cost, and for the one design choice that removes the problem, which is how far the learner looks ahead before trusting its own value estimates. Deep reinforcement learning experiments follow these predictions closely. Notably, giving the agent memory of past observations does not help, while changing how far it looks ahead does.

Temporal graph learning has become essential for analyzing real-world systems whose interactions continuously evolve over time, including financial transaction networks, communication systems, and online social platforms. However, learning from large-scale temporal graphs remains computationally challenging when networks are dense and rapidly changing. To address this limitation, we propose a network-curvature-inspired edge sparsification framework for dynamic graph learning. Our proposed method, TRicci, extends classical Forman-Ricci curvature to directed weighted temporal graphs by capturing structural support, temporal recency, and local interaction competition. Experiments on 9 transaction networks and 3 temporal graph benchmark datasets demonstrate that the proposed framework preserves predictive performance across multiple graph-level prediction tasks. The results show that TRicci sparsifies temporal graphs by approximately 80% while reducing end-to-end downstream training and inference time by an average of 55.94%, without substantial degradation in predictive performance. Our findings suggest that temporal curvature can serve as a principled basis for scalable temporal graph learning by preserving predictive temporal-structural information under substantial sparsification.

How can an analyst decide whether a nonlinear dimensionality reduction embedding can be trusted? Existing diagnostics provide only partial answers: projection glyphs characterize local sensitivity, map-continuity scores measure local conditioning, and transport-based analyses reveal path-dependent inconsistencies. However, these methods appear unrelated and provide no common framework for understanding when they agree or not. We show that they are all derived from a single geometric object induced by every differentiable embedding, whether defined implicitly through optimization or explicitly by a learned mapping. This framework provides two complementary geometric views of an embedding. The differential view explains local behavior: its first-order term recovers projection glyphs, while its second-order curvature quantifies how far their linear approximation remains reliable. The integral view follows the same geometry along high dimensional paths and determines whether an embedding depends only on the current state or also on the path taken to reach it. We further show that map-continuity is a prerequisite for the other analyses. The framework is theoretically complete for diagnostics derived from the embedding geometry, and we prove the integral view irreducible: no amount of local measurement at any number of points, to any order of derivative, reproduces what it detects. Classical rank-based metrics form a complementary class based on finite-scale neighborhood relationships. Experiments on synthetic and real datasets validate theoretical predictions, demonstrate accurate curvature-based trust estimates on single-cell embeddings, and show that the integral analysis distinguishes single-valued embeddings from path-dependent optimization-based embeddings in ways that existing pointwise diagnostics cannot.

Muon has recently emerged as a promising alternative to AdamW for language model pretraining by orthogonalizing momentum matrices using Newton-Schulz iterations. Although Muon mitigates gradient anisotropy, it does not explicitly account for the curvature geometry of the loss landscape and may therefore remain sensitive to curvature anisotropy. We bridge this gap by proposing MALT (Muon Augmented by Lightweight Two-sided Preconditioning), which uses lightweight diagonal preconditioners to reduce the sensitivity of Muon to curvature anisotropy. Specifically, MALT uses two-sided diagonal preconditioners with low memory and computational overhead to approximately capture the curvature geometry of the loss landscape. It orthogonalizes the preconditioned momentum using Newton-Schulz iterations and maps the result back to define the update direction, while norm grafting controls the update magnitude. To improve the robustness of MALT to stochastic gradient noise, we further propose MALTER (MALT with Adaptive stEpsize Rescaling). Convergence guarantees are provided for MALT in the stochastic non-convex setting. Experiments on GPT-2 Small, Medium, and Large pretraining show that the proposed methods outperform Muon while maintaining nearly the same memory footprint and wall-clock time.

Positive quadratic networks admit the low-rank representation f_U(x)=x^top UU^top x, where Uinmathbb{R}^{dtimes r} is identifiable only up to right orthogonal multiplication, representing a rank-r PSD matrix Q=UU^top. We study how this quotient structure governs training dynamics, curvature, recovery, and interpolation bias. On the full-column-rank stratum, we identify mathbb{R}^{dtimes r}_*/O(r) with the rank-r PSD manifold. For smooth objectives L(U)=ell(UU^top), the Euclidean factor gradient is horizontal. Thus, factor gradient flow projects exactly to quotient Riemannian gradient flow, while finite-step gradient descent induces an exact congruence recursion for the predictor. For quadratic regression, we derive the effective Hessian at interpolators as the empirical measurement Gram form restricted to the tangent space relative to the quotient metric. Under Gaussian rank-one measurements, we compute population curvature, prove uniform deviation bounds for the empirical normal operator, construct a spectral initializer, and establish local exponential convergence for gradient flow and linear convergence for small-step descent. Recovery guarantees are explicit but conservative due to reliance on full-space second-moment control. In underdetermined commuting regimes, factor gradient flow becomes an exact entropy mirror flow in joint spectral coordinates. Strictly positive initializations converge to Bregman projections onto the interpolation set. With isotropic initialization q(0)=varepsilon^2mathbf{1}, predictors approach the minimum-trace solution set as varepsilondownarrow0, resolving nonuniqueness via weighted entropy within the invariant joint spectral algebra. Finite-step descent selects interpolants differing from continuous-time Bregman projections by O(eta). Numerical experiments verify these quotient identities, curvature predictions, recovery behaviors, and selection laws.

Sampling-based methods offer a principled approach to uncertainty quantification in Bayesian neural networks. Their practical use, however, is often challenged by the computational cost of exploring high-dimensional and multimodal posterior distributions. To overcome these difficulties, Bayesian Deep Ensembles, i.e., warmstarting the sampling from several optimized solutions, have proven to be an effective strategy. In this paper, we demonstrate that curvature estimates computed during the warmstart as a byproduct in adaptive optimizers such as AdamW can inform the sampling phase at negligible additional cost. Specifically, our proposed preconditioned sampling strategy based on optimizer-derived geometries can substantially reduce or even eliminate the need for a lengthy sampling burn-in phase and leads to greater numerical stability. This approach consistently maintains or improves predictive performance and uncertainty quantification without any additional computational costs. We confirm the consistency of our findings across various datasets and network architectures.

High-curvature regions in 3D point clouds encapsulate critical fine-grained geometric semantics yet exhibit a distinct long-tail sparsity in their spatial distribution. The inherent limitations of polynomial volume growth in Euclidean space frequently render these intricate geometric features challenging to adequately resolve within a uniform-scale feature space. Consequently, these regions are frequently overshadowed by smooth global features dominated by low-curvature regions, thereby limiting the discriminative capacity of the network. To address this issue, we propose PointCHR, a curvature-aware hyperbolic rectification (CHR) for point cloud analysis. Utilising the property of exponential volume expansion in the vicinity of hyperbolic manifolds, CHR presents a learnable curvature-guided radial rectification mechanism. By adaptively projecting high-curvature points towards boundary regions endowed with larger effective embedding capacities, PointCHR effectively mitigates the representation crowding problem inherent in Euclidean settings. Extensive experimentation has demonstrated that PointCHR significantly enhances the ability of backbone to capture fine-grained geometric details, achieving state-of-the-art performance across multiple benchmarks.

