arXiv subjects

Yin Tat Lee

Publications and source records attributed to Yin Tat Lee.

103 records · Page 6Linked to original sources

arXiv2015

Efficient Inverse Maintenance and Faster Algorithms for Linear Programming

In this paper, we consider the following inverse maintenance problem: given $A \in \mathbb{R}^{n\times d}$ and a number of rounds $r$, we receive a $n\times n$ diagonal matrix $D^{(k)}$ at round $k$ and we wish to maintain an efficient linear system solver for $A^{T}D^{(k)}A$ under the assumption $D^{(k)}$ does not change too rapidly. This inverse maintenance problem is the computational bottleneck in solving multiple optimization problems. We show how to solve this problem with $\tilde{O}(nnz(A)+d^ω)$ preprocessing time and amortized $\tilde{O}(nnz(A)+d^{2})$ time per round, improving upon previous running times for solving this problem. Consequently, we obtain the fastest known running times for solving multiple problems including, linear programming and computing a rounding of a polytope. In particular given a feasible point in a linear program with $d$ variables, $n$ constraints, and constraint matrix $A\in\mathbb{R}^{n\times d}$, we show how to solve the linear program in time $\tilde{O}(nnz(A)+d^{2})\sqrt{d}\log(ε^{-1}))$. We achieve our results through a novel combination of classic numerical techniques of low rank update, preconditioning, and fast matrix multiplication as well as recent work on subspace embeddings and spectral sparsification that we hope will be of independent interest.

cs.DS↗

arXiv2015

Sparsified Cholesky Solvers for SDD linear systems

We show that Laplacian and symmetric diagonally dominant (SDD) matrices can be well approximated by linear-sized sparse Cholesky factorizations. We show that these matrices have constant-factor approximations of the form $L L^{T}$, where $L$ is a lower-triangular matrix with a number of nonzero entries linear in its dimension. Furthermore linear systems in $L$ and $L^{T}$ can be solved in $O (n)$ work and $O(\log{n}\log^2\log{n})$ depth, where $n$ is the dimension of the matrix. We present nearly linear time algorithms that construct solvers that are almost this efficient. In doing so, we give the first nearly-linear work routine for constructing spectral vertex sparsifiers---that is, spectral approximations of Schur complements of Laplacian matrices.

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arXiv2015

Constructing Linear-Sized Spectral Sparsification in Almost-Linear Time

We present the first almost-linear time algorithm for constructing linear-sized spectral sparsification for graphs. This improves all previous constructions of linear-sized spectral sparsification, which requires $Ω(n^2)$ time. A key ingredient in our algorithm is a novel combination of two techniques used in literature for constructing spectral sparsification: Random sampling by effective resistance, and adaptive constructions based on barrier functions.

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arXiv2015

A geometric alternative to Nesterov's accelerated gradient descent

We propose a new method for unconstrained optimization of a smooth and strongly convex function, which attains the optimal rate of convergence of Nesterov's accelerated gradient descent. The new algorithm has a simple geometric interpretation, loosely inspired by the ellipsoid method. We provide some numerical evidence that the new method can be superior to Nesterov's accelerated gradient descent.

math.OC↗

arXiv2015

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 of G into dimension O((1/epsilon^2) n polylog(n)). Using this sketch, at any point, the algorithm can output a (1 +/- epsilon) spectral sparsifier for G with high probability. While O((1/epsilon^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 Omega((1/epsilon^2) n^(5/3)) [AGM14]. To achieve our result, we show that, using a coarse sparsifier of G and a linear sketch of G's 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 ell_2/ell_2 sparse recovery, a natural extension of the ell_0 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 A^T A given a stream of updates to rows in A.

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arXiv2015

Improved Cheeger's Inequality and Analysis of Local Graph Partitioning using Vertex Expansion and Expansion Profile

