Related papers: Simultaneous computation of the row and column ran…
This is the second in a series of papers on rank decompositions of the matrix multiplication tensor. We present new rank $23$ decompositions for the $3\times 3$ matrix multiplication tensor $M_{\langle 3\rangle}$. All our decompositions…
This paper presents a hierarchical low-rank decomposition algorithm assuming any matrix element can be computed in $O(1)$ time. The proposed algorithm computes rank-revealing decompositions of sub-matrices with a blocked adaptive cross…
Large Language Models (LLM) have been widely used in reranking. Computational overhead and large context lengths remain a challenging issue for LLM rerankers. Efficient reranking usually involves selecting a subset of the ranked list from…
A new scaling and recovering algorithm is proposed for simultaneously computing the matrix $\varphi$-functions that arise in exponential integrator methods for the numerical solution of certain first-order systems of ordinary differential…
We present a recursive algorithm for multi-coefficient inversion in nonlinear Helmholtz equations with polynomial-type nonlinearities, utilizing the linearized Dirichlet-to-Neumann map as measurement data. To achieve effective recursive…
This work studies the combinatorial optimization problem of finding an optimal core tensor shape, also called multilinear rank, for a size-constrained Tucker decomposition. We give an algorithm with provable approximation guarantees for its…
Traditional low-rank approximation is a powerful tool to compress the huge data matrices that arise in simulations of partial differential equations (PDE), but suffers from high computational cost and requires several passes over the PDE…
Estimation of the precision matrix (or inverse covariance matrix) is of great importance in statistical data analysis and machine learning. However, as the number of parameters scales quadratically with the dimension $p$, computation…
Linear matrix equations, such as the Sylvester and Lyapunov equations, play an important role in various applications, including the stability analysis and dimensionality reduction of linear dynamical control systems and the solution of…
We present a new paradigm for speeding up randomized computations of several frequently used functions in machine learning. In particular, our paradigm can be applied for improving computations of kernels based on random embeddings. Above…
We give a new approach to the dictionary learning (also known as "sparse coding") problem of recovering an unknown $n\times m$ matrix $A$ (for $m \geq n$) from examples of the form \[ y = Ax + e, \] where $x$ is a random vector in $\mathbb…
This work considers a computationally and statistically efficient parameter estimation method for a wide class of latent variable models---including Gaussian mixture models, hidden Markov models, and latent Dirichlet allocation---which…
We study several variants of decomposing a symmetric matrix into a sum of a low-rank positive semidefinite matrix and a diagonal matrix. Such decompositions have applications in factor analysis and they have been studied for many decades.…
We consider low-rank reconstruction of a matrix using its columns and we present asymptotically optimal algorithms for both spectral norm and Frobenius norm reconstruction. The main tools we introduce to obtain our r esults are: (i) the use…
We propose a unified framework to solve general low-rank plus sparse matrix recovery problems based on matrix factorization, which covers a broad family of objective functions satisfying the restricted strong convexity and smoothness…
Low rank tensor decompositions are a powerful tool for learning generative models, and uniqueness results give them a significant advantage over matrix decomposition methods. However, tensors pose significant algorithmic challenges and…
What learning algorithms can be run directly on compressively-sensed data? In this work, we consider the question of accurately and efficiently computing low-rank matrix or tensor factorizations given data compressed via random projections.…
We revisit the problem of robust principal component analysis with features acting as prior side information. To this aim, a novel, elegant, non-convex optimization approach is proposed to decompose a given observation matrix into a…
We present a novel framework for Linear Combination of Unitaries (LCU)-style decomposition tailored to structured sparse matrices, which frequently arise in the numerical solution of partial differential equations (PDEs). While LCU is a…
In this paper, we develop a nonconvex approach to the problem of low-rank and sparse matrix decomposition. In our nonconvex method, we replace the rank function and the $l_{0}$-norm of a given matrix with a non-convex fraction function on…