Related papers: Shifted and extrapolated power methods for tensor …
Finding a Z-eigenpair of a symmetric tensor is equivalent to finding a KKT point of a sphere constrained minimization problem. Based on this equivalency, in this paper, we first propose a class of iterative methods to get a Z-eigenpair of a…
While Spectral Methods have long been used for Principal Component Analysis, this survey focusses on work over the last 15 years with three salient features: (i) Spectral methods are useful not only for numerical problems, but also discrete…
We study the $\ell_p$ regression problem, which requires finding $\mathbf{x}\in\mathbb R^{d}$ that minimizes $\|\mathbf{A}\mathbf{x}-\mathbf{b}\|_p$ for a matrix $\mathbf{A}\in\mathbb R^{n \times d}$ and response vector…
Tensors are a fundamental data structure for many scientific contexts, such as time series analysis, materials science, and physics, among many others. Improving our ability to produce and handle tensors is essential to efficiently address…
We propose a new numerical algorithm for computing the tensor rank decomposition or canonical polyadic decomposition of higher-order tensors subject to a rank and genericity constraint. Reformulating this computational problem as a system…
This paper is concerned with the design and analysis of a fully adaptive eigenvalue solver for linear symmetric operators. After transforming the original problem into an equivalent one formulated on $\ell_2$, the space of square summable…
In this paper, we discuss how a suitable family of tensor kernels can be used to efficiently solve nonparametric extensions of $\ell^p$ regularized learning methods. Our main contribution is proposing a fast dual algorithm, and showing that…
This paper introduces the notion of tubular eigenvalues of third-order tensors with respect to T-products of tensors and analyzes their properties. A focus of the paper is to discuss relations between tubular eigenvalues and two alternative…
Recently, Ko\c{c} proposed a neat and efficient algorithm for computing \[ x = a^{-1} \pmod {p^k} \] for a prime $p$ based on the exact solution of linear equations using $p$-adic expansions. The algorithm requires only addition and right…
The theory and computation of tensors with different tensor products play increasingly important roles in scientific computing and machine learning. Different products aim to preserve different algebraic properties from the matrix algebra,…
High-dimensional partial differential equations (PDEs) are ubiquitous in economics, science and engineering. However, their numerical treatment poses formidable challenges since traditional grid-based methods tend to be frustrated by the…
In past few decades, tensor algebra also known as multi-linear algebra has been developed and customized as a tool to be used for various engineering applications. In particular, with the help of a special form of tensor contracted product,…
Approximation of high-dimensional functions is a problem in many scientific fields that is only feasible if advantageous structural properties, such as sparsity in a given basis, can be exploited. A relevant tool for analysing sparse…
We provide new methods to straightforwardly obtain compact and analytic expressions for epsilon-expansions of functions appearing in both field and string theory amplitudes. An algebraic method is presented to explicitly solve for…
Well known connections exist between the singular value decomposition of a matrix A and the Schur decomposition of its symmetric embedding sym(A) = [ 0 A; A' 0]. In particular, if s is a singular value of A then +s and -s are eigenvalues of…
The outer inverse of tensors plays increasingly significant roles in computational mathematics, numerical analysis, and other generalized inverses of tensors. In this paper, we compute outer inverses with prescribed ranges and kernels of a…
Simulation of quantum systems is challenging due to the exponential size of the state space. Tensor networks provide a systematically improvable approximation for quantum states. 2D tensor networks such as Projected Entangled Pair States…
We resume the recent successes of the grid-based tensor numerical methods and discuss their prospects in real-space electronic structure calculations. These methods, based on the low-rank representation of the multidimensional functions and…
This paper surveys randomized algorithms in numerical linear algebra for low-rank decompositions of matrices and tensors. The survey begins with a review of classical matrix algorithms that can be accelerated by randomized dimensionality…
In this paper, we study the power iteration algorithm for the spiked tensor model, as introduced in [44]. We give necessary and sufficient conditions for the convergence of the power iteration algorithm. When the power iteration algorithm…