arXiv subjects

Richard Peng

Publications and source records attributed to Richard Peng.

At least 73 records · Page 4Linked to original sources

arXiv2017

On Computing Min-Degree Elimination Orderings

We study faster algorithms for producing the minimum degree ordering used to speed up Gaussian elimination. This ordering is based on viewing the non-zero elements of a symmetric positive definite matrix as edges of an undirected graph, and aims at reducing the additional non-zeros (fill) in the matrix by repeatedly removing the vertex of minimum degree. It is one of the most widely used primitives for pre-processing sparse matrices in scientific computing. Our result is in part motivated by the observation that sub-quadratic time algorithms for finding min-degree orderings are unlikely, assuming the strong exponential time hypothesis (SETH). This provides justification for the lack of provably efficient algorithms for generating such orderings, and leads us to study speedups via degree-restricted algorithms as well as approximations. Our two main results are: (1) an algorithm that produces a min-degree ordering whose maximum degree is bounded by $Δ$ in $O(m Δ\log^3{n})$ time, and (2) an algorithm that finds an $(1 + ε)$-approximate marginal min-degree ordering in $O(m \log^{5}n ε^{-2})$ time. Both of our algorithms rely on a host of randomization tools related to the $\ell_0$-estimator by [Cohen `97]. A key technical issue for the final nearly-linear time algorithm are the dependencies of the vertex removed on the randomness in the data structures. To address this, we provide a method for generating a pseudo-deterministic access sequence, which then allows the incorporation of data structures that only work under the oblivious adversary model.

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arXiv2017

Determinant-Preserving Sparsification of SDDM Matrices with Applications to Counting and Sampling Spanning Trees

We show variants of spectral sparsification routines can preserve the total spanning tree counts of graphs, which by Kirchhoff's matrix-tree theorem, is equivalent to determinant of a graph Laplacian minor, or equivalently, of any SDDM matrix. Our analyses utilizes this combinatorial connection to bridge between statistical leverage scores / effective resistances and the analysis of random graphs by [Janson, Combinatorics, Probability and Computing `94]. This leads to a routine that in quadratic time, sparsifies a graph down to about $n^{1.5}$ edges in ways that preserve both the determinant and the distribution of spanning trees (provided the sparsified graph is viewed as a random object). Extending this algorithm to work with Schur complements and approximate Choleksy factorizations leads to algorithms for counting and sampling spanning trees which are nearly optimal for dense graphs. We give an algorithm that computes a $(1 \pm δ)$ approximation to the determinant of any SDDM matrix with constant probability in about $n^2 δ^{-2}$ time. This is the first routine for graphs that outperforms general-purpose routines for computing determinants of arbitrary matrices. We also give an algorithm that generates in about $n^2 δ^{-2}$ time a spanning tree of a weighted undirected graph from a distribution with total variation distance of $δ$ from the $w$-uniform distribution .

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arXiv2017

Concave Flow on Small Depth Directed Networks

Small depth networks arise in a variety of network related applications, often in the form of maximum flow and maximum weighted matching. Recent works have generalized such methods to include costs arising from concave functions. In this paper we give an algorithm that takes a depth $D$ network and strictly increasing concave weight functions of flows on the edges and computes a $(1 - ε)$-approximation to the maximum weight flow in time $mD ε^{-1}$ times an overhead that is logarithmic in the various numerical parameters related to the magnitudes of gradients and capacities. Our approach is based on extending the scaling algorithm for approximate maximum weighted matchings by [Duan-Pettie JACM`14] to the setting of small depth networks, and then generalizing it to concave functions. In this more restricted setting of linear weights in the range $[w_{\min}, w_{\max}]$, it produces a $(1 - ε)$-approximation in time $O(mD ε^{-1} \log( w_{\max} /w_{\min}))$. The algorithm combines a variety of tools and provides a unified approach towards several problems involving small depth networks.

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arXiv2017

Density Independent Algorithms for Sparsifying $k$-Step Random Walks

We give faster algorithms for producing sparse approximations of the transition matrices of $k$-step random walks on undirected, weighted graphs. These transition matrices also form graphs, and arise as intermediate objects in a variety of graph algorithms. Our improvements are based on a better understanding of processes that sample such walks, as well as tighter bounds on key weights underlying these sampling processes. On a graph with $n$ vertices and $m$ edges, our algorithm produces a graph with about $n\log{n}$ edges that approximates the $k$-step random walk graph in about $m + n \log^4{n}$ time. In order to obtain this runtime bound, we also revisit "density independent" algorithms for sparsifying graphs whose runtime overhead is expressed only in terms of the number of vertices.