A short fine-tuning run can undo the safety guards of an open-weight model---retraining a refusal-trained assistant to aid weapons development or produce hate speech. Preventing such harmful fine-tuning while retaining benign adaptability remains difficult: the only prior method with an explicit curvature certificate, spectral deformation, inflates curvature globally and thereby obstructs benign adaptation along with harmful adaptation. We propose HarmAlign, which applies function-preserving spectral deformation along a estimated contrastive activation subspace. We derive finite-sample bounds for the estimated subspace energy and the resulting local harmful-distribution curvature lower bound. A stability--progress dichotomy for constant-step gradient descent turns the certified curvature into conditional convergence-rate control. Empirically, within a fixed-architecture, finite-budget first-order threat model, HarmAlign blocks direct fine-tuning and three data- or objective-adaptive attacks across a hazardous-knowledge relearning setting and a harmful-assistance fine-tuning setting, while the protected benign tasks remain trainable. The block persists across the tested first-order optimizer variants over every attack checkpoint, and under out-of-distribution harmful fine-tuning, and it extends to important cases in our threat model: accidental safety degradation and emergent misalignment.

We study adaptive gradient descent for continuously differentiable, possibly nonconvex objectives under one-sided Hölder regularity. Unlike classical Hölder- or Lipschitz-gradient assumptions, which control the full gradient variation, our condition bounds only the directional term appearing in the descent inequality. This can allow less conservative step sizes when large gradient changes are orthogonal to, or favorable along, the update direction. We propose an adaptive scalar-step method based on an estimate of positive one-sided Hölder curvature, combined with a simple sufficient-decrease safeguard. For nonconvex objectives on a convex region containing the accepted update segments, we prove an explicit best-iterate stationarity bound with a rate determined by the Hölder exponent. Unlike predetermined diminishing step-size schemes, the method adapts to the local descent geometry. We evaluate the approach on two full-batch benchmarks designed to separate directional curvature from full gradient variation. On a binary classification problem, the method achieves the lowest final cross-entropy, objective value, and gradient norm, together with the largest classification margin among the compared scalar gradient methods. On a nonconvex Hölder regression problem, it attains the lowest final objective gap and gradient norm. These results indicate that one-sided Hölder curvature is an effective adaptive step-size signal when full-gradient variation is inflated by directions that do not hinder descent.

Curvature notions on graphs, particularly Ollivier-Ricci and Forman, have emerged as powerful tools for addressing fundamental issues in Graph Neural Networks (GNNs) such as oversmoothing and oversquashing, but rely almost exclusively on local edge-level comparisons and therefore fail to certify how information actually propagates over long distances. We introduce Entropic Curvature, a global, transport-based curvature obtained by extending the Lott-Sturm-Villani framework to graphs through the displacement convexity of entropy along Wasserstein geodesics. We define a tractable Weak Entropic Curvature proxy that lower-bounds the global entropic curvature, and from it derive (i) a Poincare-type inequality controlling oversmoothing, (ii) a transport-entropy generalization bound, and (iii) an expansion paradox proving that sparsity, strong spectral expansion, and positive entropic curvature cannot coexist in large graphs, unifying oversmoothing and oversquashing as opposite ends of a single curvature spectrum. We translate the theory into three practical mechanisms, the E-Gate aggregator, the ENT structural encoding, and Midpoint-Completion Rewiring (MCR), and benchmark them against SDRF, FoSR, BORF, LCP, and Graph Ricci Flow on six node-classification benchmarks, and graph-classification.

We study decentralized online optimization for strongly geodesically convex (strongly g-convex) losses on Riemannian manifolds with bounded sectional curvature, including positively curved manifolds. In centralized Riemannian optimization, strong g-convexity tightens the optimal regret from $O(\sqrt{T})$ to $O(\log T)$, where $T$ is the time horizon; in the decentralized Riemannian setting, however, existing methods address only g-convex losses, leaving the strongly g-convex regime unexplored. One challenge is that the required decaying step size in the centralized regime is incompatible with existing network-error analyses, which typically assume a fixed step size. First, we provide a general network-error analysis for time-varying schedules. Next, we build on this analysis to establish the first $O(\log T)$ static regret bound for decentralized online Riemannian gradient descent, matching the minimax-optimal rate for strongly-convex Euclidean online optimization. Finally, we prove the same $O(\log T)$ regret bound for the two-point bandit feedback setting using novel strong subconvexity arguments for the smoothed versions of the loss functions.

We establish a $\widetildeΩ(d^{5/4}\sqrt T)$ lower bound on the minimax expected regret of stochastic bandit convex optimization of $1$-Lipschitz functions on the Euclidean ball. This presents the first nontrivial regret lower bound that grows faster than $d\sqrt{T}$ for this problem, establishing that stochastic bandit convex optimization is fundamentally harder than linear bandits. The hard class of convex functions we construct takes the following form in dimension $2d$: for an action $a = (a^1,a^2) \in \mathbb{B}^{2d}_2$, each function is the scaled soft maximum of a "tube", $r^{-1} \| W^\star a^1 - \frac{r}{8\varepsilon} a^2 \|_2$ (hyperparameterized by $\varepsilon,r$), and a squared distance function, $\frac12 \| a^1 - u^\star \|_2^2 - \frac12 \| u^\star \|_2^2$. Here, $W^\star \in \mathbb{R}^{d \times d}$ is an unknown linear transformation, and $u^\star \in \mathbb{R}^{d}$ is an unknown vector which must be learned to minimize the function. Observations are informative about $u^\star$ only when the learner's action lies near the tube determined by $W^\star$, satisfying $a^2 \approx \frac{8\varepsilon}{r} W^\star a^1$: thus the learner must either find this tube without knowing $W^\star$, or spend observations learning useful directions of $W^\star$. Formally, our regret analysis exploits this tradeoff by bounding the posterior spread of Fisher information matrices obtained under an adaptive sequence of actions. Together, these ingredients give a sample complexity lower bound of $\widetildeΩ(d^{5/2}/\varepsilon^2)$ to find an $\varepsilon$-optimal action, which translates to an $\widetildeΩ (d^{5/4} \sqrt{T})$ regret lower bound. We also extend this lower bound to the unconstrained setting where the action space is $\mathbb{R}^d$.

