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1,352 records · Page 7Linked to original sources
arXiv2026

Overcoming the spatial order barrier for nonlinear SPDEs with additive space-time white noise

We introduce a fully discrete numerical scheme for semilinear SPDEs with additive space-time white noise that overcomes the previous order barrier for the spatial convergence rate. The scheme achieves a strong convergence rate of $M^{-1+ε}$ in time and $N^{-3/2+ε}$ in space for any $ε>0$, where $M^{-1}$ and $N^{-1}$ are the temporal, respectively the spatial, meshsizes. This substantially improves the standard spatial error bounds of order $N^{-1/2}$ in the literature.

math.NA↗
arXiv2026

On cancellative pairs of families of subsets

A pair $(\mathcal{A}, \mathcal{B})$ of families of subsets of $[n]$ is cancellative if whenever $A, A' \in \mathcal{A}, B \in \mathcal{B}$ satisfy $A \cup B=A' \cup B$, then $A=A'$, and whenever $A \in \mathcal{A}, B, B' \in \mathcal{B}$ satisfy $A \cup B=A \cup B'$, then $B=B'$. We show that for every cancellative pair $(\mathcal{A}, \mathcal{B})$, the inequality $|\mathcal{A}||\mathcal{B}| \le 2.25^n$ holds, matching Tolhuizen's $(2.25-o(1))^n$ lower bound construction.

math.CO↗
arXiv2026

Exact affine conditioning beyond Gaussians: a unique characterization of the ensemble Kalman update

The analysis step of the stochastic ensemble Kalman filter, called the ensemble Kalman update (EnKU), is widely used for approximating posterior distributions in inverse problems and data assimilation. The EnKU approximates the posterior distribution $π_{X\mid Y=y_\star}$ by pushing forward the joint distribution $(X,Y)\simπ$ through an affine map $L^{\mathrm{EnKU}}_{π,y_\star}(x,y)$ that depends only on the covariance structure of $π$ and the observation $y_\star$. While the EnKU yields the exact posterior for Gaussian $π$ in the mean-field, this property alone does not uniquely determine the EnKU. In fact, there are infinitely many affine maps $L_{π, y_\star}$ that achieve such exact conditioning. In this paper, we offer a novel characterization of the EnKU among all such affine maps. We first exhaustively characterize the set ${E}^{\mathrm{EnKU}}$ of joint distributions for which the EnKU yields exact conditioning, showing that it is much larger than the set of Gaussians. Next, we show that except for a small class of highly symmetric distributions within ${E}^{\mathrm{EnKU}}$, the EnKU is the {unique} exact affine conditioning map. Further, we characterize the largest possible set of distributions ${F}$ for which a distribution-dependent, weakly observation-dependent, affine map exists, a class of transports that naturally includes the EnKU. We show that ${F}={E}^{\mathrm{EnKU}}\cup{S}_{\mathrm{nl-dec}}$ with a small symmetry class ${S}_{\mathrm{nl-dec}}$, meaning that for affine conditioning beyond the Gaussian setting, the EnKU has an exact set that is essentially maximally large.

math.ST↗
arXiv2026

Tight Bounds for Linear and Non-Linear Contraction of Divergences via Duality

We develop a novel framework for bounding the contraction of information divergences, using duality and associated norms in Orlicz spaces. By working in the dual space, we obtain a principled approach to bounding both distribution-dependent strong data-processing inequality (SDPI) constants and \(F_φ\)-curves of divergences. Our bounds are either available in closed form or reducible to one-dimensional convex optimisation problems, in contrast to the infinite-dimensional optimisation problems that characterise SDPIs. These bounds depend on the densities of the reverse kernels with respect to a reference measure. To the best of our knowledge, they are the first universal closed-form bounds on distribution-dependent SDPI constants. We establish tightness for the \(χ^2\)-divergence on several important channel classes, including full-rank binary kernels. We apply our results to several settings. In particular, we derive bounds on the mixing times of Markov chains, including chains with heavy-tailed stationary distributions; obtain improved bounds on burn-in periods for Markov chain Monte Carlo; and strengthen concentration-of-measure bounds for dependent random variables.

cs.IT↗
arXiv2026

Sharp Restricted Isometry Thresholds for Global Minima of Rank-Restricted Matrix LASSO