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arXiv2017

Partitioning Well-Clustered Graphs: Spectral Clustering Works!

In this paper we study variants of the widely used spectral clustering that partitions a graph into k clusters by (1) embedding the vertices of a graph into a low-dimensional space using the bottom eigenvectors of the Laplacian matrix, and (2) grouping the embedded points into k clusters via k-means algorithms. We show that, for a wide class of graphs, spectral clustering gives a good approximation of the optimal clustering. While this approach was proposed in the early 1990s and has comprehensive applications, prior to our work similar results were known only for graphs generated from stochastic models. We also give a nearly-linear time algorithm for partitioning well-clustered graphs based on computing a matrix exponential and approximate nearest neighbor data structures.

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arXiv2016

A Framework for Analyzing Resparsification Algorithms

A spectral sparsifier of a graph $G$ is a sparser graph $H$ that approximately preserves the quadratic form of $G$, i.e. for all vectors $x$, $x^T L_G x \approx x^T L_H x$, where $L_G$ and $L_H$ denote the respective graph Laplacians. Spectral sparsifiers generalize cut sparsifiers, and have found many applications in designing graph algorithms. In recent years, there has been interest in computing spectral sparsifiers in semi-streaming and dynamic settings. Natural algorithms in these settings often involve repeated sparsification of a graph, and accumulation of errors across these steps. We present a framework for analyzing algorithms that perform repeated sparsifications that only incur error corresponding to a single sparsification step, leading to better results for many resparsification-based algorithms. As an application, we show how to maintain a spectral sparsifier in the semi-streaming setting: We present a simple algorithm that, for a graph $G$ on $n$ vertices and $m$ edges, computes a spectral sparsifier of $G$ with $O(n \log n)$ edges in a single pass over $G$, using only $O(n \log n)$ space, and $O(m \log^2 n)$ total time. This improves on previous best semi-streaming algorithms for both spectral and cut sparsifiers by a factor of $\log{n}$ in both space and runtime. The algorithm extends to semi-streaming row sampling for general PSD matrices. We also use our framework to combine a spectral sparsification algorithm by Koutis with improved spanner constructions to give a parallel algorithm for constructing $O(n\log^2{n}\log\log{n})$ sized spectral sparsifiers in $O(m\log^2{n}\log\log{n})$ time. This is the best known combinatorial graph sparsification algorithm.The size of the sparsifiers is only a factor $\log{n}\log\log{n}$ more than ones produced by numerical routines.

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arXiv2016

Faster Algorithms for Computing the Stationary Distribution, Simulating Random Walks, and More

In this paper, we provide faster algorithms for computing various fundamental quantities associated with random walks on a directed graph, including the stationary distribution, personalized PageRank vectors, hitting times, and escape probabilities. In particular, on a directed graph with $n$ vertices and $m$ edges, we show how to compute each quantity in time $\tilde{O}(m^{3/4}n+mn^{2/3})$, where the $\tilde{O}$ notation suppresses polylogarithmic factors in $n$, the desired accuracy, and the appropriate condition number (i.e. the mixing time or restart probability). Our result improves upon the previous fastest running times for these problems; previous results either invoke a general purpose linear system solver on a $n\times n$ matrix with $m$ non-zero entries, or depend polynomially on the desired error or natural condition number associated with the problem (i.e. the mixing time or restart probability). For sparse graphs, we obtain a running time of $\tilde{O}(n^{7/4})$, breaking the $O(n^{2})$ barrier of the best running time one could hope to achieve using fast matrix multiplication. We achieve our result by providing a similar running time improvement for solving directed Laplacian systems, a natural directed or asymmetric analog of the well studied symmetric or undirected Laplacian systems. We show how to solve such systems in time $\tilde{O}(m^{3/4}n+mn^{2/3})$, and efficiently reduce a broad range of problems to solving $\tilde{O}(1)$ directed Laplacian systems on Eulerian graphs. We hope these results and our analysis open the door for further study into directed spectral graph theory.

