Speeding Up Sparsification using Inner Product Search Data Structures

TL;DR

This paper introduces a framework using specialized data structures to accelerate sparsification processes, achieving faster algorithms for spectral sparsifiers, discrepancy problems, and experimental design, with broad applicability.

Contribution

It presents the first deterministic algorithm for spectral sparsifier construction with improved runtime and develops efficient inner product search data structures for various iterative problems.

Findings

Deterministic spectral sparsifier algorithm with $O(d^{ ext{omega}+1})$ time

Fast algorithms for Kadison-Singer discrepancy problems

Speed-up of key swapping in experimental design

Abstract

We present a general framework that utilizes different efficient data structures to improve various sparsification problems involving an iterative process. We also provide insights and characterization for different iterative process, and answer that when should we use which data structures in what type of problem. We obtain improved running time for the following problems. * For constructing linear-sized spectral sparsifier (Batson, Spielman and Srivastava, 2012), all the existing deterministic algorithms require $\Omega(d^4)$ time. In this work, we provide the first deterministic algorithm that breaks that barrier which runs in $O(d^{\omega+1})$ time, where $\omega$ is the exponent of matrix multiplication. * For one-sided Kadison-Singer-typed discrepancy problem, we give fast algorithms for both small and large number of iterations. * For experimental design problem, we speed up a key swapping process. In the heart of our work is the design of a variety of different inner product search data structures that have efficient initialization, query and update time, compatible to dimensionality reduction and robust against adaptive adversary.