In two-player zero-sum games whose Nash equilibria form a convex set, regularized solvers such as Regularized Nash Dynamics (R-NaD) empirically select the maximum-entropy member: the information projection (I-projection) of a uniform reference onto the Nash set. On a panel of small games this match is exact, with one apparent exception: in Kuhn poker R-NaD lands at bluff coordinate 0.180 while the maximum-entropy member sits at 0.201, a coordinate gap of about 0.021, even though R-NaD attains 99.7 percent of the maximum entropy. We ask whether this gap is a genuine selection bias or an artifact, and answer it quantitatively. We show that for selection on a one-dimensional Nash manifold the coordinate gap factorizes as $\mathrm{gap} \approx \sqrt{2δ/κ}$, where $δ$ is the entropy shortfall of the solver and $κ$ is the curvature of the entropy landscape at its peak. Across five games this relation holds to within $2 \times 10^{-4}$ (under 1 percent relative error). The four matrix games have $δ\approx 0$ (R-NaD reaches the maximum-entropy member exactly) and therefore no gap regardless of curvature; only the sequential game (Kuhn) has $δ> 0$. A causal sweep of the magnet strength drives $δ\to 0$ and the gap toward zero along the predicted curve (fitted scaling exponent 0.50, $R^2 > 0.999999$, against the exact prediction of 1/2), until the dynamics destabilize at a stability floor: behavior consistent with a removable shortfall and inconsistent with a fixed bias. We quantify the curvature half of the law from measured curvatures and flag a moving-target pitfall in the natural Tsallis-entropy experiment. The Kuhn gap is thus the curvature shadow of a small, removable entropy shortfall on an unusually flat peak; the I-projection account is upheld up to a flatness-limited residual.

Protein fold classification can be approached via sequence-based representations or structural descriptors, but direct comparisons between lightweight handcrafted descriptors and pretrained protein language model embeddings remain limited. We investigate discrete Ricci curvature on Calpha contact graphs as a lightweight structural descriptor for fold classification. Each protein domain is represented by a 22-dimensional fixed-length feature derived from summary statistics and quantiles of Ollivier-Ricci and Forman-Ricci edge curvature distributions. We evaluate on CATH top-10 Topology classification and on the ASTRAL 40%-identity SCOPe top-10 Fold benchmark, comparing against geometry, contact-graph statistics, persistent homology, and mean-pooled ESM-2 (150M) baselines. On both datasets, lightweight structural descriptors substantially outperform mean-pooled ESM-2 embeddings, with a larger performance gap on the ASTRAL 40% SCOPe benchmark. Ricci alone uses 22 dimensions, or 3.4% of the ESM-2 baseline dimensionality, and already outperforms mean-pooled ESM-2 on both datasets. Combining Ricci with persistent homology yields the strongest performance, achieving macro-F1 of 0.71 on CATH and 0.68 on SCOPe with a 112-dimensional feature vector. These results identify a regime where lightweight interpretable graph descriptors offer a practical alternative to pretrained protein language model embeddings.

Sharpness-Aware Minimization (SAM) improves generalization by minimizing the worst-case loss in a local parameter neighborhood. Standard SAM implicitly allocates its global perturbation budget across parameter blocks according to instantaneous minibatch gradient norms. Such an allocation can be noisy and may not reflect the sensitivity that blocks accumulate throughout training. We propose Gradient-Energy Adaptive Radius SAM (GEAR-SAM), which maintains an exponential moving average (EMA) of squared block gradients as a lightweight, curvature-related sensitivity signal and allocates the fixed SAM budget through a closed-form constrained optimization. GEAR-SAM preserves the global SAM radius, requires no Hessian-vector products or explicit Fisher estimation, and adds only scalar state beyond SAM. Experiments on image classification, transfer learning, noisy-label learning, and partition studies demonstrate improved generalization and robustness across architectures and tasks. More broadly, GEAR-SAM provides a dynamic view of sharpness-aware optimization: a fixed perturbation budget should be redistributed as the sensitivity of functional network blocks evolves during training.

We study linear regression when the regressor is latent and observed only through multiple noisy measurements, each a smooth but possibly nonlinear function of the latent variable. The problem is acute in the measurement of occupational exposure to artificial intelligence, where competing scores yield downstream estimates that differ by a factor of eleven. A regression on any single measurement recovers a source-specific coefficient rather than the structural one. We fix the latent scale by requiring the consensus measurement function to be linear and bound the remaining curvature heterogeneity across sources relative to slope. Under this bound, the structural coefficient lies in a closed-form interval centered at a symmetric cross-source estimator. The interval is invariant to unknown source loadings, and its half-width is second order in the curvature bound and sharp to the same order. With at least four measurements, the bound is estimable from the joint distribution of the sources through a split-instrument auxiliary regression, and Imbens-Manski confidence intervals with the Stoye critical value attain uniform coverage over the curvature class, including at the point-identified boundary. The application matches six exposure measures to an American Community Survey panel of 8.88 million person-year observations for 2015 to 2024. The post-2022 employment coefficient changes sign between the language-model measures and the Webb patent-text measure, and an ex ante factor-analytic rule separates the Webb measure as a distinct construct. The five retained sources yield a loading-invariant consensus coefficient of -0.239, with a partial-identification half-width of 1.23 percent of the point estimate, or 1.88 percent at the one-sided 95 percent upper bound on the curvature. We read the application as measurement reconciliation rather than as a causal estimate of AI displacement.

Nonlinear least-squares optimization is central to regression, physics-informed neural networks, and other machine-learning tasks. Such problems have a natural geometric interpretation, model predictions form a manifold in data space, while the chosen parameterization can introduce parameter-effects curvature that becomes a dominant source of nonlinearity. This exposes a limitation of the Levenberg-Marquardt (LM) method, its tangent-space step is applied as a straight update in parameter coordinates. Geodesic acceleration gives a second-order correction, but its removal of parameter-effect curvature is exact only in the infinitesimal-step limit. We propose a Riemann-normal-coordinate Levenberg-Marquardt method (RNC-LM) to improve this consistency for finite optimization steps. By reformulating the geodesic equation, RNC-LM extends geodesic acceleration to arbitrary-order corrections and constructs finite-step updates with progressively higher reparameterization consistency. A line search along the resulting RNC curve controls the traveled distance while keeping the cost close to standard LM. The method eliminates the tangential component of residual acceleration order by order in a moving tangent frame, making the actual objective reduction more consistent with the linear model prediction of LM. On classical nonlinear least-squares benchmarks, RNC-LM improves convergence and robustness in curved valleys and rank-deficient problems. On a reaction-diffusion PINN failure-mode benchmark, it reduces the relative L2 error to the order of 1e-3 and recovers a physically meaningful solution. On a large-scale machine-learning potential-energy-surface fitting task, it achieves a 34-fold speedup over standard LM.

