The power of randomized algorithms : from numerical linear algebra to biological systems

Abstract

In this thesis we study simple, randomized algorithms from a dual perspective. The first part of the work considers how randomized methods can be used to accelerate the solution of core problems in numerical linear algebra. In particular, we give a randomized low-rank approximation algorithm for positive semidefinite matrices that runs in sublinear time, significantly improving upon what is possible with traditional deterministic methods. We also discuss lower bounds on low-rank approximation and spectral summarization problems that attempt to explain the importance of randomization and approximation in accelerating linear algebraic computation. The second part of the work considers how the theory of randomized algorithms can be used more generally as a tool to understand how complexity emerges from low-level stochastic behavior in biological systems. We study population density- estimation in ant colonies, which is a key primitive in social decision-making and task allocation. We define a basic computational model and show how agents in this model can estimate their density using a simple random-walk-based algorithm. We also consider simple randomized algorithms for computational primitives in spiking neural networks, focusing on fast winner-take-all networks.