Navin Goyal

Fourier PCA and robust tensor decomposition

Fourier PCA is Principal Component Analysis of a matrix obtained from higher order derivatives of the logarithm of the Fourier transform of a distribution. To make this algorithmic, we develop a robust tensor decomposition method; this is also of independent interest. Our main application is the first provably polynomial-time algorithm for underdetermined ICA, i.e., learning an n × m matrix A from observations y = Ax where x is drawn from an unknown product distribution with arbitrary non-Gaussian components. The number of component distributions m can be arbitrarily higher than the dimension n and the columns of A only need to satisfy a natural and efficiently verifiable nondegeneracy condition. As a second application, we give an alternative algorithm for learning mixtures of spherical Gaussians with linearly independent means. These results also hold in the presence of Gaussian noise.

The VPN conjecture is true

We consider the following network design problem. We are given an undirected graph G=(V,E) with edges costs c(e) and a set of terminal nodes W. A hose demand matrix for W is any symmetric matrix [Dij] such that for each i, ∑ j ≠ i Dij ≤ 1. We must compute the minimum cost edge capacities that are able to support the oblivious routing of every hose matrix in the network. An oblivious routing template, in this context, is a simple path Pij for each pair i,j ∈ W. Given such a template, if we are to route a demand matrix D, then for each i,j we send Dij units of flow along each Pij. Fingerhut et al. and Gupta et al. obtained a 2-approximation for this problem, using a solution template in the form of a tree. It has been widely asked and subsequently conjectured [Italiano 2006] that this solution actually results in the optimal capacity for the single path VPN design problem; this has become known as the VPN conjecture. The conjecture has previously been proven for some restricted classes of graphs [Hurkens 2005, Grandoni 2007, Fiorini 2007]. Our main theorem establishes that this conjecture is true in general graphs. This also gives the first polynomial time algorithm for the single path VPN problem. We also show that the multipath version of the conjecture is false.

Lower Bounds for the Noisy Broadcast Problem

We prove the first nontrivial (superlinear) lower bound in the noisy broadcast model of distributed computation. In this model, there are n + 1 processors P/sub 0/, P/sub 1/, ..., P/sub n/. Each P/sub i/, for i /spl ges/ 1, initially has a private bit x/sub i/ and the goal is for P/sub 0/ to learn f (x/sub l/, ..., x/sub n/) for some specified function f. At each time step, a designated processor broadcasts some function of its private bit and the bits it has heard so far. Each broadcast is received by the other processors but each reception may be corrupted by noise. In this model, Gallager (1988) gave a noise-resistant protocol that allows P/sub 0/ to learn the entire input in O(n log log n) broadcasts. We prove that Gallagers protocol is optimal up to a constant factor. Our lower bound follows from a lower bound in a new model, the generalized noisy decision tree model, which may be of independent interest.

Expanders via Random Spanning Trees

Motivated by the problem of routing reliably and scalably in a graph, we introduce the notion of a splicer, the union of spanning trees of a graph. We prove that for any bounded-degree n-vertex graph, the union of two random spanning trees approximates the expansion of every cut of the graph to within a factor of O(log n). For the random graph Gn, p, for p = Ω(log n/n), we give a randomized algorithm for constructing two spanning trees whose union is an expander. This is suggested by the case of the complete graph, where we prove that two random spanning trees give an expander. The construction of the splicer is elementary; each spanning tree can be produced independently using an algorithm by Aldous and Broder: A random walk in the graph with edges leading to previously unvisited vertices included in the tree. Splicers also turn out to have applications to graph cut-sparsification where the goal is to approximate every cut using only a small subgraph of the original graph. For random graphs, splicers provide simple algorithms for sparsifiers of size O(n) that approximate every cut to within a factor of O(log n).

A Combinatorial Shape Matching Algorithm for Rigid Protein Docking

The protein docking problem is to predict the structure of protein-protein complexes from the structures of individual proteins. It is believed that shape complementarity plays a dominant role in protein docking. Recently, it has been shown empirically by Bespamaytnikh et al [4] that the shape complementarity (measured by a score function) is sufficient for the bound protein docking problem, in which proteins are taken directly from the known protein-protein complex and reassembled, treating each protein as a rigid body. In this paper, we study the shape complementarity measured by their score function from a theoretical point of view. We give a combinatorial characterization of the docked configuration achieved by the maximum score. This leads to a simple polynomial time algorithm to find such a configuration. The arrangement of spheres inspired by the combinatorial characterization plays an essential role in an efficient local search heuristic of Choi et al [7] for rigid protein docking. We also show that our general idea can be used to give simple algorithms for some point pattern matching problems in any dimension.

Lower bounds for the noisy broadcast problem

We prove the first nontrivial (superlinear) lower bound in the noisy broadcast model of distributed computation. In this model, there are n + 1 processors P/sub 0/, P/sub 1/, ..., P/sub n/. Each P/sub i/, for i /spl ges/ 1, initially has a private bit x/sub i/ and the goal is for P/sub 0/ to learn f (x/sub l/, ..., x/sub n/) for some specified function f. At each time step, a designated processor broadcasts some function of its private bit and the bits it has heard so far. Each broadcast is received by the other processors but each reception may be corrupted by noise. In this model, Gallager (1988) gave a noise-resistant protocol that allows P/sub 0/ to learn the entire input in O(n log log n) broadcasts. We prove that Gallagers protocol is optimal up to a constant factor. Our lower bound follows from a lower bound in a new model, the generalized noisy decision tree model, which may be of independent interest.

