Cameron Musco

Uniform Sampling for Matrix Approximation

Random sampling has become a critical tool in solving massive matrix problems. For linear regression, a small, manageable set of data rows can be randomly selected to approximate a tall, skinny data matrix, improving processing time significantly. For theoretical performance guarantees, each row must be sampled with probability proportional to its statistical leverage score. Unfortunately, leverage scores are difficult to compute. A simple alternative is to sample rows uniformly at random. While this often works, uniform sampling will eliminate critical row information for many natural instances. We take a fresh look at uniform sampling by examining what information it does preserve. Specifically, we show that uniform sampling yields a matrix that, in some sense, well approximates a large fraction of the original. While this weak form of approximation is not enough for solving linear regression directly, it is enough to compute a better approximation. This observation leads to simple iterative row sampling algorithms for matrix approximation that run in input-sparsity time and preserve row structure and sparsity at all intermediate steps. In addition to an improved understanding of uniform sampling, our main proof introduces a structural result of independent interest: we show that every matrix can be made to have low coherence by reweighting a small subset of its rows.

Single Pass Spectral Sparsification in Dynamic Streams

We present the first single pass algorithm for computing spectral sparsifiers of graphs in the dynamic semi-streaming model. Given a single pass over a stream containing insertions and deletions of edges to a graph, G, our algorithm maintains a randomized linear sketch of the incidence matrix into dimension O((1/aepsi;2) n polylog(n)). Using this sketch, the algorithm can output a (1 +/- aepsi;) spectral sparsifier for G with high probability. While O((1/aepsi;2) n polylog(n)) space algorithms are known for computing cut sparsifiers in dynamic streams [AGM12b, GKP12] and spectral sparsifiers in insertion-only streams [KL11], prior to our work, the best known single pass algorithm for maintaining spectral sparsifiers in dynamic streams required sketches of dimension a#x03A9;((1/aepsi;2) n(5/3)) [AGM14]. To achieve our result, we show that, using a coarse sparsifier of G and a linear sketch of Gs incidence matrix, it is possible to sample edges by effective resistance, obtaining a spectral sparsifier of arbitrary precision. Sampling from the sketch requires a novel application of ell2/ell2 sparse recovery, a natural extension of the ell0 methods used for cut sparsifiers in [AGM12b]. Recent work of [MP12] on row sampling for matrix approximation gives a recursive approach for obtaining the required coarse sparsifiers. Under certain restrictions, our approach also extends to the problem of maintaining a spectral approximation for a general matrix AT A given a stream of updates to rows in A.

Ant-inspired density estimation via random walks

Significance Highly complex distributed algorithms are ubiquitous in nature: from the behavior of social insect colonies and bird flocks, to cellular differentiation in embryonic development, to neural information processing. In our research, we study biological computation theoretically, combining a scientific perspective, which seeks to better understand the systems being studied, with an engineering perspective, which takes inspiration from these systems to improve algorithm design. In this work, we focus on the problem of population density estimation in ant colonies, demonstrating that extremely simple algorithms, similar to those used by ants, solve the problem with strong theoretical guarantees and have a number of interesting computational applications. Many ant species use distributed population density estimation in applications ranging from quorum sensing, to task allocation, to appraisal of enemy colony strength. It has been shown that ants estimate local population density by tracking encounter rates: The higher the density, the more often the ants bump into each other. We study distributed density estimation from a theoretical perspective. We prove that a group of anonymous agents randomly walking on a grid are able to estimate their density within a small multiplicative error in few steps by measuring their rates of encounter with other agents. Despite dependencies inherent in the fact that nearby agents may collide repeatedly (and, worse, cannot recognize when this happens), our bound nearly matches what would be required to estimate density by independently sampling grid locations. From a biological perspective, our work helps shed light on how ants and other social insects can obtain relatively accurate density estimates via encounter rates. From a technical perspective, our analysis provides tools for understanding complex dependencies in the collision probabilities of multiple random walks. We bound the strength of these dependencies using local mixing properties of the underlying graph. Our results extend beyond the grid to more general graphs, and we discuss applications to size estimation for social networks, density estimation for robot swarms, and random walk-based sampling for sensor networks.

Online Row Sampling

Finding a small spectral approximation for a tall

Ant-Inspired Density Estimation via Random Walks: Extended Abstract

n \times d

Input sparsity time low-rank approximation via ridge leverage score sampling

matrix

Distributed House-Hunting in Ant Colonies

A

Minimizing Polarization and Disagreement in Social Networks

is a fundamental numerical primitive. For a number of reasons, one often seeks an approximation whose rows are sampled from those of

Spectrum Approximation Beyond Fast Matrix Multiplication: Algorithms and Hardness

A

Sublinear Time Low-Rank Approximation of Positive Semidefinite Matrices

. Row sampling improves interpretability, saves space when