Generalization remains a pivotal challenge in deep learning, where traditional optimizers like Stochastic Gradient Descent (SGD) often converge to sharp minima, leading to overfitting and reduced performance on unseen data. Building on Sharpness-Aware Minimization (SAM), for seeking flat minima associated with improved generalization, we propose the Extragradient-Inspired Sharpness-Aware Minimization (EISAM), a novel optimizer that enhances generalization via the extragradient technique. EISAM uses a two-step update process: a prediction step investigating the geometry of the loss landscape and a perturbation step that refines updates with a base optimizer. This approach achieves better generalization performance than SAM. Crucially, EISAM reduces sensitivity to the perturbation radius, enhancing robustness, and simplifying the tuning across diverse settings. Extensive experiments on benchmark datasets demonstrate that EISAM consistently outperforms SGD, Adaptive Moment Estimation (Adam), and SAM in test accuracy and training efficiency across various architectures. Theoretical analysis further confirms that EISAM tightens the generalization bound by steering parameters toward flatter minima with reduced curvature. Accompanied by a thorough hyperparameter analysis, EISAM offers practical tuning guidance, establishing it as a robust, scalable, and broadly applicable optimization solution that advances both the theory and practice in deep learning.

Under a fixed privacy budget, the utility of differentially private (DP) training is ultimately determined by its optimization efficiency. Standard first-order DP optimizers such as DP-SGD rely solely on local gradients and ignore the underlying loss curvature. This geometric blindness causes severe zigzagging in ill-conditioned landscapes, squandering precious privacy budgets on inefficient iterations. Practitioners are thus trapped in a bind: either stop training prematurely or inject massive per-step noise, both of which critically compromise final model utility. Natural Gradient Descent (NGD) resolves this by preconditioning gradients with curvature, aligning updates with the loss geometry and extracting more efficient signal from every noisy step, offering a principled pathway to break the privacy-utility bottleneck. Despite its theoretical appeal, directly integrating NGD with DP introduces fundamental challenges: curvature estimation itself consumes prohibitive privacy budgets, isotropic DP operations conflict with the anisotropic scaling of NGD, and the inverse curvature catastrophically amplify parameter updates in flat directions, causing training instability. We propose DP-NGD, a practical framework that systematically addresses these obstacles by decoupling curvature estimation from private data, reconciling isotropic DP constraints with anisotropic second-order optimization via a whitened-space mechanism, and dynamically clamping the curvature to stabilize training. Extensive experiments on standard benchmarks demonstrate that DP-NGD achieves state-of-the-art accuracy, breaking through the utility ceilings of first-order baselines while delivering up to a $10\times$ convergence speedup under the same privacy budget.

Whether a hyperbolic representation model uses its geometry cannot be inferred from curvature alone: what matters is the dimensionless operating point $\sqrt{c}ρ$ and whether the radial and cone mechanisms are operational there. We develop necessary-condition diagnostics and audit three published hyperbolic vision-language families -- MERU, HyCoCLIP, and PHyCLIP -- across released checkpoints and matched interventions. All converged checkpoints remain near-Euclidean ($H(u)\approx1$; none reaches $\sqrt{c}ρ>1$), and releasing the curvature floor changes $c$ and norms without leaving this regime or substantially degrading downstream performance. Entailment cones are inactive or saturated, and graded traversal fails under controlled readouts, including the models' native distance metrics. External parent-child ordering shows no shuffle-controlled pair-specific radial signal at quantified sensitivity; the only surviving pair-specific signal, a statistically detectable but small residual on the GRIT box-to-full-caption relation, remains non-operative under the evaluated readouts. Taxonomy correlations show no detectable norm contribution beyond cosine, and coarse-retrieval gains co-vary with box/compositional supervision without establishing an active radial mechanism. Gradient diagnostics expose a low-curvature, wide-cone shortcut in the entailment objective. A closed-form aperture identity places the saturation edge at $\sqrt{c}ρ\le2K$: with the floor released, all trained relation-level parent means lie at or below this edge, leaving the parent cones fully or nearly saturated. Entailment-off runs pass the edge and continue contracting. The shortcut is the dominant accelerator of collapse, not its sole cause. These audited formulations do not show an operative radial/cone mechanism under our diagnostics. We distill the audit into a five-number geometry report for hierarchy claims.

Detecting communities in heterophilic graphs -- where connected nodes often belong to different classes -- is hard for unsupervised methods: classical modularity and spectral methods are feature agnostic, while deep graph-clustering methods rely on contrastive or generative machinery that is opaque. We propose Curvature-Guided Sheaf Diffusion (CGSD), a fully unsupervised community-detection algorithm that uses the discrete Forman--Ricci curvature of each edge as its single topological signal, propagated through every stage of an end-to-end pipeline. CGSD makes three concrete contributions: (i)~a curvature-gated sheaf-diffusion encoder that gates edge messages by $σ(κ_e)$ and is trained from three label-free structural losses (modularity, anti-collapse, curvature-weighted reconstruction); (ii)~a curvature-aware spectral clusterer (CSpec) that re-weights the $k$-NN affinity of the embedding by $σ(ακ_{e^*})$ before Ng--Jordan--Weiss; and (iii)~a unified label-free evaluation against nine truly-unsupervised baselines. On five heterophilic benchmarks (Cora, Cornell, Texas, Wisconsin, Chameleon), CGSD wins outright on Wisconsin and Chameleon and is competitive on the remaining three against nine unsupervised baselines. The gain over the strongest baseline is driven by the clusterer, not the encoder: on the same embedding, CSpec improves mean NMI from $0.091$ with $K$-Means to $0.107$ ($+15\%$, paired $t$-test $p=0.008$). The mechanism is interpretable: intra-community and inter-community curvature distributions are visibly separated. Code is open-sourced at https://github.com/woodywff/cgsd.

Hessian spectral properties are a standard tool in analysing neural-network training, with eigenvalues linked to sharpness, generalization, and optimization dynamics. Eigenvalues quantify curvature magnitude, while eigenvectors identify which parameters generate that curvature. In this work, we study how the leading Hessian eigenvectors evolve during training and how they affect the learning trajectories. We track the training dynamics of multilayer perceptrons on a classification problem and measure eigenvector dynamics through two complementary statistics: (i) displacement over time, inspired by analyses of glassy systems, and (ii) localization via the inverse participation ratio. The metrics are compared against a random null model of the Hessian induced by the architecture. Our results reveal clear optimizer-dependent behaviour. SGD leads to progressively more stable leading curvature directions, while Adam exhibits substantially stronger reorganization of eigenvectors throughout training. We also observe a localization phenomenon under Adam, where a small subset of parameters contributes disproportionately to the leading curvature directions. These results suggest that Hessian eigenvector dynamics capture key differences in optimizer behaviour and the resulting training trajectories.

We introduce a pre-registered screening rule that decides, before any implementation, whether an evolutionary / population / lifecycle outer loop over neural-network parameters or structure is worth building. Such outer loops cost 10^2-10^3x their gradient inner loop, yet whether they beat a cheap single-shot alternative is usually discovered only after the expense is paid. Our rule computes, at a Phase-0 gate, a single number: the recovery R = s/G, the best single-shot gradient/curvature statistic's gain s divided by the best gain G of any cheap method evaluated, and prescribes skipping the outer loop when R >= 90%. We validate the rule on a within-lab series of pre-registered outer-loop bets (two analyzed cases plus a disclosed file drawer): in both analyzed cases a static or single-shot computation captured the effect on the project's own metric, the gate fired (R approximately 1.0 in both cases; approximately 0.95 under a stricter metric on one), and the outer loop was abandoned, including one case where a companion factorial decomposition localizes the apparent win to a static substrate change with the evolutionary lifecycle contributing no detectable gain. On one project the gate cost about 50-70 GPU-hours and screened out an estimated 400+ GPU-hours (first cell only) plus weeks of implementation, a 6-8x saving. The rule is prospectively falsifiable: a task with R < 90% where the outer loop still fails to beat single-shot would refute it.

