We consider the problem of low-rank rectangular matrix completion in the regime where the matrix $M$ of size $n\times m$ is "long", i.e., the aspect ratio $m/n$ diverges to infinity. Such matrices are of particular interest in the study of tensor completion, where they arise from the unfolding of a low-rank tensor. In the case where the sampling probability is $\frac{d}{\sqrt{mn}}$, we propose a new spectral algorithm for recovering the singular values and left singular vectors of the original matrix $M$ based on a variant of the standard non-backtracking operator of a suitably defined bipartite weighted random graph, which we call a non-backtracking wedge operator. When $d$ is above a Kesten-Stigum-type sampling threshold, our algorithm recovers a correlated version of the singular value decomposition of $M$ with quantifiable error bounds. This is the first result in the regime of bounded $d$ for weak recovery and the first for weak consistency when $d\to\infty$ arbitrarily slowly without any polylog factors. As an application, for low-rank orthogonal $k$-tensor completion, we efficiently achieve weak recovery with sample size $O(n^{k/2})$, and weak consistency with sample size $\omega(n^{k/2})$.
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We extend generalized functional linear models under independence to a situation in which a functional covariate is related to a scalar response variable that exhibits spatial dependence. For estimation, we apply basis expansion and truncation for dimension reduction of the covariate process followed by a composite likelihood estimating equation to handle the spatial dependency. We develop asymptotic results for the proposed model under a repeating lattice asymptotic context, allowing us to construct a confidence interval for the spatial dependence parameter and a confidence band for the parameter function. A binary conditionals model is presented as a concrete illustration and is used in simulation studies to verify the applicability of the asymptotic inferential results.
We perturb a real matrix $A$ of full column rank, and derive lower bounds for the smallest singular values of the perturbed matrix, in terms of normwise absolute perturbations. Our bounds, which extend existing lower-order expressions, demonstrate the potential increase in the smallest singular values, and represent a qualitative model for the increase in the small singular values after a matrix has been downcast to a lower arithmetic precision. Numerical experiments confirm the qualitative validity of this model and its ability to predict singular values changes in the presence of decreased arithmetic precision.
We develop a sparse hierarchical $hp$-finite element method ($hp$-FEM) for the Helmholtz equation with rotationally invariant variable coefficients posed on a two-dimensional disk or annulus. The mesh is an inner disk cell (omitted if on an annulus domain) and concentric annuli cells. The discretization preserves the Fourier mode decoupling of rotationally invariant operators, such as the Laplacian, which manifests as block diagonal mass and stiffness matrices. Moreover, the matrices have a sparsity pattern independent of the order of the discretization and admit an optimal complexity factorization. The sparse $hp$-FEM can handle radial discontinuities in the right-hand side and in rotationally invariant Helmholtz coefficients. We consider examples such as a high-frequency Helmholtz equation with radial discontinuities, the time-dependent Schr\"odinger equation, and an extension to a three-dimensional cylinder domain, with a quasi-optimal solve, via the Alternating Direction Implicit (ADI) algorithm.
We give an algorithm that given a graph $G$ with $n$ vertices and $m$ edges and an integer $k$, in time $O_k(n^{1+o(1)}) + O(m)$ either outputs a rank decomposition of $G$ of width at most $k$ or determines that the rankwidth of $G$ is larger than $k$; the $O_k(\cdot)$-notation hides factors depending on $k$. Our algorithm returns also a $(2^{k+1}-1)$-expression for cliquewidth, yielding a $(2^{k+1}-1)$-approximation algorithm for cliquewidth with the same running time. This improves upon the $O_k(n^2)$ time algorithm of Fomin and Korhonen [STOC 2022]. The main ingredient of our algorithm is a fully dynamic algorithm for maintaining rank decompositions of bounded width: We give a data structure that for a dynamic $n$-vertex graph $G$ that is updated by edge insertions and deletions maintains a rank decomposition of $G$ of width at most $4k$ under the promise that the rankwidth of $G$ never grows above $k$. The amortized running time of each update is $O_k(2^{\sqrt{\log n} \log \log n})$. The data structure furthermore can maintain whether $G$ satisfies some fixed ${\sf CMSO}_1$ property within the same running time. We also give a framework for performing ``dense'' edge updates inside a given set of vertices $X$, where the new edges inside $X$ are described by a given ${\sf CMSO}_1$ sentence and vertex labels, in amortized $O_k(|X| \cdot 2^{\sqrt{\log n} \log \log n})$ time. Our dynamic algorithm generalizes the dynamic treewidth algorithm of Korhonen, Majewski, Nadara, Pilipczuk, and Soko{\l}owski [FOCS 2023].
The main reason for query model's prominence in complexity theory and quantum computing is the presence of concrete lower bounding techniques: polynomial and adversary method. There have been considerable efforts to give lower bounds using these methods, and to compare/relate them with other measures based on the decision tree. We explore the value of these lower bounds on quantum query complexity and their relation with other decision tree based complexity measures for the class of symmetric functions, arguably one of the most natural and basic sets of Boolean functions. We show an explicit construction for the dual of the positive adversary method and also of the square root of private coin certificate game complexity for any total symmetric function. This shows that the two values can't be distinguished for any symmetric function. Additionally, we show that the recently introduced measure of spectral sensitivity gives the same value as both positive adversary and approximate degree for every total symmetric Boolean function. Further, we look at the quantum query complexity of Gap Majority, a partial symmetric function. It has gained importance recently in regard to understanding the composition of randomized query complexity. We characterize the quantum query complexity of Gap Majority and show a lower bound on noisy randomized query complexity (Ben-David and Blais, FOCS 2020) in terms of quantum query complexity. Finally, we study how large certificate complexity and block sensitivity can be as compared to sensitivity for symmetric functions (even up to constant factors). We show tight separations, i.e., give upper bounds on possible separations and construct functions achieving the same.
