SEARCH · Search arXiv

Results for “math.DG”

Search indexed arXiv papers on artificial intelligence, large language models, computer vision and robotics. Read source abstracts and follow links to arXiv.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

1,315 records · Page 6Linked to original sources

arXiv2026

Exposing Finite-Depth, Finite-Shot Guarantees for Constrained Quantum Optimization via Fejér Filtering

Constrained quantum optimization algorithms need quantitative guarantees that connect circuit resources to the probability of actually sampling feasible or optimal solutions in finitely many shots. We establish such a connection by exposing a positive sampling law in which mixer-driven exploration and spectral selection can be controlled separately. We show that after removing interference between distinct cost eigenspaces as an analytic device, the measurement distribution becomes the normalized product of a mixer-induced exploration envelope and a Fejér spectral weight, with the former describing how the mixer spreads probability over the encoded manifold and the latter enhancing the target cost phase while suppressing spectrally separated nontarget phases. In this model, finite-shot success becomes a tractable competition between target weight and off-target leakage, yielding an explicit lower bound on the probability of sampling an optimum. For the primary bound, we rescale the cost Hamiltonian to an integer-valued spectrum, placing the wrapped cost phases on a controlled lattice for Fejér filtering. We then define $δ$ as the minimum circular separation between the optimal phase and every nontarget phase. The single-shot success probability $q_0$ satisfies \[ q_0 \ge \frac{x}{1+x}, \qquad x = (p+1)^2 \sin^2\!\left(\fracδ{2}\right) C_β, \] where $p$ is the filter order and $C_β$ is the mixer-envelope mass on the optimal set, exposing a finite-resource compensation law in which weaker phase separation or smaller envelope mass can be compensated by increased filter order and additional shots. The same filtering principle exposes a feasibility guarantee when applied to penalty phases. We further prove analogous bounds for nonlattice spectra through off-target suppression, extending our results beyond exact lattice normalization.

quant-ph↗

arXiv2026

Exact Limits of Random Projections for Preserving Geometry: Distance Recovery, Nearest-Neighbor Rankings, and Covariance Shape in Gaussian Models

The Johnson-Lindenstrauss (JL) lemma guarantees that a random projection of $n$ points to $m=O(\varepsilon^{-2}\log n)$ dimensions preserves pairwise squared distances within relative error $\varepsilon$ with high probability, and this dimension order is asymptotically optimal. In high dimensions, however, distances concentrate around a baseline while key geometric information lies in much smaller fluctuations. We show that the JL bound can therefore be uninformative about retained geometry: an independent Gaussian replacement map can satisfy it even though the replacement cloud is independent of the original data. We then ask how well any decoder can recover a feature $f(D)$ of a squared distance $D$ from a linear sketch. Under squared-error loss, the optimal decoder is conditional expectation, so recovery defines a linear operator whose singular values quantify feature recovery. For isotropic Gaussian data ($Σ=σ^2 I_d$), we diagonalize this operator in closed form. For fixed $k$ with $m,d-m\to\infty$, its $k$th singular value satisfies $\ell_k\approx(m/ d)^{k/2}$. This yields three sharp consequences. A rank-$m$ sketch retains at most an $m/d$ fraction of the variance of any feature of one squared distance. If $m\to\infty$ and $m/d\to0$, the expected Kendall correlation is $\frac{2}π\sqrt{m/d}(1+o(1))$; for fixed $q$, nearest- neighbor agreement tends to $1/q$. Yet one projection can satisfy the JL bound while mean Kendall correlation vanishes when $\log n\ll m\ll d$. After removing scale, Haar-averaged retained covariance-shape information is $(m/d)^2$. Thus JL distance preservation does not quantify the geometry available for comparison or inference.