We determine the sharp restricted isometry threshold for recovery at global minima of the rank-restricted matrix LASSO. For target rank $r_{\star}$, if the rank-$k$ RIP constant satisfies $δ<δ_{\mathrm{sharp}}(k/r_{\star})$, where $δ_{\mathrm{sharp}}(t)=t/(4-t)$ for $0<t<4/3$ and $δ_{\mathrm{sharp}}(t)=\sqrt{(t-1)/t}$ for $t\ge4/3$, then every global minimizer has Frobenius error $\lesssim\sqrt{r_{\star}}λ$ for all $λ\gtrsim\|\mathcal{A}^{*}(ξ)\|_{\mathrm{op}}$ and at every search rank $r\ge r_{\star}$. The constants depend only on the RIP constant and $t=k/r_{\star}$, and in particular are independent of the search rank. When the rank restriction is inactive, the result specializes to the ordinary convex matrix LASSO. We also obtain the analogous results for sparsity-restricted vector LASSO. Conversely, we show that the threshold $δ<δ_{\mathrm{sharp}}(k/r_{\star})$ cannot be improved, due to the existence of counterexamples whose global minimizers fail to recover the ground truth.

stat.ML↗
arXiv2026

A discontinuous Galerkin approximation of the Dean--Kawasaki equation

We introduce and analyse an arbitrary order spatial discontinuous Galerkin (dG) method for the Dean--Kawasaki equation, a highly singular SPDE modelling density fluctuations of $N$ diffusing particles in the regime of large particle number $N \gg 1$. Our starting point is a general procedure for discretising multiplicative, divergence-form noise on finite element spaces whilst preserving its cross-variation structure at the discrete level; the construction is explicit, elementwise, applies to continuous and discontinuous spaces alike, and extends to general mobilities. Using it, we prove weak error estimates of order $O(h^p)$ between fluctuations of the semi-discrete scheme and those of the underlying particle system, together with a correction that is exponentially small in the scaling regime $Nh^d \gg 1$ and arises because the scheme does not preserve positivity. The resulting method is locally and globally conservative and applies on unstructured simplicial meshes. Quantitative numerical experiments support the analysis, and further experiments illustrate the method beyond the scope of the theory: indicator function observables, external and interaction potentials, convection-dominated regimes and reflecting boundary conditions.

math.NA↗
arXiv2026

TopoAlign: A Framework for Aligning Code to Math via Topological Decomposition

Large Language Models (LLMs) excel at both informal and formal (e.g. Lean 4) mathematical reasoning but still struggle with autoformalisation, the task of transforming informal into formal mathematical statements. Yet, the performance of current Math LLMs is constrained by the scarcity of large-scale corpora, particularly those containing pairs of informal and formal statements. Interestingly, the formal languages used in autoformalisation share structural similarities with programming languages, and code data is available at scale. However, current models trained on code do not transfer effectively to formal math, due to structural and syntactic differences between them. To address this, we propose TopoAlign, a framework that unlocks widely available code repositories as training resources for Math LLMs. TopoAlign decomposes code into docstrings, main functions, and dependency functions, and reassembles these components into analogues that structurally mirror formal statements. We train three state-of-the-art models, DeepSeek-Math, Qwen-3 and Herald, and evaluate them on the MiniF2F, Putnam, and ProofNet benchmarks. TopoAlign provides substantial gains for DeepSeek-Math, improving performance by 17.77% on BEq@10 and 68.82% on typecheck@10, and also measurably improves Herald by 0.12% on BEq@10 and 1.09% on typecheck@10 despite introducing no new mathematical knowledge.

cs.CL↗
arXiv2026

Covering 1024 syndromes with 50 columns

We exhibit a binary linear $[50,40]_2$ code of covering radius $2$, so $\ell_2(10,2)\le 50$, one column below the Kaikkonen--Rosendahl length $51$ that has stood since 2003 and that still seeds the $R=2$ family of Davydov--Marcugini--Pambianco (arXiv:2511.02542). The new matrix admits a $(2,0)$-partition into ten blocks, so Construction $\mathrm{QM}_2^2$ propagates it to exhaustively verified codes of lengths $815$ and $1631$ at $r=18$ and $r=20$, and to the family $n=51\cdot 2^{r/2-5}-1$ of asymptotic density $2601/2048$. The matrices, verifiers, and source are at https://github.com/wustep/maths, pin problems/covering/share/2026-08-24/ at commit 736a38f.

cs.IT↗
arXiv2026

Rate of convergence of a fully discrete structure-preserving midpoint scheme for the stochastic Landau--Lifshitz--Gilbert equation