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arXiv2016

Almost-Linear-Time Algorithms for Markov Chains and New Spectral Primitives for Directed Graphs

In this paper we introduce a notion of spectral approximation for directed graphs. While there are many potential ways one might define approximation for directed graphs, most of them are too strong to allow sparse approximations in general. In contrast, we prove that for our notion of approximation, such sparsifiers do exist, and we show how to compute them in almost linear time. Using this notion of approximation, we provide a general framework for solving asymmetric linear systems that is broadly inspired by the work of [Peng-Spielman, STOC`14]. Applying this framework in conjunction with our sparsification algorithm, we obtain an almost linear time algorithm for solving directed Laplacian systems associated with Eulerian Graphs. Using this solver in the recent framework of [Cohen-Kelner-Peebles-Peng-Sidford-Vladu, FOCS`16], we obtain almost linear time algorithms for solving a directed Laplacian linear system, computing the stationary distribution of a Markov chain, computing expected commute times in a directed graph, and more. For each of these problems, our algorithms improves the previous best running times of $O((nm^{3/4} + n^{2/3} m) \log^{O(1)} (n κε^{-1}))$ to $O((m + n2^{O(\sqrt{\log{n}\log\log{n}})}) \log^{O(1)} (n κε^{-1}))$ where $n$ is the number of vertices in the graph, $m$ is the number of edges, $κ$ is a natural condition number associated with the problem, and $ε$ is the desired accuracy. We hope these results open the door for further studies into directed spectral graph theory, and will serve as a stepping stone for designing a new generation of fast algorithms for directed graphs.

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arXiv2016

On Fully Dynamic Graph Sparsifiers

We initiate the study of dynamic algorithms for graph sparsification problems and obtain fully dynamic algorithms, allowing both edge insertions and edge deletions, that take polylogarithmic time after each update in the graph. Our three main results are as follows. First, we give a fully dynamic algorithm for maintaining a $ (1 \pm ε) $-spectral sparsifier with amortized update time $poly(\log{n}, ε^{-1})$. Second, we give a fully dynamic algorithm for maintaining a $ (1 \pm ε) $-cut sparsifier with \emph{worst-case} update time $poly(\log{n}, ε^{-1})$. Both sparsifiers have size $ n \cdot poly(\log{n}, ε^{-1})$. Third, we apply our dynamic sparsifier algorithm to obtain a fully dynamic algorithm for maintaining a $(1 + ε)$-approximation to the value of the maximum flow in an unweighted, undirected, bipartite graph with amortized update time $poly(\log{n}, ε^{-1})$.

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arXiv2016

An Empirical Study of Cycle Toggling Based Laplacian Solvers

We study the performance of linear solvers for graph Laplacians based on the combinatorial cycle adjustment methodology proposed by [Kelner-Orecchia-Sidford-Zhu STOC-13]. The approach finds a dual flow solution to this linear system through a sequence of flow adjustments along cycles. We study both data structure oriented and recursive methods for handling these adjustments. The primary difficulty faced by this approach, updating and querying long cycles, motivated us to study an important special case: instances where all cycles are formed by fundamental cycles on a length $n$ path. Our methods demonstrate significant speedups over previous implementations, and are competitive with standard numerical routines.

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arXiv2016

Faster and Simpler Width-Independent Parallel Algorithms for Positive Semidefinite Programming

This paper studies the problem of finding an $(1+ε)$-approximate solution to positive semidefinite programs. These are semidefinite programs in which all matrices in the constraints and objective are positive semidefinite and all scalars are non-negative. We present a simpler \NC parallel algorithm that on input with $n$ constraint matrices, requires $O(\frac{1}{ε^3} log^3 n)$ iterations, each of which involves only simple matrix operations and computing the trace of the product of a matrix exponential and a positive semidefinite matrix. Further, given a positive SDP in a factorized form, the total work of our algorithm is nearly-linear in the number of non-zero entries in the factorization.

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arXiv2016

Scalable Constrained Clustering: A Generalized Spectral Method

We present a simple spectral approach to the well-studied constrained clustering problem. It captures constrained clustering as a generalized eigenvalue problem with graph Laplacians. The algorithm works in nearly-linear time and provides concrete guarantees for the quality of the clusters, at least for the case of 2-way partitioning. In practice this translates to a very fast implementation that consistently outperforms existing spectral approaches both in speed and quality.

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arXiv2015

Sparsified Cholesky and Multigrid Solvers for Connection Laplacians

We introduce the sparsified Cholesky and sparsified multigrid algorithms for solving systems of linear equations. These algorithms accelerate Gaussian elimination by sparsifying the nonzero matrix entries created by the elimination process. We use these new algorithms to derive the first nearly linear time algorithms for solving systems of equations in connection Laplacians, a generalization of Laplacian matrices that arise in many problems in image and signal processing. We also prove that every connection Laplacian has a linear sized approximate inverse. This is an LU factorization with a linear number of nonzero entries that is a strong approximation of the original matrix. Using such a factorization one can solve systems of equations in a connection Laplacian in linear time. Such a factorization was unknown even for ordinary graph Laplacians.

