Daniel A. Spielman

Samuel I. Daitch, Daniel A. Spielman

We present faster approximation algorithms for generalized network flow problems. A generalized flow is one in which the flow out of an edge differs from the flow into the edge by a constant factor. We limit ourselves to the lossy case, when these factors are at most 1. Our algorithm uses a standard interior-point algorithm to solve a linear program formulation of the network flow problem. The system of linear equations that arises at each step of the interior-point algorithm takes the form of a symmetric M-matrix. We present an algorithm for solving such systems in nearly linear time. The algorithm relies on the Spielman-Teng nearly linear time algorithm for solving linear systems in diagonally-dominant matrices. For a graph with m edges, our algorithm obtains an additive epsilon approximation of the maximum generalized flow and minimum cost generalized flow in time tildeO(m^(3/2) * log(1/epsilon)). In many parameter ranges, this improves over previous algorithms by a factor of approximately m^(1/2). We also obtain a similar improvement for exactly solving the standard min-cost flow problem.

Rasmus Kyng, Anup Rao, Sushant Sachdeva et al.

We develop fast algorithms for solving regression problems on graphs where one is given the value of a function at some vertices, and must find its smoothest possible extension to all vertices. The extension we compute is the absolutely minimal Lipschitz extension, and is the limit for large $p$ of $p$-Laplacian regularization. We present an algorithm that computes a minimal Lipschitz extension in expected linear time, and an algorithm that computes an absolutely minimal Lipschitz extension in expected time $\widetilde{O} (m n)$. The latter algorithm has variants that seem to run much faster in practice. These extensions are particularly amenable to regularization: we can perform $l_{0}$-regularization on the given values in polynomial time and $l_{1}$-regularization on the initial function values and on graph edge weights in time $\widetilde{O} (m^{3/2})$.

Daniel A. Spielman, Huan Wang, John Wright

We consider the problem of learning sparsely used dictionaries with an arbitrary square dictionary and a random, sparse coefficient matrix. We prove that $O (n \log n)$ samples are sufficient to uniquely determine the coefficient matrix. Based on this proof, we design a polynomial-time algorithm, called Exact Recovery of Sparsely-Used Dictionaries (ER-SpUD), and prove that it probably recovers the dictionary and coefficient matrix when the coefficient matrix is sufficiently sparse. Simulation results show that ER-SpUD reveals the true dictionary as well as the coefficients with probability higher than many state-of-the-art algorithms.