Online high-dimensional regression requires algorithms that can update sequentially while preserving structural sparsity. We propose \textit{Adaptive Iterative Hard Thresholding (AIHT)}, an online sparse-regression framework that alternates stochastic subgradient updates with adaptively scheduled hard-thresholding steps. The key idea is to separate support discovery from local refinement: early in the learning process, AIHT delays thresholding so that weak but informative coordinates have time to accumulate signal, while later it increases the projection frequency to stabilize the sparse estimator and exploit local curvature. We develop the theory for high-dimensional online quantile regression, a challenging setting in which the loss is nonsmooth and the data may exhibit heterogeneity or heavy-tailed noise. Under restricted curvature and gradient-leakage conditions, AIHT remains in an inflated sparse cone, exhibits a two-phase convergence behavior, and attains logarithmic regret for the sliding-window objective. Simulations for online quantile regression, together with threshold-scheduling ablations, support the proposed mechanism and illustrate its advantage over standard online sparse-learning baselines.

We study the spectral perturbation of the empirical Fisher Information Matrix (FIM) of a parametric statistical model under two structured perturbations: departure of the input from a reference (in-distribution) ensemble, and finite-precision (quantized) perturbation of the model's parameters. For the first, under an explicit local curvature-monotonicity hypothesis on the dominant eigenvalue lambda_max of the FIM, we show departure from a reference manifold provably elevates lambda_max relative to a calibration baseline (Proposition 3.2), and discuss why this hypothesis is required, since curvature need not increase monotonically under every perturbation. Our principal result is a directional eigenvalue perturbation bound, via Weyl's inequality, showing lambda_max under a quantization noise perturbation is lower bounded by its unperturbed value up to a third-order remainder, and, under a mild genericity condition, strictly exceeds it at leading order (Theorem 4.3). We give two tractable approximations to lambda_max -- one heuristic, one with a rigorous two-sided bound -- and a completeness result for a threshold-based partition of an augmented state space. These results motivate using sigma_t = lambda_max(F_t)/lambda_base as a runtime monitoring statistic for deployed language models: the quantization result offers a mechanism for an empirical observation of our own, where a calibration threshold for this statistic was approximately 244 times larger than a preliminary full-precision estimate on a 4-bit quantized model, a single measurement rather than a value derived in closed form. We report supporting measurements (twelve models, n=1,080 trajectories) broadly consistent with our predictions, discuss the scope and limitations of every result, and state as an open problem the closed-form prediction of the quantization inflation magnitude our bound does not supply.

We study linear TD(0) under Markovian sampling, where data are generated along a single trajectory. We provide high-probability guarantees for a plain unprojected TD(0) algorithm with Polyak-Ruppert (PR) averaging, using a single stepsize schedule $η_t \propto \frac{1}{τ_{\mathrm{mix}}\log(t)\sqrt{t}}$ that depends on the mixing time but requires no prior knowledge of the curvature parameter $ω$. Our first result shows that such a choice of the stepsize guarantees that the TD(0) iterates are automatically and uniformly bounded with high probability, without projections and without any stability argument based on $ω$. Building on this result, we establish a simultaneous high-probability convergence guarantee for the PR average: the same stepsize yields both a robust curvature-free $\widetilde{\mathcal{O}}\!\left(\frac{τ_{\mathrm{mix}}}{\sqrt{T}}\right)$ rate and a fast curvature-dependent $\widetilde{\mathcal{O}}\!\left(\frac{τ_{\mathrm{mix}}^2}{ωT}\right)$rate, with the bound taking the minimum of the two. The core technical ingredient is a Poisson-equation toolkit for geometrically mixing Markov chains, which decomposes Markov noise into a martingale term plus a controlled remainder and enables a new self-bounding inductive argument for pathwise stability.

Graph Neural Networks (GNNs) have emerged as a powerful paradigm for learning on graph-structured data by iteratively propagating and aggregating information across edges. However, conventional message passing schemes often suffer from over-squashing, whereby exponentially large neighborhoods are compressed into fixed-dimensional embeddings, impeding effective long-range dependency learning. In this work, we introduce Ramanujan Propagation, a graph rewiring strategy that leverages Ramanujan graphs to alleviate topological bottlenecks in GNNs. We first establish that suitably chosen Ramanujan graphs guarantee non-negative resistance curvature, which mitigates over-squashing and facilitates efficient information flow. We then propose an algorithmic framework to construct a Ramanujan rewired graph that preserves the local connectivity of the original graph. Our experiments demonstrate that our method outperforms nine state-of-the-art rewiring techniques. These results establish Ramanujan graphs as a rigorous structural prior for scalable, topology-aware message passing in GNNs.

Surface reconstruction from point clouds is important for consumer-grade 3D capture, including AR/VR and indoor scanning. Local-patch Unsigned Distance Field (UDF) methods are lightweight and generalizable, but their accuracy depends on the support radius, traditionally fixed or selected by a one-dimensional curvature heuristic that cannot capture heterogeneous local geometry. We propose a learned per-query radius selector that predicts a continuous support radius and plugs into a frozen LoSF-UDF backbone. The selector is trained using off-grid target radii obtained by parabolic interpolation of cached UDF error curves. Experiments show improved fine-scale reconstruction accuracy.

Curvature of planar curves serves as a key regularization term for computing second-order minimal paths, due to its tight relevance to desirable geometric properties such as smoothness, rigidity, and elasticity. In this paper, we tackle a more challenging problem in computational physics and geometry problem: tracking minimal paths whose curvature is constrained by arbitrary upper and lower bounds. For that purpose, we propose a new curvature-bounded geodesic model, developed under the Hamilton-Jacobi-Bellman (HJB) partial differential equation (PDE) framework. It provides strong geometric control over minimal paths by enforcing curvature range constraints, whose paths are smooth and of bounded curvature limitation. We also present a discretization scheme for the Hamiltonian and the HJB PDE incorporating curvature bounds, allowing efficient solver for estimating numerical solutions to the model. Finally, we illustrate the capability of the proposed curvature-bounded geodesic model in applications of robot path planning and curvilinear structures tracking from images. Numerical experiments demonstrate that the proposed curvature-bounded geodesic model serves as a powerful and robust tool for finding satisfactory paths.