The Weisfeiler-Leman dimension of a graph $G$ is the least number $k$ such that the $k$-dimensional Weisfeiler-Leman algorithm distinguishes $G$ from every other non-isomorphic graph. The dimension is a standard measure of the descriptive complexity of a graph and recently finds various applications in particular in the context of machine learning. In this paper, we study the computational complexity of computing the Weisfeiler-Leman dimension. We observe that in general the problem of deciding whether the Weisfeiler-Leman dimension of $G$ is at most $k$ is NP-hard. This is also true for the more restricted problem with graphs of color multiplicity at most 4. Therefore, we study parameterized and approximate versions of the problem. We give, for each fixed $k\geq 2$, a polynomial-time algorithm that decides whether the Weisfeiler-Leman dimension of a given graph of color multiplicity at most $5$ is at most $k$. Moreover, we show that for these color multiplicities this is optimal in the sense that this problem is P-hard under logspace-uniform $\text{AC}_0$-reductions. Furthermore, for each larger bound $c$ on the color multiplicity and each fixed $k \geq 2$, we provide a polynomial-time approximation algorithm for the abelian case: given a relational structure with abelian color classes of size at most $c$, the algorithm outputs either that its Weisfeiler-Leman dimension is at most $(k+1)c$ or that it is larger than $k$.
We present an efficient preconditioner for two-by-two block system of linear equations with the coefficient matrix $ \begin{pmatrix} F & -G^H G & F \end{pmatrix}$ where $F\in\mathbb{C}^{n\times n}$ is Hermitian positive definite and $G\in\mathbb{C}^{n\times n}$ is positive semidefinite. Spectral analysis of the preconditioned matrix is analyzed. In each iteration of a Krylov subspace method, like GMRES, for solving the preconditioned system in conjunction with proposed preconditioner two subsystems with Hermitian positive definite coefficient matrix should be solved which can be accomplished exactly using the Cholesky factorization or inexactly using the conjugate gradient method. Application of the proposed preconditioner to the systems arising from finite element discretization of PDE-constrained optimization problems is presented. Numerical results are given to demonstrate the efficiency of the preconditioner.
In this paper, we examine the relationship between the stability of the dynamical system $x^{\prime}=f(x)$ and the computability of its basins of attraction. We present a computable $C^{\infty}$ system $x^{\prime}=f(x)$ that possesses a computable and stable equilibrium point, yet whose basin of attraction is robustly non-computable in a neighborhood of $f$ in the sense that both the equilibrium point and the non-computability of its associated basin of attraction persist when $f$ is slightly perturbed. This indicates that local stability near a stable equilibrium point alone is insufficient to guarantee the computability of its basin of attraction. However, we also demonstrate that the basins of attraction associated with a structurally stable - globally stable - planar system defined on a compact set are computable. Our findings suggest that the global stability of a system and the compactness of the domain play a pivotal role in determining the computability of its basins of attraction.
The uniform interpolation property in a given logic can be understood as the definability of propositional quantifiers. We mechanise the computation of these quantifiers and prove correctness in the Coq proof assistant for three modal logics, namely: (1) the modal logic K, for which a pen-and-paper proof exists; (2) G\"odel-L\"ob logic GL, for which our formalisation clarifies an important point in an existing, but incomplete, sequent-style proof; and (3) intuitionistic strong L\"ob logic iSL, for which this is the first proof-theoretic construction of uniform interpolants. Our work also yields verified programs that allow one to compute the propositional quantifiers on any formula in this logic.
Wavelet compressed, modified Hilbert transform in the space-time discretization of the heat equation
On a finite time interval $(0,T)$, we consider the multiresolution Galerkin discretization of a modified Hilbert transform $\mathcal H_T$ which arises in the space-time Galerkin discretization of the linear diffusion equation. To this end, we design spline-wavelet systems in $(0,T)$ consisting of piecewise polynomials of degree $\geq 1$ with sufficiently many vanishing moments which constitute Riesz bases in the Sobolev spaces $ H^{s}_{0,}(0,T)$ and $ H^{s}_{,0}(0,T)$. These bases provide multilevel splittings of the temporal discretization spaces into "increment" or "detail" spaces of direct sum type. Via algebraic tensor-products of these temporal multilevel discretizations with standard, hierarchic finite element spaces in the spatial domain (with standard Lagrangian FE bases), sparse space-time tensor-product spaces are obtained, which afford a substantial reduction in the number of the degrees of freedom as compared to time-marching discretizations. In addition, temporal spline-wavelet bases allow to compress certain nonlocal integrodifferential operators which appear in stable space-time variational formulations of initial-boundary value problems, such as the heat equation and the acoustic wave equation. An efficient preconditioner is proposed that affords linear complexity solves of the linear system of equations which results from the sparse space-time Galerkin discretization.
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