相关论文: Revisiting CUR Perturbation Analysis: A Local Tang…

In this paper, we focus on the fixed TT-rank and precision problems of finding an approximation of the tensor train (TT) decomposition of a tensor. Note that the TT-SVD and TT-cross are two well-known algorithms for these two problems.…

Updating a truncated Singular Value Decomposition (SVD) is crucial in representation learning, especially when dealing with large-scale data matrices that continuously evolve in practical scenarios. Aligning SVD-based models with fast-paced…

An optimization-based approach for the Tucker tensor approximation of parameter-dependent data tensors and solutions of tensor differential equations with low Tucker rank is presented. The problem of updating the tensor decomposition is…

Subsampled natural gradient descent (SNG) has been used to enable high-precision scientific machine learning, but standard analyses based on stochastic preconditioning fail to provide insight into realistic small-sample settings. We…

Reconstruction-based inference assigns a class by comparing class-wise reconstruction residuals; Sparse Representation Classification (SRC) is a canonical instance whose reliability depends on the geometry of the learned representation. We…

Optimization problems with rank constraints arise in many applications, including matrix regression, structured PCA, matrix completion and matrix decomposition problems. An attractive heuristic for solving such problems is to factorize the…

In the realm of tensor optimization, the low-rank Tucker decomposition is crucial for reducing the number of parameters and for saving storage. We explore the geometry of Tucker tensor varieties -- the set of tensors with bounded Tucker…

In this work, we analyze a sublinear-time algorithm for selecting a few rows and columns of a matrix for low-rank approximation purposes. The algorithm is based on an initial uniformly random selection of rows and columns, followed by a…

We show that given an estimate $\widehat{A}$ that is close to a general high-rank positive semi-definite (PSD) matrix $A$ in spectral norm (i.e., $\|\widehat{A}-A\|_2 \leq \delta$), the simple truncated SVD of $\widehat{A}$ produces a…

A CUR factorization is often utilized as a substitute for the singular value decomposition (SVD), especially when a concrete interpretation of the singular vectors is challenging. Moreover, if the original data matrix possesses properties…

This paper develops a spatially resolved perturbation theory for singular vectors under high-dimensional separable noise and applies it to data-driven matrix recovery. In the asymptotic regime where the matrix dimensions are proportional…

We consider quenched random perturbations of skew products of rotations on the unit circle over uniformly expanding maps on the unit circle. It is known that if the skew product satisfies a certain condition (shown to be generic in the case…

The manuscript describes efficient algorithms for the computation of the CUR and ID decompositions. The methods used are based on simple modifications to the classical truncated pivoted QR decomposition, which means that highly optimized…

Parametric system identification methods estimate the parameters of explicitly defined physical systems from data. Yet, they remain constrained by the need to provide an explicit function space, typically through a predefined library of…

In intelligent transportation systems, traffic data imputation, estimating the missing value from partially observed data is an inevitable and challenging task. Previous studies have not fully considered traffic data's multidimensionality…

Pseudospectra and structured pseudospectra are important tools for the analysis of matrices. Their computation, however, can be very demanding for all but small matrices. A new approach to compute approximations of pseudospectra and…

ConicCurv is a new derivative-free algorithm to estimate the curvature of a plane curve from a sample of data points. It is based on a known tangent estimator method grounded on classic results of Projective Geometry and B\'ezier rational…

Distributed optimization plays an important role in modern large-scale machine learning and data processing systems by optimizing the utilization of computational resources. One of the classical and popular approaches is Local Stochastic…

We develop two new sets of stable, rank-adaptive Dynamically Orthogonal Runge-Kutta (DORK) schemes that capture the high-order curvature of the nonlinear low-rank manifold. The DORK schemes asymptotically approximate the truncated singular…

Curvature influences generalization, robustness, and how reliably neural networks respond to small input perturbations. Existing sharpness metrics are typically defined in parameter space (e.g., Hessian eigenvalues) and can be expensive,…