Truncating the loop series expansion for Belief Propagation

Recently, M. Chertkov and V.Y. Chernyak derived an exact expression for the partition sum (normalization constant) corresponding to a graphical model, which is an expansion around the Belief Propagation solution. By adding correction terms to the BP free energy, one for each "generalized loop" in the factor graph, the exact partition sum is obtained. However, the usually enormous number of generalized loops generally prohibits summation over all correction terms. In this article we introduce Truncated Loop Series BP (TLSBP), a particular way of truncating the loop series of M. Chertkov and V.Y. Chernyak by considering generalized loops as compositions of simple loops. We analyze the performance of TLSBP in different scenarios, including the Ising model, regular random graphs and on Promedas, a large probabilistic medical diagnostic system. We show that TLSBP often improves upon the accuracy of the BP solution, at the expense of increased computation time. We also show that the performance of TLSBP strongly depends on the degree of interaction between the variables. For weak interactions, truncating the series leads to significant improvements, whereas for strong interactions it can be ineffective, even if a high number of terms is considered.Comment: 31 pages, 12 figures, submitted to Journal of Machine Learning Researc

Quasiconvex Programming

We define quasiconvex programming, a form of generalized linear programming in which one seeks the point minimizing the pointwise maximum of a collection of quasiconvex functions. We survey algorithms for solving quasiconvex programs either numerically or via generalizations of the dual simplex method from linear programming, and describe varied applications of this geometric optimization technique in meshing, scientific computation, information visualization, automated algorithm analysis, and robust statistics.Comment: 33 pages, 14 figure

Earthquake Arrival Association with Backprojection and Graph Theory

The association of seismic wave arrivals with causative earthquakes becomes progressively more challenging as arrival detection methods become more sensitive, and particularly when earthquake rates are high. For instance, seismic waves arriving across a monitoring network from several sources may overlap in time, false arrivals may be detected, and some arrivals may be of unknown phase (e.g., P- or S-waves). We propose an automated method to associate arrivals with earthquake sources and obtain source locations applicable to such situations. To do so we use a pattern detection metric based on the principle of backprojection to reveal candidate sources, followed by graph-theory-based clustering and an integer linear optimization routine to associate arrivals with the minimum number of sources necessary to explain the data. This method solves for all sources and phase assignments simultaneously, rather than in a sequential greedy procedure as is common in other association routines. We demonstrate our method on both synthetic and real data from the Integrated Plate Boundary Observatory Chile (IPOC) seismic network of northern Chile. For the synthetic tests we report results for cases with varying complexity, including rates of 500 earthquakes/day and 500 false arrivals/station/day, for which we measure true positive detection accuracy of > 95%. For the real data we develop a new catalog between January 1, 2010 - December 31, 2017 containing 817,548 earthquakes, with detection rates on average 279 earthquakes/day, and a magnitude-of-completion of ~M1.8. A subset of detections are identified as sources related to quarry and industrial site activity, and we also detect thousands of foreshocks and aftershocks of the April 1, 2014 Mw 8.2 Iquique earthquake. During the highest rates of aftershock activity, > 600 earthquakes/day are detected in the vicinity of the Iquique earthquake rupture zone

