We investigate some Bernstein–Gelfand–Gelfand complexes consisting of Sobolev spaces on bounded Lipschitz domains in $${\mathbb {R}}^{n}$$ . In particular, we compute the cohomology of the conformal deformation complex and the conformal Hessian complex in the Sobolev setting. The machinery does not require algebraic injectivity/surjectivity conditions between the input spaces...
We study the computational complexity of (deterministic or randomized) algorithms based on point samples for approximating or integrating functions that can be well approximated by neural networks. Such algorithms (most prominently stochastic gradient descent and its variants) are used extensively in the field of deep learning. One of the most important problems in this field...
We present a Lohner-type algorithm for rigorous integration of systems of delay differential equations (DDEs) with multiple delays, and its application in computation of Poincaré maps, to study the dynamics of some bounded, eternal solutions. The algorithm is based on a piecewise Taylor representation of the solutions in the phase space, and it exploits the smoothing of solutions...
We study the representative volume element (RVE) method, which is a method to approximately infer the effective behavior $$a_{\textrm{hom}}$$ of a stationary random medium. The latter is described by a coefficient field a(x) generated from a given ensemble $$\langle \cdot \rangle $$ and the corresponding linear elliptic operator $$-\nabla \cdot a\nabla $$ . In line with the...
We consider the problem of estimating expectations with respect to a target distribution with an unknown normalising constant, and where even the un-normalised target needs to be approximated at finite resolution. This setting is ubiquitous across science and engineering applications, for example in the context of Bayesian inference where a physics-based model governed by an...
We study symmetries of bases and spanning sets in finite element exterior calculus, using representation theory. We want to know which vector-valued finite element spaces have bases invariant under permutation of vertex indices. The permutations of vertex indices correspond to the symmetry group of the simplex. That symmetry group is represented on simplicial finite element...
We present here a novel approach to handling curved meshes in polytopal methods within the framework of hybrid high-order methods. The hybrid high-order method is a modern numerical scheme for the approximation of elliptic PDEs. An extension to curved meshes allows for the strong enforcement of boundary conditions on curved domains and for the capture of curved geometries that...
This note remedies an error in our paper tilted Conormal Spaces and Whitney Stratifications (Found. Comput. Math., 2022).
The discrete distribution of the length of longest increasing subsequences in random permutations of n integers is deeply related to random matrix theory. In a seminal work, Baik, Deift and Johansson provided an asymptotics in terms of the distribution of the scaled largest level of the large matrix limit of GUE. As a numerical approximation, however, this asymptotics is...
A periodic lattice in Euclidean space is the infinite set of all integer linear combinations of basis vectors. Any lattice can be generated by infinitely many different bases. This ambiguity was partially resolved, but standard reductions remain discontinuous under perturbations modelling atomic displacements. This paper completes a continuous classification of 2-dimensional...
Computing spectra is a central problem in computational mathematics with an abundance of applications throughout the sciences. However, in many applications gaining an approximation of the spectrum is not enough. Often it is vital to determine geometric features of spectra such as Lebesgue measure, capacity or fractal dimensions, different types of spectral radii and numerical...
Suppose $$A\!=\!\{a_1,\ldots ,a_{n+2}\}\!\subset \!\mathbb {Z}^n$$ has cardinality $$n+2$$ , with all the coordinates of the $$a_j$$ having absolute value at most d, and the $$a_j$$ do not all lie in the same affine hyperplane. Suppose $$F\!=\!(f_1,\ldots ,f_n)$$ is an $$n\times n$$ polynomial system with generic integer coefficients at most H in absolute value, and A the union...
We study the computation of the zero set of the Bargmann transform of a signal contaminated with complex white noise, or, equivalently, the computation of the zeros of its short-time Fourier transform with Gaussian window. We introduce the adaptive minimal grid neighbors algorithm (AMN), a variant of a method that has recently appeared in the signal processing literature, and...
The protection of private information is of vital importance in data-driven research, business and government. The conflict between privacy and utility has triggered intensive research in the computer science and statistics communities, who have developed a variety of methods for privacy-preserving data release. Among the main concepts that have emerged are anonymity and...
We exhibit a randomized algorithm which, given a square matrix $$A\in \mathbb {C}^{n\times n}$$ with $$\Vert A\Vert \le 1$$ and $$\delta >0$$ , computes with high probability an invertible V and diagonal D such that $$ \Vert A-VDV^{-1}\Vert \le \delta $$ using $$O(T_\mathsf {MM}(n)\log ^2(n/\delta ))$$ arithmetic operations, in finite arithmetic with $$O(\log ^4(n/\delta )\log n...
In this article, we introduce a new family of lattice polytopes with rational linear precision. For this purpose, we define a new class of discrete statistical models that we call multinomial staged tree models. We prove that these models have rational maximum likelihood estimators (MLE) and give a criterion for these models to be log-linear. Our main result is then obtained by...
In this review, we discuss approaches for learning causal structure from data, also called causal discovery. In particular, we focus on approaches for learning directed acyclic graphs and various generalizations which allow for some variables to be unobserved in the available data. We devote special attention to two fundamental combinatorial aspects of causal structure learning...
In certain polytopal domains $$\varOmega $$ , in space dimension $$d=2,3$$ , we prove exponential expressivity with stable ReLU Neural Networks (ReLU NNs) in $$H^1(\varOmega )$$ for weighted analytic function classes. These classes comprise in particular solution sets of source and eigenvalue problems for elliptic PDEs with analytic data. Functions in these classes are locally...
We describe a new algorithm for computing Whitney stratifications of complex projective varieties. The main ingredients are (a) an algebraic criterion, due to Lê and Teissier, which reformulates Whitney regularity in terms of conormal spaces and maps, and (b) a new interpretation of this conormal criterion via ideal saturations, which can be practically implemented on a computer...
The classical notion of retraction map used to approximate geodesics is extended and rigorously defined to become a powerful tool to construct geometric integrators and it is called discretization map. Using the geometry of the tangent and cotangent bundles, we are able to tangently and cotangent lift such a map so that these lifts inherit the same properties as the original one...
We describe the first gradient methods on Riemannian manifolds to achieve accelerated rates in the non-convex case. Under Lipschitz assumptions on the Riemannian gradient and Hessian of the cost function, these methods find approximate first-order critical points faster than regular gradient descent. A randomized version also finds approximate second-order critical points. Both...
We construct numerical approximations for Mean Field Games with fractional or nonlocal diffusions. The schemes are based on semi-Lagrangian approximations of the underlying control problems/games along with dual approximations of the distributions of agents. The methods are monotone, stable, and consistent, and we prove convergence along subsequences for (i) degenerate equations...
Geometric, robust-to-noise features of curves in Euclidean space are of great interest for various applications such as machine learning and image analysis. We apply Fels–Olver’s moving-frame method (for geometric features) paired with the log-signature transform (for robust features) to construct a set of integral invariants under rigid motions for curves in $${\mathbb {R}}^d...
Quiver representations arise naturally in many areas across mathematics. Here we describe an algorithm for calculating the vector space of sections, or compatible assignments of vectors to vertices, of any finite-dimensional representation of a finite quiver. Consequently, we are able to define and compute principal components with respect to quiver representations. These...
Formal power series products appear in nonlinear control theory when systems modeled by Chen–Fliess series are interconnected to form new systems. In fields like adaptive control and learning systems, the coefficients of these formal power series are estimated sequentially with real-time data. The main goal is to prove the continuity and analyticity of such products with respect...