cs.LG↗

arXiv2026

Equation free data-driven modelling of chaotic processes

We introduce a method for constructing predictive models of non cyclic physical processes directly from time-series data, without assuming an underlying differential equation. The observations define a discrete evolution rule whose recurrent behaviour captures the essential dynamics of the process. Analysing this behaviour across multiple geometric scales leads to probabilistic models in the form of Markov chains. Hyperbolicity criteria identify when these models provide a consistent statistical description of the data. The method is inspired by, and illustrated through, the analysis of a biological imaging data set referred to as the Cell Process.

math.DS↗

arXiv2026

Residual neural networks overcome the curse of dimensionality for semilinear heat equations

Rigorous results show that feedforward neural networks can overcome the curse of dimensionality in the numerical approximation of high-dimensional partial differential equations (PDEs), but comparatively little is known about residual neural networks (ResNets) in the nonlinear PDE setting. We prove that ResNets overcome the curse of dimensionality in the numerical approximation of solutions of semilinear heat equations with globally Lipschitz continuous, gradient-independent nonlinearities: under polynomial growth and network approximability hypotheses on the PDE data, there exist $η\in(0,\infty)$ and ResNets $Ψ_{d,\varepsilon}$, $d\in\mathbb{N}$, $\varepsilon\in(0,1]$, with at most $ηd^η\varepsilon^{-η}$ parameters whose realizations approximate the solution in dimension $d$ with an $L^2$-error of at most $\varepsilon$. The proof represents one deterministic realization of a multilevel Picard estimator by a ResNet whose shortcut connections transmit the spatial variable and a scalar accumulator, while the residual branches successively add the summands of the estimator. For ridge-sum initial conditions, admissible sigmoidal activations, and globally Lipschitz truncations of the nonlinearity, we obtain, for every $ξ>0$, the explicit bound $C_ξd^{4+ξ}\varepsilon^{-(3+ξ)}$ on the number of parameters.

math.NA↗

arXiv2026

Discrete Gromov-Wasserstein Duality: Algorithms and Isomorphism Testing

The Gromov-Wasserstein (GW) distance provides a principled framework for aligning metric measure (mm) spaces based solely on their intrinsic structure. Its ability to identify isomorphic representations of distributions across spaces renders it valuable for comparing data where equality up to isomorphism occurs naturally such as in graphs or, more generally, distributions on graphs. Recently, a type of dual form for the GW distance between Euclidean distributions with the squared Euclidean or inner product costs was derived, spurring the development of new statistical and algorithmic results for this setting. This work furnishes a novel duality result for GW distances with and without entropic regularization that is applicable to all finitely supported mm spaces. Leveraging this result, we derive the sample complexity of empirical GW distances between finite mm spaces, as well as limit distributions under proper centering and scaling. Furthermore, we propose new algorithms for solving the regularized GW problem which are subject to formal convergence guarantees. These statistical and algorithmic advancements give rise to a principled and efficient framework for testing whether two distributions on the set of graphs with a fixed number of nodes are isomorphic based on samples.

math.ST↗

arXiv2026

A Projected Semiexplicit Integrator for Dissipative Systems with Configuration-Dependent Kinetic Energy: Contact-Herglotz Formulation and Benchmarks

Contact Hamiltonian dynamics gives dissipative mechanics an intrinsic action variable, but explicit contact splittings reach only kinetic energies whose terms are exactly integrable: frozen-coordinate diagonal metrics (the spherical pendulum, a torus particle) are included, while dense metrics with momentum cross terms, with the double pendulum as flagship, are not. We introduce a projected Pihajoki-contact integrator for this non-separable setting, combining phase-space duplication, symmetric projection onto the physical diagonal, and constant-friction damping half-steps, with the action factor carried by an exact Herglotz update. As in the projected extended-phase-space framework it builds on, the construction needs no binding parameter, returns the copies to the diagonal at every step, and confines the nonlinear solve to the $2n$ projection variables. For constant friction the step rescales $ω=dη$ by the exact factor $e^{-γτ}$ when the projection is solved exactly (a classical conformally symplectic identity, realized here for this class), while time-symmetry, consistency, and smoothness yield an $O(τ^3)$ one-step contact-form residual, a bound not specific to the contact form. On the damped double pendulum, spherical pendulum, and torus particle the method is second-order accurate, reproduces the contact decay law, and controls long-time energy and contact drift in coarse or stiff regimes where the Tao baseline and the unprojected average lose the solution. A head-to-head with exact-contactomorphism splittings delimits the niche: where a frozen-coordinate splitting exists it preserves the contact form exactly and wins at matched cost; for the dense double-pendulum metric the realizable alternative is first-order with a prohibitive constant and the projected method prevails. The contact-form estimate is local, one-step, and constant-friction.