Optimal bandwidth reservation schedule in cellular networks

Efficient bandwidth allocation strategy with simultaneous fulfillment of QoS requirement of a user in a mobile cellular network is still a critical and an important practical issue. We explore the problem of finding the reservation schedule that would minimize the amount of time for which bandwidth has to be allocated in a cell while meeting the QoS constraint. With the knowledge about the arrival and residence time distribution of a user in a cell, the above problem can be optimally solved using a dynamic programming based approach in polynomial time. To be able to use the solution, we provide a mechanism for constructing the arrival/residence time distribution based on the measurement of hand-off events in a cell. The above solution allows us to propose an optimal time based bandwidth reservation and call admission scheme. By being scalable and distributed, the proposed scheme justifies for practical implementation. Simulations results are also presented to show the effectiveness of the scheme to achieve the target QoS level and optimal bandwidth utilization.

Dynamic vs. Oblivious Routing in Network Design

Consider the robust network design problem of finding a minimum cost network with enough capacity to route all traffic demand matrices in a given polytope. We investigate the impact of different routing models in this robust setting: in particular, we compare oblivious routing, where the routing between each terminal pair must be fixed in advance, to dynamic routing, where routings may depend arbitrarily on the current demand. Our main result is a construction that shows that the optimal cost of such a network based on oblivious routing (fractional or integral) may be a factor of Ω(log n) more than the cost required when using dynamic routing. This is true even in the important special case of the asymmetric hose model. This answers a question in (Chekuri, SIGACT News 38(3):106–128, 2007), and is tight up to constant factors. Our proof technique builds on a connection between expander graphs and robust design for single-sink traffic patterns (Chekuri et al., Networks 50(1):50–54, 2007).

Lower Bounds for the Average and Smoothed Number of Pareto-Optima

Smoothed analysis of multiobjective 0-1 linear optimization has drawn con- siderable attention recently. The goal is to give bounds for the number of Pareto-optimal solutions (i. e., solutions with the property that no other solution is at least as good in all the coordinates and better in at least one) for multiobjective optimization problems. In this article we prove several lower bounds for the expected number of Pareto optima. Our basic result is a lower bound of Wd(n d 1 ) for optimization problems with d objectives and n variables under fairly general conditions on the distributions of the linear objectives. Our proof relates the problem of finding lower bounds for the number of Pareto optima to results in discrete geometry and geometric probability about arrangements of hyperplanes. We use our basic result to derive the following results: (1) To our knowledge, the first lower bound for natural multiobjective optimization problems. We illustrate this on the maximum spanning tree problem with randomly chosen edge weights. Our technique is sufficiently flexible to yield such lower bounds also for other standard objective functions studied in this setting (such as multiobjective shortest path, TSP, matching). (2) A smoothed lower bound of minfWd(n d 1:5 f d ); 2 Qd(n) g for f -smooth instances of the 0-1 knapsack problem with d profits.

The VPN Conjecture Is True

We consider the following network design problem. We are given an undirected graph <i>G</i> = (<i>V</i>,<i>E</i>) with edge costs <i>c</i>(<i>e</i>) and a set of terminal nodes <i>W</i> ⊆ <i>V</i>. A <i>hose</i> demand matrix is any symmetric matrix <i>D</i>, indexed by the terminals, such that for each <i>i</i> ∈ <i>W</i>, ∑_j≠i <i>D</i>_ij ≤ 1. We must compute the minimum-cost edge capacities that are able to support the oblivious routing of every hose matrix in the network. An <i>oblivious routing</i> template, in this context, is a simple path <i>P</i>_ij for each pair <i>i,j</i> ∈ <i>W</i>. Given such a template, if we are to route a demand matrix <i>D</i>, then for each <i>i,j</i>, we send <i>D</i>_ij units of flow along each <i>P</i>_ij. Fingerhut et al. [1997] and Gupta et al. [2001] obtained a 2-approximation for this problem, using a solution template in the form of a tree. It has been widely asked and subsequently conjectured [Italiano et al. 2006] that this solution actually results in the optimal capacity for the single-path VPN design problem; this has become known as the <i>VPN Conjecture</i>. The conjecture has previously been proven for some restricted classes of graphs [Fingerhut et al. 1997; Fiorini et al. 2007; Grandoni et al. 2008; Hurkens et al. 2007]. Our main theorem establishes that this conjecture is true in general graphs. This also has the implication that the single-path VPN problem is solvable in polynomial time. A natural fractional version of the conjecture had also been proposed [Hurkens et al. 2007]. In this version, the routing may split flow between many paths, in specified proportions. We demonstrate that this multipath version of the conjecture is in fact false. The multipath and single path versions of the VPN problem are essentially direct analogues of the randomized and nonrandomized versions of oblivious routing schemes for minimizing congestion for permutation routing [Borodin and Hopcroft 1982; Valiant 1982].