We prove two generalizations of the Cheeger's inequality. The first generalization relates the second eigenvalue to the edge expansion and the vertex expansion of the graph G, $λ_2 = Ω(ϕ^V(G) ϕ(G))$, where $ϕ^V(G)$ denotes the robust vertex expansion of G and $ϕ(G)$ denotes the edge expansion of G. The second generalization relates the second eigenvalue to the edge expansion and the expansion profile of G, for all $k \ge 2$, $λ_2 = Ω(ϕ_k(G) ϕ(G) / k)$, where $ϕ_k(G)$ denotes the k-way expansion of G. These show that the spectral partitioning algorithm has better performance guarantees when $ϕ^V(G)$ is large (e.g. planted random instances) or $ϕ_k(G)$ is large (instances with few disjoint non-expanding sets). Both bounds are tight up to a constant factor. Our approach is based on a method to analyze solutions of Laplacian systems, and this allows us to extend the results to local graph partitioning algorithms. In particular, we show that our approach can be used to analyze personal pagerank vectors, and to give a local graph partitioning algorithm for the small-set expansion problem with performance guarantees similar to the generalizations of Cheeger's inequality. We also present a spectral approach to prove similar results for the truncated random walk algorithm. These show that local graph partitioning algorithms almost match the performance of the spectral partitioning algorithm, with the additional advantages that they apply to the small-set expansion problem and their running time could be sublinear. Our techniques provide common approaches to analyze the spectral partitioning algorithm and local graph partitioning algorithms.

cs.DS↗

arXiv2015

Path Finding I :Solving Linear Programs with Õ(sqrt(rank)) Linear System Solves

In this paper we present a new algorithm for solving linear programs that requires only $\tilde{O}(\sqrt{rank(A)}L)$ iterations to solve a linear program with $m$ constraints, $n$ variables, and constraint matrix $A$, and bit complexity $L$. Each iteration of our method consists of solving $\tilde{O}(1)$ linear systems and additional nearly linear time computation. Our method improves upon the previous best iteration bound by factor of $\tildeΩ((m/rank(A))^{1/4})$ for methods with polynomial time computable iterations and by $\tildeΩ((m/rank(A))^{1/2})$ for methods which solve at most $\tilde{O}(1)$ linear systems in each iteration. Our method is parallelizable and amenable to linear algebraic techniques for accelerating the linear system solver. As such, up to polylogarithmic factors we either match or improve upon the best previous running times in both depth and work for different ratios of $m$ and $rank(A)$. Moreover, our method matches up to polylogarithmic factors a theoretical limit established by Nesterov and Nemirovski in 1994 regarding the use of a "universal barrier" for interior point methods, thereby resolving a long-standing open question regarding the running time of polynomial time interior point methods for linear programming.

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arXiv2015

Path Finding II : An Õ(m sqrt(n)) Algorithm for the Minimum Cost Flow Problem

In this paper we present an $\tilde{O}(m\sqrt{n}\log^{O(1)}U)$ time algorithm for solving the maximum flow problem on directed graphs with $m$ edges, $n$ vertices, and capacity ratio $U$. This improves upon the previous fastest running time of $O(m\min\left(n^{2/3},m^{1/2}\right)\log\left(n^{2}/m\right)\log U)$ achieved over 15 years ago by Goldberg and Rao. In the special case of solving dense directed unit capacity graphs our algorithm improves upon the previous fastest running times of of $O(\min\{m^{3/2},mn^{^{2/3}}\})$ achieved by Even and Tarjan and Karzanov over 35 years ago and of $\tilde{O}(m^{10/7})$ achieved recently by Mądry. We achieve these results through the development and application of a new general interior point method that we believe is of independent interest. The number of iterations required by this algorithm is better than that predicted by analyzing the best self-concordant barrier of the feasible region. By applying this method to the linear programming formulations of maximum flow, minimum cost flow, and lossy generalized minimum cost flow and applying analysis by Daitch and Spielman we achieve running time of $\tilde{O}(m\sqrt{n}\log^{O(1)}(U/ε))$ for these problems as well. Furthermore, our algorithm is parallelizable and using a recent nearly linear time work polylogarithmic depth Laplacian system solver of Spielman and Peng we achieve a $\tilde{O}(\sqrt{n}\log^{O(1)}(U/ε))$ depth algorithm and $\tilde{O}(m\sqrt{n}\log^{O(1)}(U/ε))$ work algorithm for solving these problems.

cs.DS↗

arXiv2014

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.