Current evaluation practices in relational learning rely heavily on flat leaderboards that average performance across heterogeneous datasets, implicitly assuming a uniform underlying structure. We show that this assumption introduces systematic bias: it obscures geometry-dependent performance variations and can lead to misleading conclusions about model generalization. In this work, we identify intrinsic geometry as a key latent factor governing model effectiveness. We demonstrate that conventional aggregated metrics mask critical performance trade-offs that only become visible when datasets are stratified by their geometric properties. To address this issue, we introduce a curvature-stratified evaluation framework that partitions datasets into positive, negative, and near-zero curvature regimes. Our benchmark evaluates 18 representative models including Graph Convolutional Networks (GCNs), Graph Foundation Models (GFMs), and tabular learning methods across 14 datasets. We find that model rankings are highly stable within each curvature regime but shift significantly across regimes, indicating that performance is fundamentally geometry-dependent rather than universally transferable. Notably, we identify regimes where GFMs offer diminishing returns compared to geometry-aligned GNNs. Based on these findings, we propose a geometry-aware evaluation protocol that yields more reliable and interpretable comparisons than standard aggregated benchmarks. We release all code, curvature-stratified dataset splits, and evaluation tools to support reproducible and rigorous assessment of future relational learning methods. Code and datasets are provided in our project homepage: https://sirbabbage.github.io/CurvBench_HOME/.

Real-time autonomous driving commonly relies on sampling-based trajectory planners that link candidate trajectories to target waypoints along the road centerline. The placement of these waypoints directly impacts both the existence and quality of feasible trajectories. Yet, its effect on planner performance remains largely unexplored. In this paper, we treat waypoint placement as a first-class design variable. We hold the trajectory primitive and candidate budget fixed, and systematically sweep three placement strategies (uniform spacing, an augmented Ramer-Douglas-Peucker variant (RDP*), and a novel curvature-conditioned allocation) across 449 configurations and five CommonRoad maps of increasing geometric complexity. Our results show that the nominal inter-waypoint spacing $d_s$ is the primary performance driver, with large differences in planner reliability attributed to placement alone. Uniform sampling at a well-tuned spacing matches or surpasses both RDP* and the centered curvature variant. The curvature variant offers a small but consistent advantage on geometrically complex roads under reliability-first and balanced weightings, while RDP* never outperforms uniform sampling. These findings suggest that $d_s$ should be treated as the dominant tuning parameter, with geometry-aware strategies reserved for curvature-rich corridors where feasibility is the limiting factor.

Estimating local mean curvature at each point of a high-dimensional dataset is a key ingredient of geometry-aware machine learning algorithms, such as the Mean Curvature Boundary Points (MCBP) method. The naive implementation of this computation, based on a local shape operator approximated from k-nearest neighbor patches, involves an explicit construction of a matrix $H$ whose trace form yields an $O(m^4)$ cost per point, rendering the approach intractable for datasets with more than a few dozen features. This paper introduces two complementary contributions that together reduce this cost by several orders of magnitude. The first contribution is an exact algebraic identity. This identity, derived from the orthogonality of the eigenvectors of the covariance matrix and the cyclicity of the trace operator, eliminates $H$ entirely and reduces the per-point cost to $O(m^2)$ after the eigendecomposition. The second contribution addresses the remaining $O(m^3)$ bottleneck of the full eigendecomposition. Since the local covariance matrix has rank at most $k-1 \ll m$, we replace it with a truncated SVD of the $k \times m$ centered data matrix, an $O(k^2 m)$ operation, and derive an analytical approximation for the contribution of the null-space eigenvectors based on the expected value of their outer product under the Haar measure. The resulting estimator has total cost $O(k^2 m + k m p^2)$, where $p = k-1$. Experiments on real-world datasets confirm speedups of 50 to 300 times relative to the original implementation, with negligible loss when the fast estimator is used to replace the original version. By providing a scalable and data-driven estimate of local curvature, the proposed method establishes curvature as a practical geometric feature for a broad range of machine learning tasks, from classical to modern deep learning pipelines.

Muon improves training efficiency over Adam in large language-model training by about two times, but the local geometric source of this advantage remains unclear. Our work takes a first step toward demystifying Muon's superiority over Adam from a curvature perspective. First, we apply a second-order Taylor approximation to the training landscape and show that Muon achieves a larger one-step loss decrease than Adam at matched validation loss. The two optimizers have comparable first-order gains, but Muon consistently incurs a smaller second-order curvature penalty. Second, we decompose this curvature penalty into the squared update norm and Normalized Directional Sharpness (NDS). We find that Muon and Adam have comparable update norms, so Muon's smaller curvature penalty is driven by lower NDS, not update scale. Third, we study how training data and model structure shape Muon's NDS advantage. Using Zipf-Probabilistic Context-Free Grammar (PCFG) data with controlled imbalance, we show that data imbalance amplifies Muon's NDS advantage over Adam. A within-/cross-layer decomposition further shows that, in the middle and late stages of training, Muon's lower NDS is mainly sustained by smaller within-layer curvature. Beyond empirical evidence, we analyze stylized quadratic problems with heterogeneous curvature and gradient alignment toward high-curvature modes. We prove that Muon attains a smaller average NDS than GD by balancing update energy across curvature groups; when curvature heterogeneity is sufficiently strong, this also yields lower local quadratic loss after the same number of steps.

Multi-objective optimization (MOO) underlies many machine learning problems, yet MOO solvers across the loss-balancing, gradient-balancing, and Pareto-based families almost universally hand their reconciled directions to Adam~\cite{kingma2015adam}. We show this coupling introduces two systematic gaps between the solver's intent and the optimizer's execution. The first is a \emph{weighting mismatch}: Adam's second-moment denominator entangles the time-varying preference vector with gradient statistics, marginalizing the preference into a history average and collapsing distinct Pareto trade-offs toward a near-uniform mixture. The second is a \emph{geometric mismatch}: Adam's adaptive metric distorts the Euclidean geometry MOO solvers assume, turning aligned objectives into apparent conflicts. To resolve both jointly, we introduce \textbf{MAdam} (Metric-Aware Multi-Objective Adam), a drop-in wrapper that leaves both solver and optimizer unchanged. MAdam preconditions the reconciled direction by the preference-conditioned curvature of the scalarized objective; on this whitened input, Adam's second moment collapses to identity, so the realized update is governed by the preference-conditioned metric. Across multi-task learning, Pareto-front recovery, physics-informed neural networks, and medical imaging, MAdam consistently improves over Adam for every solver family.

