Related papers: Improved Rectangular Matrix Multiplication using P…
We investigate the power iteration algorithm for the tensor PCA model introduced in Richard and Montanari (2014). Previous work studying the properties of tensor power iteration is either limited to a constant number of iterations, or…
Motivated by problems in algebraic complexity theory (e.g., matrix multiplication) and extremal combinatorics (e.g., the cap set problem and the sunflower problem), we introduce the geometric rank as a new tool in the study of tensors and…
The learnable, linear neural network layers between tensor power spaces of $\mathbb{R}^{n}$ that are equivariant to the orthogonal group, $O(n)$, the special orthogonal group, $SO(n)$, and the symplectic group, $Sp(n)$, were characterised…
Recent work in machine learning community proposed multiple methods for performing lossy compression (quantization) of large matrices. This quantization is important for accelerating matrix multiplication (main component of large language…
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…
In this paper, we introduce novel fast matrix inversion algorithms that leverage triangular decomposition and recurrent formalism, incorporating Strassen's fast matrix multiplication. Our research places particular emphasis on triangular…
Fast matrix multiplication can be described as searching for low-rank decompositions of the matrix--multiplication tensor. We design a neural architecture, \textsc{StrassenNet}, which reproduces the Strassen algorithm for $2\times 2$…
We observe that any $T(n)$ time algorithm (quantum or classical) for several central linear algebraic problems, such as computing $\det(A)$, $tr(A^3)$, or $tr(A^{-1})$ for an $n \times n$ integer matrix $A$, yields a $O(T(n)) + \tilde…
Fast matrix multiplication algorithms may be useful, provided that their running time is good in practice. Particularly, the leading coefficient of their arithmetic complexity needs to be small. Many sub-cubic algorithms have large leading…
A cumbersome operation in numerical analysis and linear algebra, optimization, machine learning and engineering algorithms; is inverting large full-rank matrices which appears in various processes and applications. This has both numerical…
Recovery of low-rank matrices from a small number of linear measurements is now well-known to be possible under various model assumptions on the measurements. Such results demonstrate robustness and are backed with provable theoretical…
Triangular factorizations are an important tool for solving integral equations and partial differential equations with hierarchical matrices ($\mathcal{H}$-matrices). Experiments show that using an $\mathcal{H}$-matrix LR factorization to…
The recently introduced collaborative nonnegative matrix factorization (CoNMF) algorithm was conceived to simultaneously estimate the number of endmembers, the mixing matrix, and the fractional abundances from hyperspectral linear mixtures.…
This paper introduces an efficient algorithm for finding the dominant generalized eigenvectors of a pair of symmetric matrices. Combining tools from approximation theory and convex optimization, we develop a simple scalable algorithm with…
The theory of matrix splitting is a useful tool for finding solution of rectangular linear system of equations, iteratively. The purpose of this paper is two-fold. Firstly, we revisit theory of weak regular splittings for rectangular…
There is a significant expansion in both volume and range of applications along with the concomitant increase in the variety of data sources. These ever-expanding trends have highlighted the necessity for more versatile analysis tools that…
The {\it matrix-chain multiplication} problem is a classic problem that is widely taught to illustrate dynamic programming. The textbook solution runs in $\theta(n^3)$ time. However, there is a complex $O(n \log n)$-time method \cite{HU82},…
We study quantum algorithms for approximating Lasserre's hierarchy values for polynomial optimization. Let $f,g_1,\ldots,g_m$ be real polynomials in $n$ variables and $f^\star$ the infimum of $f$ over the semialgebraic set $S(g)=\{x:…
Tensor decomposition serves as a powerful primitive in statistics and machine learning, and has numerous applications in problems such as learning latent variable models or mixture of Gaussians. In this paper, we focus on using power…
We give new, smaller constructions of constant-depth linear circuits for computing any matrix which is the Kronecker power of a fixed matrix. A standard argument (e.g., the mixed product property of Kronecker products, or a generalization…