math-ph↗

arXiv2026

Deciding superellipticity and computing the Weierstrass normal form

Let $ \mathcal{S}_{g,n} \subset \mathcal{M}_g $ be the locus of curves of genus $ g \geq 2 $ admitting a model $ y^n = h(x) $ with $ h $ separable; such curves $C$ have a cyclic group $ C_n \leq \operatorname{Aut}(C) $ of order $ n $ with $ C/C_n \cong \mathbb{P}^1 $. % We give an algorithm which, given an absolutely irreducible plane model $ F(x,y) = 0 $ of a curve $ C $ over a field $ k_0 $ of characteristic zero, decides for which $ n $ the curve lies in $ \mathcal{S}_{g,n} $ and returns a model $ y^n = h(x) $ together with the birational transformation to it.

math.AG↗

arXiv2026

The best approximation pair problem relative to two subsets in a normed space

In the classical best approximation pair (BAP) problem, one is given two nonempty, closed, convex and disjoint subsets in a finite- or an infinite-dimensional Hilbert space, and the goal is to find a pair of points, each from each subset, which realizes the distance between the subsets. Motivated by our recent algorithm for solving the BAP problem [Censor, Mansour, Reem, J. Approx. Theory (2024)], we discuss the problem in more general normed spaces and with possibly non-convex subsets, and focus our attention on the fundamental issues of uniqueness and existence of the solution to the problem. We present several sufficient geometric conditions for the (at most) uniqueness of a BAP. These conditions are related to the structure and the relative orientation of the boundaries of the subsets and to the norm. We also present many sufficient conditions for the existence of a BAP. In general, the paper re-examines several aspects related to the BAP problem, including the historical one, and shows, probably for the first time, how wide is the scope of the BAP problem in terms of the scientific communities which are involved in it (frequently independently) and in terms of its applications.

math.OC↗

arXiv2026

SABER-Math: Automated Benchmark for Information Retrieval Evaluation in Mathematics

As agentic AI systems tackle more complex mathematical tasks, they increasingly rely on information retrieval (IR) to search problem databases, theorem libraries, and educational resources. However, choosing the right retriever remains difficult, as it is infeasible to directly isolate its effect on downstream performance. On the other hand, existing retrieval-specific benchmarks often fail to capture fine-grained mathematical relevance, penalizing relevant documents. We address this gap by introducing SABER-Math, the first fully automated benchmark for evaluating mathematical IR without expert annotation. Starting from 283K high-school-level math problems with solutions, SABER-Math builds challenging reranking tasks in three steps: (i) first, LLMs extract concise solution summaries and mathematical topics for each problem; (ii) then, per-query relevant documents are discovered using ontology topic-based and lexical solutions-summary-based similarities, and (iii) finally, a Swiss-style LLM preference tournament produces fine-grained relevance ratings for the documents. We evaluate lexical retrievers, specialized mathematical retrieval systems, and recent embedding models. We find that while modern embedding models substantially outperform classical and math-specific baselines, even the strongest systems struggle in symbol-heavy domains like Algebra and Calculus. Importantly, we show that general-purpose IR benchmarks such as MTEB do not reliably predict mathematical performance, especially for recent embedding models, highlighting the need for math-specific retrieval benchmarks.

