Determinantal Point Processes in Randomized Numerical Linear Algebra
This work bridges two mathematical areas to enhance algorithms for matrix problems in scientific computing and data science, though it is incremental as it builds on existing RandNLA techniques.
The paper explores connections between Determinantal Point Processes (DPPs) and Randomized Numerical Linear Algebra (RandNLA), leading to new guarantees and improved algorithms for tasks like least squares regression and low-rank approximation, such as DPP-based unbiased estimators for least squares and optimal algorithms for the Nyström method.
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.