The existing optimizers for deep neural networks (DNNs) typically rely on either the $\ell_2$ norm or the $\ell_\infty$ norm, resulting in optimizers that do not adapt well to substantial changes in curvature across parameter dimensions. Generally, the training process of DNNs often exhibits strong curvature anisotropy in the early period, whereas in the later period, the training process of DNNs tends to move toward flatter regions with weaker anisotropy. Particularly, optimizers based on the $\ell_2$-norm are usually dominated by high-curvature directions, restricting updates of optimizers along with lower curvature direction and thus leading to a slower convergence rate. While optimizers based on the $\ell_\infty$-norm are prone to oscillations in flatter regions, due to the coordinate-wise updates of the same magnitude. To address these two extreme cases generated by $\ell_2$ and $\ell_\infty$ norms, we propose a novel $\ell_p$-norm scheme with a dynamical value of $p$ and incorporate it into stochastic gradient descent (SGD) and SGD with momentum (SGDM), leading to two novel optimizers with better generalization performance: ${\ell_p}$-SGD (LPSGD) and ${\ell_p}$-SGDM (LPSGDM). Particularly, the resulting optimizers suppress the dominance of high-curvature directions in the early period by utilizing a large $p$ ($p>2$), followed by a gradual decrease of $p$ toward 2 to enable more stable and refined updates, where the latter process is motivated by the cosine annealing strategy. We establish theoretical guarantees of the resulting algorithms and analyze that both LPSGD and LPSGDM achieve an $O(T^{-1/2})$ convergence rate for the nonconvex setting. Extensive experiments are conducted on benchmark datasets, including CIFAR-10, CIFAR-100, and ImageNet-1K, with multiple DNNs such as VGG-11, ResNet-18, and ResNet-50.

Linear attention reduces the quadratic cost of softmax attention by maintaining a recurrent fast-weight state, but it consistently lags on in-context retrieval and long-context tasks. Existing remedies act on the write side of memory through gating, delta updates, or kernel feature maps, but the read step is left unchanged: every past key contributes additively to the output, so useful targets are diluted by the bulk of stored vectors. We borrow one specific piece of softmax's geometry to construct a cheap read-time contraction of the query. A second-order Taylor expansion of the softmax log-partition at the isotropic-attention point gives a local quadratic model whose curvature coincides with the running key covariance, a quantity that can be maintained with the same recurrent/chunkwise mechanism as the linear-attention state. The associated linear operator contracts the query along the high-variance directions of memory before it reads the state. We call this mechanism Curvature-Conditioned Query (CCQ). CCQ modifies only the read step and is composable with any linear-attention backbone. Attached to GLA and Gated DeltaNet, it improves perplexity, zero-shot downstream accuracy, S-NIAH retrieval at and beyond the training context, length-extrapolation perplexity from 4K to 20K, and LongBench accuracy.

Compact Vision Transformers are attractive for medical imaging in low-data and resource-constrained settings, but most existing variants assume that Euclidean latent geometry is sufficient for organizing image representations. We introduce hZACH-ViT, a family of curved-geometry extensions of ZACH-ViT, a compact zero-token Vision Transformer that removes positional embeddings and the class token and relies on global average pooling over patch representations. To isolate the role of geometry, we preserve the verified ZACH-ViT backbone and modify only the final representation space and prototype-based classifier head, enabling a controlled comparison between Euclidean, hyperbolic, and spherical latent geometries. We evaluate Poincaré, Klein, and spherical hZACH-ViT heads on seven MedMNIST datasets under an identical few-shot protocol with 50 samples per class and five random seeds. The completed benchmark contains 770 training runs spanning seven datasets, three non-Euclidean geometries, seven curvature magnitudes, and a Euclidean baseline. Across all seven datasets, the best non-Euclidean hZACH-ViT configuration improves over Euclidean ZACH-ViT, with an average gain of +0.021 in the dataset-specific primary metric and the largest improvement on OCTMNIST (+0.055 MacroF1). Fixed low-curvature configurations retain positive gains on the majority of datasets, and low curvature values (c = 0.1 or 0.2) account for six of the seven dataset-level winners. Rather than identifying a universally optimal manifold, our results establish geometry and curvature as dataset-dependent model-selection variables, with fixed low-curvature analyses confirming that gains persist beyond exhaustive per-dataset tuning.

Many machine learning techniques rely on approximating a loss function's curvature, but this is notoriously hard to do at the scale of modern deep networks. Surprisingly, no previous work has exploited the curvature constraints that arise from well known weight-space symmetries in loss landscapes. By analytically averaging over group actions that leave the loss invariant, we construct structured Hessian approximations from single gradients that can be tractably estimated, stored, and inverted. The choice of user-specified symmetry group directly governs the trade-off between approximation accuracy and computational cost. Moreover, our framework provides a unifying theoretical lens for viewing existing methods; in particular, a specific choice of symmetry group recovers Shampoo/Muon-like curvature estimates. We validate our method on a range of network architectures, and deploy it to second-order optimization benchmarks, including a small language model. Our curvature estimation framework might find applications in other machine learning problems such as uncertainty estimation, continual learning, compression/pruning, training data attribution, and more.

Curvature-penalized geodesic models have proven their effectiveness in image segmentation by computing globally optimal curves. Unfortunately, these models remain susceptible to shortcuts when delineating objects with complex shapes and image intensity distributions, as they lack mechanisms to enforce shape-aware tangent constraints. To address this limitation, we propose a unified geodesic framework that integrates tangent-constrained priors with curvature penalization. The key idea is to formulate tangent admissibility directly within the orientation-lifted space, where path tangents are restricted to spatially varying angular sectors derived from intrinsic shape representatives (ISR) such as skeletons or interior landmarks. This formulation gives rise to a family of tangent-constrained Finslerian metrics, extending the classical curvature-penalized geodesic models while enforcing mandatory tangent constraints. The resulting Hamilton-Jacobi-Bellman (HJB) partial differential equations (PDEs) admit efficient numerical solutions via variants of the fast marching method, preserving the single-pass computational complexity. Experiments on synthetic, natural, and medical images demonstrate that the proposed geodesic framework indeed improves robustness against weak boundaries and topological shortcuts, yielding segmentation results with enhanced shape fidelity compared to existing geodesic models.

Diffusion models can unintentionally memorize training samples, raising concerns about privacy and copyright. While recent methods can detect memorization, they often rely on global or model-specific signals and provide limited insight into where memorization appears within a generated image. We provide a geometric characterization of local memorization as a coordinate-wise variance collapse. However, such collapse can also arise from intrinsic data constraints rather than overfitting. To isolate overfitting-driven memorization, we propose curvature-difference methods that subtract the curvature of an underfitted baseline, either the unconditional model or a less-trained version of itself. We further derive a score-difference proxy that provides a geometric explanation for the widely used score-difference-based detection metric. Experiments on Stable Diffusion, evaluated against ground-truth memorization masks, show that our method outperforms the prior attention-based localization method. Code is available at https://github.com/Gwangho99/mem-curv-diff.