Weighted Min-Cut: Sequential, Cut-Query and Streaming Algorithms

Consider the following 2-respecting min-cut problem. Given a weighted graph $G$ and its spanning tree $T$, find the minimum cut among the cuts that contain at most two edges in $T$. This problem is an important subroutine in Karger's celebrated randomized near-linear-time min-cut algorithm [STOC'96]. We present a new approach for this problem which can be easily implemented in many settings, leading to the following randomized min-cut algorithms for weighted graphs. * An $O(m\frac{\log^2 n}{\log\log n} + n\log^6 n)$-time sequential algorithm: This improves Karger's $O(m \log^3 n)$ and $O(m\frac{(\log^2 n)\log (n^2/m)}{\log\log n} + n\log^6 n)$ bounds when the input graph is not extremely sparse or dense. Improvements over Karger's bounds were previously known only under a rather strong assumption that the input graph is simple [Henzinger et al. SODA'17; Ghaffari et al. SODA'20]. For unweighted graphs with parallel edges, our bound can be improved to $O(m\frac{\log^{1.5} n}{\log\log n} + n\log^6 n)$. * An algorithm requiring $\tilde O(n)$ cut queries to compute the min-cut of a weighted graph: This answers an open problem by Rubinstein et al. ITCS'18, who obtained a similar bound for simple graphs. * A streaming algorithm that requires $\tilde O(n)$ space and $O(\log n)$ passes to compute the min-cut: The only previous non-trivial exact min-cut algorithm in this setting is the 2-pass $\tilde O(n)$-space algorithm on simple graphs [Rubinstein et al., ITCS'18] (observed by Assadi et al. STOC'19). In contrast to Karger's 2-respecting min-cut algorithm which deploys sophisticated dynamic programming techniques, our approach exploits some cute structural properties so that it only needs to compute the values of $\tilde O(n)$ cuts corresponding to removing $\tilde O(n)$ pairs of tree edges, an operation that can be done quickly in many settings.Comment: Updates on this version: (1) Minor corrections in Section 5.1, 5.2; (2) Reference to newer results by GMW SOSA21 (arXiv:2008.02060v2), DEMN STOC21 (arXiv:2004.09129v2) and LMN 21 (arXiv:2102.06565v1

Efficient Algorithms with Asymmetric Read and Write Costs

In several emerging technologies for computer memory (main memory), the cost of reading is significantly cheaper than the cost of writing. Such asymmetry in memory costs poses a fundamentally different model from the RAM for algorithm design. In this paper we study lower and upper bounds for various problems under such asymmetric read and write costs. We consider both the case in which all but O(1) memory has asymmetric cost, and the case of a small cache of symmetric memory. We model both cases using the (M,omega)-ARAM, in which there is a small (symmetric) memory of size M and a large unbounded (asymmetric) memory, both random access, and where reading from the large memory has unit cost, but writing has cost omega >> 1. For FFT and sorting networks we show a lower bound cost of Omega(omega*n*log_{omega*M}(n)), which indicates that it is not possible to achieve asymptotic improvements with cheaper reads when omega is bounded by a polynomial in M. Moreover, there is an asymptotic gap (of min(omega,log(n)/log(omega*M)) between the cost of sorting networks and comparison sorting in the model. This contrasts with the RAM, and most other models, in which the asymptotic costs are the same. We also show a lower bound for computations on an n*n diamond DAG of Omega(omega*n^2/M) cost, which indicates no asymptotic improvement is achievable with fast reads. However, we show that for the minimum edit distance problem (and related problems), which would seem to be a diamond DAG, we can beat this lower bound with an algorithm with only O(omega*n^2/(M*min(omega^{1/3},M^{1/2}))) cost. To achieve this we make use of a "path sketch" technique that is forbidden in a strict DAG computation. Finally, we show several interesting upper bounds for shortest path problems, minimum spanning trees, and other problems. A common theme in many of the upper bounds is that they require redundant computation and a tradeoff between reads and writes

Rectangular Layouts and Contact Graphs

Contact graphs of isothetic rectangles unify many concepts from applications including VLSI and architectural design, computational geometry, and GIS. Minimizing the area of their corresponding {\em rectangular layouts} is a key problem. We study the area-optimization problem and show that it is NP-hard to find a minimum-area rectangular layout of a given contact graph. We present O(n)-time algorithms that construct $O(n^2)$-area rectangular layouts for general contact graphs and $O(n\log n)$-area rectangular layouts for trees. (For trees, this is an $O(\log n)$-approximation algorithm.) We also present an infinite family of graphs (rsp., trees) that require $\Omega(n^2)$ (rsp., $\Omega(n\log n)$) area. We derive these results by presenting a new characterization of graphs that admit rectangular layouts using the related concept of {\em rectangular duals}. A corollary to our results relates the class of graphs that admit rectangular layouts to {\em rectangle of influence drawings}.Comment: 28 pages, 13 figures, 55 references, 1 appendi