The stochastic Landau--Lifshitz--Gilbert (sLLG) equation is a strongly nonlinear stochastic PDE with a non-convex pointwise constraint arising in the theory of micromagnetics. We analyse a fully discrete, structure-preserving finite element approximation of the sLLG equation with coloured multiplicative Stratonovich noise on a bounded interval. The method utilises continuous piecewise affine finite elements, mass lumping, and midpoint time discretisation to preserve the unit-length constraint exactly at the finite element nodes. Under suitable regularity assumptions on the initial data and the noise, we establish uniform higher-moment stability and develop an error analysis for the scheme. The analysis exploits the geometric structure of the equation and the stochastic midpoint discretisation. For every $γ\in(0,\frac12)$, we prove first-order spatial convergence and temporal convergence of order $γ$ in the natural discrete energy norm, locally in mean square on events of arbitrarily large probability and, consequently, in probability. To the best of our knowledge, this is the first convergence-rate result for a fully discrete structure-preserving finite element scheme solving the stochastic Landau--Lifshitz--Gilbert equation.

math.NA↗
arXiv2026

Moments of crosscorrelation demerit factors of binary sequences

Families of sequences with low mutual aperiodic crosscorrelation assist the design of systems for multi-user asynchronous communications and multiple-input multiple-output radar. The crosscorrelation demerit factor of a pair of sequences is the sum of the squared magnitudes of their crosscorrelation values at every shift when the sequences are normalized to unit Euclidean norm, and the merit factor is the reciprocal of the demerit factor. For each positive integer $\ell$, we endow the $2^{2 \ell}$ pairs of binary sequences of length $\ell$ with uniform probability measure and study the distribution of their crosscorrelation demerit factors. Sarwate showed that the mean value is always $1$ regardless of length $\ell$. We develop a method for finding an exact formula for the $p$th central moment (for any positive integer $p$) as a function of $\ell$. Formulae for the variance and third central moment ($p=2$ and $3$) are then obtained by hand calculations, while the fourth through sixth central moments are obtained by computer-assisted calculations. Our theory also shows that all the central moments must be strictly positive for $p\geq 2$ and $\ell \geq 3$.

cs.IT↗
arXiv2026

Distinguishing classes of intersection graphs of homothets or similarities of two convex disks

For smooth convex disks $A$, i.e., convex compact subsets of the plane with non-empty interior and with at most one tangent at every boundary point, we classify the classes $G^{\text{hom}}(A)$ and $G^{\text{sim}}(A)$ of intersection graphs that can be obtained from homothets and similarities of $A$, respectively. Namely, we prove that $G^{\text{hom}}(A)=G^{\text{hom}}(B)$ if and only if $A$ and $B$ are affine equivalent, and $G^{\text{sim}}(A)=G^{\text{sim}}(B)$ if and only if $A$ and $B$ are similar.

cs.CG↗
arXiv2026

A fully globalized solver for discretized inverse elliptic coefficient problems with exact data

We consider finite-dimensional nonlinear inverse problems arising from finite element discretizations of elliptic inverse coefficient problems such as the Calderón problem with finitely many measurements and unknowns. Such inverse coefficient problems are notorious for their nonlinearity and ill-posedness, and numerical solvers tend to depend strongly on good initial values. In this work, we develop a new locally convergent algorithm with an explicit residual criterion that ensures convergence to the inverse problem solution, and a globalized variant that is guaranteed to automatically switch to the faster locally convergent algorithm after finitely many global search steps.

math.NA↗
arXiv2026

Geometric Ergodicity of Affine Invariant Ensemble Langevin and its Discrete Time Variants

Affine-invariant ensemble samplers are widely used in Bayesian applications. However, their quantitative convergence theory, in particular geometric ergodicity, remains a basic open question. We study the affine invariant ensemble Langevin dynamics, an interacting particle system that uses the empirical covariance of the whole ensemble as a preconditioner. While effective in practice, theoretical understanding of this method is not available beyond plain qualitative convergence in total variation; a central difficulty is that the empirical covariance can approach singularity. This paper addresses this challenge. For potentials with bounded Hessian that are strongly convex outside a ball, we prove geometric ergodicity using a novel Lyapunov function that combines an inverse-covariance barrier with a coercive exponential energy. We then show that directly applying the Euler--Maruyama scheme can diverge with positive probability, even for a one-dimensional Gaussian target. This motivates a covariance-trace time regularization. We prove geometric ergodicity of the regularized diffusion and, for sufficiently small step size, of its unadjusted Euler--Maruyama discretization. We also show that the invariant distributions of the discretization converge weakly to the product target distribution as the step size tends to zero.

math.ST↗
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↗