cs.IR↗

arXiv2026

Learning Latent Graph Geometry via Fixed-Point Schrödinger-Type Activation: A Theoretical Study

We study neural architectures in which each hidden layer is defined by the stationary state of a dissipative Schrödinger-type dynamics on a learned latent graph. On stable branches, the local stationary problem defines a differentiable implicit graph layer. To learn the graph itself, we optimize over the stratified moduli space of weighted graphs and equip each stratum with a non-degenerate Kähler-Hessian metric that keeps natural-gradient descent and face crossing well posed. We then show that a multilayer stationary network is equivalent to an exact global stationary problem on a supra-graph, and that it admits a penalized global relaxation whose stationary states converge to the exact one as the penalty parameter tends to infinity. Reverse-mode differentiation is recovered as the adjoint of the exact global system, and the penalized adjoint converges to it in the same limit. Finally, under finite-dimensional strong-monotonicity and admissible-lift assumptions, the corresponding represented hypothesis classes coincide among resolvent feed-forward networks, graph-stationary networks, supra-graph stationary systems, and sheaf-based architectures with unitary connection. The resulting structural identifications yield complexity bounds controlled by sparse graph or supra-graph geometry rather than dense ambient connectivity.

cs.LG↗

arXiv2026

Convergence of implicit schemes for Hamilton-Jacobi-Bellman quasi-variational inequalities

In [Azimzadeh, P., and P. A. Forsyth. "Weakly chained matrices, policy iteration, and impulse control." SIAM J. Num. Anal. 54.3 (2016): 1341-1364], we outlined the theory and implementation of computational methods for implicit schemes for Hamilton-Jacobi-Bellman quasi-variational inequalities (HJBQVIs). No convergence proofs were given therein. This work closes the gap by giving rigorous proofs of convergence. We do so by introducing the notion of nonlocal consistency and appealing to a Barles-Souganidis type analysis. Our results rely only on a well-known comparison principle and are independent of the specific form of the intervention operator.

math.NA↗

arXiv2026

A proof of Ross's conjecture for two-site moving-target search

A target moves between two sites according to a discrete-time Markov chain with a $2\times2$ transition matrix $M$. At each epoch one site is searched at positive cost, and a search may overlook a target that is present. Ross conjectured that an optimal policy is threshold in the posterior probability that the target is at site~1. MacPhee and Jordan proved the conjecture throughout the nonpositive-determinant ($\det M\le0$) regime and for part of the positive-determinant ($\det M>0$) regime, leaving the remaining cases open. We prove threshold optimality throughout the positive-determinant regime, completing Ross's conjecture for all parameter values.

math.PR↗

arXiv2026

Quadratic Point Estimate Method for Uncertainty Quantification with Dependent Non-Gaussian Inputs

As an extension of the Point Estimate Method (PEM) to evaluate probabilistic moments of quantities of interest (QoI) in general $n$-dimensional spaces, the Quadratic Point Estimate Method (QPEM) has been recently developed. This new method is defined to fully represent up to fifth-order input moments in the Gaussian space, providing general analytical expressions for sample locations and weights, without requiring any numerical optimization. The QPEM can significantly improve the estimation accuracy of the output QoI moments, in relation to PEM-based methods whose numbers of sigma points grow linearly with the problem dimension, while at the same time having an affordable and competitive computational cost up to a considerable number of dimensions. The QPEM is further enhanced in this work by enabling copula integration into the framework, which enables effective modeling of the joint input probability density function by estimating marginals and the dependence structure of the involved random variables. The validity and efficient performance of the copula-based QPEM are showcased against numerous other sampling methods in various examples considering two practical scenarios: (i) when the joint dependence structure can be inferred from data, and (ii) when only marginal distributions and correlation matrices are known.

math.NA↗

arXiv2026

Low-rank matrix recovery landscapes beyond RIP with application to rank-one measurements

