Determinantal Point Processes in Randomized Numerical Linear Algebra

05/07/2020
by   Michał Dereziński, et al.
0

Randomized Numerical Linear Algebra (RandNLA) uses randomness to develop improved algorithms for matrix problems that arise in scientific computing, data science, machine learning, etc. Determinantal Point Processes (DPPs), a seemingly unrelated topic in pure and applied mathematics, is a class of stochastic point processes with probability distribution characterized by sub-determinants of a kernel matrix. Recent work has uncovered deep and fruitful connections between DPPs and RandNLA which lead to new guarantees and improved algorithms that are of interest to both areas. We provide an overview of this exciting new line of research, including brief introductions to RandNLA and DPPs, as well as applications of DPPs to classical linear algebra tasks such as least squares regression, low-rank approximation and the Nyström method. For example, random sampling with a DPP leads to new kinds of unbiased estimators for least squares, enabling more refined statistical and inferential understanding of these algorithms; a DPP is, in some sense, an optimal randomized algorithm for the Nyström method; and a RandNLA technique called leverage score sampling can be derived as the marginal distribution of a DPP. We also discuss recent algorithmic developments, illustrating that, while not quite as efficient as standard RandNLA techniques, DPP-based algorithms are only moderately more expensive.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
12/24/2017

Lectures on Randomized Numerical Linear Algebra

This chapter is based on lectures on Randomized Numerical Linear Algebra...
research
12/17/2019

A literature survey of matrix methods for data science

Efficient numerical linear algebra is a core ingredient in many applicat...
research
02/04/2020

Randomized Numerical Linear Algebra: Foundations Algorithms

This survey describes probabilistic algorithms for linear algebra comput...
research
01/03/2018

Randomized Linear Algebra Approaches to Estimate the Von Neumann Entropy of Density Matrices

The von Neumann entropy, named after John von Neumann, is the extension ...
research
06/21/2022

Algorithmic Gaussianization through Sketching: Converting Data into Sub-gaussian Random Designs

Algorithmic Gaussianization is a phenomenon that can arise when using ra...
research
09/01/2020

A heuristic independent particle approximation to determinantal point processes

A determinantal point process is a stochastic point process that is comm...
research
11/09/2020

Quantum-Inspired Algorithms from Randomized Numerical Linear Algebra

We create classical (non-quantum) dynamic data structures supporting que...

Please sign up or login with your details

Forgot password? Click here to reset