The curvature exponent $α$ in $h_k \propto σ_k^α$ -- governing how Hessian eigenvalues scale with gradient singular values -- varies systematically across layer types ($α\approx 2$ for convolutions, $\approx 1$ for transformer attention, $< 1$ for MLP up-projections). Why? We prove the Spectral Alignment Decomposition: $α= 2 + d\logΦ_k / d\logσ_k$, where $Φ_k$ measures alignment between Kronecker factor eigenbases and gradient singular directions. This reduces "why does $α$ vary?" to a geometric question we answer for LayerNorm, residual connections, and softmax heads. The decomposition implies a spectral transfer identity $s = αγ$ linking curvature exponent, effective gradient rank-decay $γ$, and Hessian decay exponent $s$. The identity is algebraic; its empirical content is that $α$ and $γ$, fit on independent data (HVPs vs. SVD), recover $s$ to ~2% median error across 93 layers, five architectures, and three datasets -- with no free parameters. A zeta-function bound on participation ratio shows curvature concentrates onto effectively one direction per layer. As a proof of concept, we derive the architecture-adaptive preconditioner $T(σ;α)$ and show that Spectral Newton -- implementing $T$ in the gradient singular basis -- outperforms AdamW on vision benchmarks where $α\approx 2$.

Continual learning requires new-task adaptation without damaging previously acquired capabilities. Recent forward-pass and zeroth-order (ZO) results show that low-query adaptation may retain better than first-order (FO) descent, but the usual view of ZO as noisy FO estimation does not explain why. We give a local randomized gradient-shaping analysis: finite differences expose a raw shape that is mean-aligned with FO, while the norm-matched comparator fixes the expected squared adaptation norm. Under this controlled comparison, forgetting depends on how the adaptation shape exposes retention curvature. For norm-matched ZO, the expected shaped retention curvature obeys an exact identity that preserves the isotropic retention floor while contracting only the anisotropic component. Projecting this identity onto the incoming gradient yields the observable FO--ZO quadratic forgetting gap: ZO improves mean forgetting precisely when the FO direction has above-average retention curvature, by a query-dependent fraction of that curvature excess. A practical finite-query accounting separates the mean mechanism from one-batch sampling and smoothing perturbations. As an algorithmic transfer, RISE applies the calibrated ZO shape to exact FO gradients inside parameter blocks. Its target is a stability--plasticity tradeoff: randomized shaping may reduce the retention exposure paid by FO, exact gradients remove finite-smoothing bias from finite-difference ZO, and blockwise sampling supplies many local shaping directions after one gradient computation. The blockwise analysis separates mean-step damage from centered random exposure, showing how block-diagonal curvature, cross-block coupling, and local shaping diagnostics specify where this exact-gradient transfer is most likely to be visible.

Submodular functions -- functions exhibiting diminishing returns -- are central to machine learning. When the objective is monotone and non-negative, the greedy algorithm achieves a tight $63\%$ approximation. But many practical objectives incorporate costs that make them negative on some inputs, and all existing multiplicative guarantees require non-negativity. Prior work handles negativity through additive bounds for the special class of decomposable functions and non-monotonicity through partial-monotonicity parameters, but these address each difficulty in isolation and neither extends the classical structural theory. We extend \emph{curvature} -- a parameter measuring how far a function deviates from linearity -- to all submodular functions, handling both non-monotonicity and negativity through a single classical concept. A greedy algorithm with pruning achieves a curvature-controlled multiplicative ratio for \emph{any} submodular function, including those taking negative values -- the first such guarantee beyond monotonicity and non-negativity. In the non-monotone regime $1 \le c_g < 2.2$, the bound strictly beats the best known uniform ratio of $0.401$ (for non-negative $f$), and it recovers the classical $(1-e^{-c_g})/c_g$ guarantee for monotone functions. A multilinear-extension variant extends the framework to general combinatorial constraints via multilinear relaxation. Experiments on cost-penalized experimental design, coverage, feature selection, and a curvature sweep on Multi-News passage selection support the theory.

Kolmogorov-Arnold networks (KANs) offer a potent combination of accuracy and interpretability, thanks to their compositions of learnable univariate activation functions. However, the activations of well-fitting KANs tend to exhibit pathologically high-curvature oscillations, making them difficult to interpret, and standard regularization penalties do not prevent this. Here we derive a basis-agnostic curvature penalty and show that penalized models can maintain accuracy while achieving substantially smoother activations. Accounting for how function composition shapes curvature, we prove an upper bound on the full model's curvature relative to the curvature penalty, and use this to motivate richer forms of penalties. Scientific machine learning is increasingly bottlenecked by the trade-off between accuracy and interpretability. Results such as ours that improve interpretability without sacrificing accuracy will further strengthen KANs as a practical tool for both prediction and insight.

Accurate quantification of the geometry of curvilinear biological structures is essential for understanding cellular mechanics and disease-related morphological alterations. Microtubule curvature is a key descriptor of filament rigidity and mechanical perturbations. However, reliable curvature extraction from fluorescence microscopy images remains challenging due to noise, low contrast, and partial filament visibility. Existing approaches rely on segmentation pipelines with pre or post-processing, which are highly sensitive to segmentation errors and often fail under adverse imaging conditions. In this work, we propose MTCurv, a deep learning framework for direct, segmenta-tion-free regression of microtubule curvature maps from noisy microscopy images. Leveraging a synthetic dataset with pixel-wise curvature annotations, we reformulated curvature estimation as a regression problem and adapted an attention-based residual U-Net. To reduce hallucinations and enforce spatial coherence, we introduced a gradient-aware loss combining Mean Squared Error with a gradient consistency term. Beyond model and loss design, we evaluated commonly used regression and image quality metrics, revealing that many perceptual and blind metrics are poorly suited for curvature estimation. Correlation-based metrics, particularly Spearman correlation, emerged as more reliable indicators of curvature prediction quality. Experiments on two datasets of increasing difficulty demonstrated that MTCurv accurately recovers local microtubule curvatures, even in the presence of background fluorescence. Ablation studies highlighted the contribution of both residual encoding and attention-based decoding. Overall, this work provides a practical tool for filament curvature analysis and methodological insights for geometry-aware regression in biomedical imaging. Datasets and code are made available.

In autoregressive large language models (LLMs), temporal straightening offers an account of how the next-token prediction objective shapes representations. Models learn to progressively straighten the representational trajectory of input sequences across layers, potentially facilitating next-token prediction via linear extrapolation. However, a direct link between this trajectory and token-level behavior has been missing. We provide such a link by relating contextual curvature-a geometric measure of how sharply the representational trajectory bends over recent context-to next-token entropy. Across two models (GPT-2 XL and Pythia-2.8B), contextual curvature is correlated with entropy, and this relationship emerges during training. Perturbation experiments reveal selective dependence: manipulating curvature through trajectory-aligned interventions reliably modulates entropy, while geometrically misaligned perturbations have no effect. Finally, regularizing representations to be straighter during training modestly reduces token-level entropy without degrading validation loss. These results identify trajectory curvature as a task-aligned representational feature that influences behavioral uncertainty in LLMs.

In this paper, we introduce the Quasi-Quadratic Gradient (QQG), a novel search direction designed to accelerate the BFGS method within the quasi-Newton framework. By defining the QQG as the product of the inverse Hessian approximation and the current gradient, we explicitly leverage local second-order curvature to rectify the search path. Theoretical analysis and empirical results demonstrate that our approach significantly outperforms vanilla BFGS in convergence speed while maintaining computational efficiency.