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arXiv2013

Probabilistic Spectral Sparsification In Sublinear Time

In this paper, we introduce a variant of spectral sparsification, called probabilistic $(\varepsilon,δ)$-spectral sparsification. Roughly speaking, it preserves the cut value of any cut $(S,S^{c})$ with an $1\pm\varepsilon$ multiplicative error and a $δ\left|S\right|$ additive error. We show how to produce a probabilistic $(\varepsilon,δ)$-spectral sparsifier with $O(n\log n/\varepsilon^{2})$ edges in time $\tilde{O}(n/\varepsilon^{2}δ)$ time for unweighted undirected graph. This gives fastest known sub-linear time algorithms for different cut problems on unweighted undirected graph such as - An $\tilde{O}(n/OPT+n^{3/2+t})$ time $O(\sqrt{\log n/t})$-approximation algorithm for the sparsest cut problem and the balanced separator problem. - A $n^{1+o(1)}/\varepsilon^{4}$ time approximation minimum s-t cut algorithm with an $\varepsilon n$ additive error.

cs.DS↗

arXiv2013

An Almost-Linear-Time Algorithm for Approximate Max Flow in Undirected Graphs, and its Multicommodity Generalizations

In this paper, we introduce a new framework for approximately solving flow problems in capacitated, undirected graphs and apply it to provide asymptotically faster algorithms for the maximum $s$-$t$ flow and maximum concurrent multicommodity flow problems. For graphs with $n$ vertices and $m$ edges, it allows us to find an $ε$-approximate maximum $s$-$t$ flow in time $O(m^{1+o(1)}ε^{-2})$, improving on the previous best bound of $\tilde{O}(mn^{1/3} poly(1/ε))$. Applying the same framework in the multicommodity setting solves a maximum concurrent multicommodity flow problem with $k$ commodities in $O(m^{1+o(1)}ε^{-2}k^2)$ time, improving on the existing bound of $\tilde{O}(m^{4/3} poly(k,ε^{-1})$. Our algorithms utilize several new technical tools that we believe may be of independent interest: - We give a non-Euclidean generalization of gradient descent and provide bounds on its performance. Using this, we show how to reduce approximate maximum flow and maximum concurrent flow to the efficient construction of oblivious routings with a low competitive ratio. - We define and provide an efficient construction of a new type of flow sparsifier. In addition to providing the standard properties of a cut sparsifier our construction allows for flows in the sparse graph to be routed (very efficiently) in the original graph with low congestion. - We give the first almost-linear-time construction of an $O(m^{o(1)})$-competitive oblivious routing scheme. No previous such algorithm ran in time better than $\tilde{Ω}(mn)$. We also note that independently Jonah Sherman produced an almost linear time algorithm for maximum flow and we thank him for coordinating submissions.

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arXiv2013

Efficient Accelerated Coordinate Descent Methods and Faster Algorithms for Solving Linear Systems

In this paper we show how to accelerate randomized coordinate descent methods and achieve faster convergence rates without paying per-iteration costs in asymptotic running time. In particular, we show how to generalize and efficiently implement a method proposed by Nesterov, giving faster asymptotic running times for various algorithms that use standard coordinate descent as a black box. In addition to providing a proof of convergence for this new general method, we show that it is numerically stable, efficiently implementable, and in certain regimes, asymptotically optimal. To highlight the computational power of this algorithm, we show how it can used to create faster linear system solvers in several regimes: - We show how this method achieves a faster asymptotic runtime than conjugate gradient for solving a broad class of symmetric positive definite systems of equations. - We improve the best known asymptotic convergence guarantees for Kaczmarz methods, a popular technique for image reconstruction and solving overdetermined systems of equations, by accelerating a randomized algorithm of Strohmer and Vershynin. - We achieve the best known running time for solving Symmetric Diagonally Dominant (SDD) system of equations in the unit-cost RAM model, obtaining an O(m log^{3/2} n (log log n)^{1/2} log (log n / eps)) asymptotic running time by accelerating a recent solver by Kelner et al. Beyond the independent interest of these solvers, we believe they highlight the versatility of the approach of this paper and we hope that they will open the door for further algorithmic improvements in the future.

cs.DS↗

arXiv2013

Improved Cheeger's Inequality: Analysis of Spectral Partitioning Algorithms through Higher Order Spectral Gap

Let ϕ(G) be the minimum conductance of an undirected graph G, and let 0=λ_1 <= λ_2 <=... <= λ_n <= 2 be the eigenvalues of the normalized Laplacian matrix of G. We prove that for any graph G and any k >= 2, ϕ(G) = O(k) λ_2 / \sqrt{λ_k}, and this performance guarantee is achieved by the spectral partitioning algorithm. This improves Cheeger's inequality, and the bound is optimal up to a constant factor for any k. Our result shows that the spectral partitioning algorithm is a constant factor approximation algorithm for finding a sparse cut if λ_k$ is a constant for some constant k. This provides some theoretical justification to its empirical performance in image segmentation and clustering problems. We extend the analysis to other graph partitioning problems, including multi-way partition, balanced separator, and maximum cut.

cs.DS↗