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arXiv2015

Approximate Undirected Maximum Flows in O(m polylog(n)) Time

We give the first O(m polylog(n)) time algorithms for approximating maximum flows in undirected graphs and constructing polylog(n) -quality cut-approximating hierarchical tree decompositions. Our algorithm invokes existing algorithms for these two problems recursively while gradually incorporating size reductions. These size reductions are in turn obtained via ultra-sparsifiers, which are key tools in solvers for symmetric diagonally dominant (SDD) linear systems.

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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

Improved Parallel Algorithms for Spanners and Hopsets

We use exponential start time clustering to design faster and more work-efficient parallel graph algorithms involving distances. Previous algorithms usually rely on graph decomposition routines with strict restrictions on the diameters of the decomposed pieces. We weaken these bounds in favor of stronger local probabilistic guarantees. This allows more direct analyses of the overall process, giving: * Linear work parallel algorithms that construct spanners with $O(k)$ stretch and size $O(n^{1+1/k})$ in unweighted graphs, and size $O(n^{1+1/k} \log k)$ in weighted graphs. * Hopsets that lead to the first parallel algorithm for approximating shortest paths in undirected graphs with $O(m\;\mathrm{polylog}\;n)$ work.

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arXiv2015

Spectral Sparsification of Random-Walk Matrix Polynomials

We consider a fundamental algorithmic question in spectral graph theory: Compute a spectral sparsifier of random-walk matrix-polynomial $$L_α(G)=D-\sum_{r=1}^dα_rD(D^{-1}A)^r$$ where $A$ is the adjacency matrix of a weighted, undirected graph, $D$ is the diagonal matrix of weighted degrees, and $α=(α_1...α_d)$ are nonnegative coefficients with $\sum_{r=1}^dα_r=1$. Recall that $D^{-1}A$ is the transition matrix of random walks on the graph. The sparsification of $L_α(G)$ appears to be algorithmically challenging as the matrix power $(D^{-1}A)^r$ is defined by all paths of length $r$, whose precise calculation would be prohibitively expensive. In this paper, we develop the first nearly linear time algorithm for this sparsification problem: For any $G$ with $n$ vertices and $m$ edges, $d$ coefficients $α$, and $ε> 0$, our algorithm runs in time $O(d^2m\log^2n/ε^{2})$ to construct a Laplacian matrix $\tilde{L}=D-\tilde{A}$ with $O(n\log n/ε^{2})$ non-zeros such that $\tilde{L}\approx_εL_α(G)$. Matrix polynomials arise in mathematical analysis of matrix functions as well as numerical solutions of matrix equations. Our work is particularly motivated by the algorithmic problems for speeding up the classic Newton's method in applications such as computing the inverse square-root of the precision matrix of a Gaussian random field, as well as computing the $q$th-root transition (for $q\geq1$) in a time-reversible Markov model. The key algorithmic step for both applications is the construction of a spectral sparsifier of a constant degree random-walk matrix-polynomials introduced by Newton's method. Our algorithm can also be used to build efficient data structures for effective resistances for multi-step time-reversible Markov models, and we anticipate that it could be useful for other tasks in network analysis.

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arXiv2014

$\ell_p$ Row Sampling by Lewis Weights

We give a simple algorithm to efficiently sample the rows of a matrix while preserving the p-norms of its product with vectors. Given an $n$-by-$d$ matrix $\boldsymbol{\mathit{A}}$, we find with high probability and in input sparsity time an $\boldsymbol{\mathit{A}}'$ consisting of about $d \log{d}$ rescaled rows of $\boldsymbol{\mathit{A}}$ such that $\| \boldsymbol{\mathit{A}} \boldsymbol{\mathit{x}} \|_1$ is close to $\| \boldsymbol{\mathit{A}}' \boldsymbol{\mathit{x}} \|_1$ for all vectors $\boldsymbol{\mathit{x}}$. We also show similar results for all $\ell_p$ that give nearly optimal sample bounds in input sparsity time. Our results are based on sampling by "Lewis weights", which can be viewed as statistical leverage scores of a reweighted matrix. We also give an elementary proof of the guarantees of this sampling process for $\ell_1$.

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