We study the problem of low-rank matrix recovery from linear measurements via the global nonconvex landscape of a low-rank factored formulation of the matrix LASSO (nuclear-norm--regularized least-squares). If the landscape is benign, that is, has no bad local optima, then practical and scalable algorithms can compute good statistical estimates. Previous state-of-the-art landscape guarantees have typically assumed that the linear measurement operator has the restricted isometry property, that is, the operator is approximately an isometry over all low-rank matrices. This is an unrealistic assumption for many applications; in particular, when the individual measurement matrices are themselves low-rank, we typically have poor upper isometry constants. To overcome this, we establish new guarantees of a benign landscape under a weaker isometry condition: rather than requiring upper isometry over all low-rank matrices, we only require it over the linear low-rank tangent space to the low-rank ground truth matrix. To illustrate the utility of this result, we apply it to the problem of matrix recovery from random rank-one linear measurements; via high-probability concentration bounds on the random measurement operator, we prove a novel landscape guarantee with statistically near-optimal sample complexity and recovery error.

math.OC↗

arXiv2024

The Ramshaw-Mesina Hybrid Algorithm applied to the Navier Stokes Equations

In 1991, Ramshaw and Mesina proposed a novel synthesis of penalty methods and artificial compression methods. When the two were balanced they found the combination was 3-4 orders more accurate than either alone. This report begins the study of their interesting method applied to the Navier-Stokes equations. We perform stability analysis, semi-discrete error analysis, and tests of the algorithm. Although most of the results for implicit time discretizations of our numerical tests comply with theirs for explicit time discretizations, the behavior in damping pressure oscillations and violations of incompressibility are different from their findings and our heuristic analysis.

math.NA↗

arXiv2026

Greedy Thiele continued-fraction approximation on continuum domains in the complex plane

We describe an adaptive greedy algorithm for Thiele continued-fraction (TCF) approximation of a function defined on a continuum domain in the complex plane. The algorithm iteratively selects interpolation nodes from an adaptively refined set of sample points on the domain boundary. We also present new algorithms for evaluating Thiele continued fractions and their accessory weights using only a single floating-point division. Numerical experiments comparing the greedy TCF method with the AAA algorithm on several challenging functions defined on the interval $[-1,1]$ and on the unit circle show that continuum TCF is consistently faster than AAA, by factors ranging from 8 to 40.

math.NA↗

arXiv2026

An Euler scheme for BSDEs via the Wiener chaos decomposition

The Euler scheme is a standard time discretization for BSDEs, but its implementation hinges on approximating conditional expectations and the associated martingale terms at each time step. We propose an implementation based on the Wiener chaos decomposition to approximate these quantities. In contrast to many numerical schemes that rely on a finite-dimensional Markovian representation, our approach accommodates arbitrary $\mathcal{F}_T$-measurable square-integrable terminal conditions. We provide a comprehensive convergence analysis under additional Malliavin regularity assumptions and illustrate the method on several numerical examples, including genuinely non-Markovian problems arising, for instance, in the pricing and hedging of contingent claims under rough-volatility models.

math.NA↗

arXiv2026

An Exposition of the $\widetilde{O}(\log^{1/4} n)$ Bound for the Komlós Problem

A conjecture of Komlós states that the combinatorial discrepancy of any matrix $A\in\mathbb R^{m\times n}$ whose columns have Euclidean norm at most one is bounded by a universal constant. We prove that the combinatorial discrepancy of every such matrix is at most $O((\log n)^{1/4}(\log\log n)^{7/4})$. This is the first asymptotic improvement over the $O(\sqrt{\log n})$ bound established by Banaszczyk [Banaszczyk, Random Struct.\ Algorithms, 1998], and it refutes a conjecture of Hajela [Hajela, European J.\ Combin., 1988] that a lower bound of order $Ω(\sqrt{\log n})$ should hold.

math.CO↗