Nisheeth K. Vishnoi

Biased normalized cuts

We present a modification of “Normalized Cuts” to incorporate priors which can be used for constrained image segmentation. Compared to previous generalizations of “Normalized Cuts” which incorporate constraints, our technique has two advantages. First, we seek solutions which are sufficiently “correlated” with priors which allows us to use noisy top-down information, for example from an object detector. Second, given the spectral solution of the unconstrained problem, the solution of the constrained one can be computed in small additional time, which allows us to run the algorithm in an interactive mode. We compare our algorithm to other graph cut based algorithms and highlight the advantages.

Approximating the exponential, the lanczos method and an Õ( m )-time spectral algorithm for balanced separator

We give a novel spectral approximation algorithm for the balanced (edge-)separator problem that, given a graph G, a constant balance b ∈ (0,1/2], and a parameter γ, either finds an Ω(b)-balanced cut of conductance O(√γ) in G, or outputs a certificate that all b-balanced cuts in G have conductance at least γ, and runs in time ~O(m). This settles the question of designing asymptotically optimal spectral algorithms for balanced separator. Our algorithm relies on a variant of the heat kernel random walk and requires, as a subroutine, an algorithm to compute exp(-L)v where L is the Laplacian of a graph related to G and v is a vector. Algorithms for computing the matrix-exponential-vector product efficiently comprise our next set of results. Our main result here is a new algorithm which computes a good approximation to exp(-A)v for a class of symmetric positive semidefinite (PSD) matrices A and a given vector v, in time roughly ~O(mA), independent of the norm of A, where mA is the number of non-zero entries of A. This uses, in a non-trivial way, the result of Spielman and Teng on inverting symmetric and diagonally-dominant matrices in ~O(mA) time. Finally, using old and new uniform approximations to e-x we show how to obtain, via the Lanczos method, a simple algorithm to compute exp(-A)v for symmetric PSD matrices that runs in time roughly O(tA⋅ √norm(A)), where tA is the time required for the computation of the vector Aw for given vector w. As an application, we obtain a simple and practical algorithm, with output conductance O(√γ), for balanced separator that runs in time O(m/√γ). This latter algorithm matches the running time, but improves on the approximation guarantee of the Evolving-Sets-based algorithm by Andersen and Peres for balanced separator.

On partitioning graphs via single commodity flows

In this paper we obtain improved upper and lower bounds for the best approximation factor for Sparsest Cut achievable in the cut-matching game framework proposed in Khandekar et al. [9]. We show that this simple framework can be used to design combinatorial algorithms that achieve O(log n) approximation factor and whose running time is dominated by a poly-logarithmic number of single-commodity max-flow computations. This matches the performance of the algorithm of Arora and Kale [2]. Moreover, we also show that it is impossible to get an approximation factor of better than Ω(√log n) in the cut-matching game framework. These results suggest that the simple and concrete abstraction of the cut-matching game may be powerful enough to capture the essential features of the complexity of Sparsest Cut.

Unique games on expanding constraint graphs are easy: extended abstract

We present an efficient algorithm to find a good solution to the Unique Games problem when the constraint graph is an expander. We introduce a new analysis of the standard SDP in this case that involves correlations among distant vertices. It also leads to a parallel repetition theorem for unique games when the graph is an expander.

The impact of noise on the scaling of collectives: a theoretical approach

The performance of parallel applications running on large clusters is known to degrade due to the interference of kernel and daemon activities on individual nodes, often referred to as noise. In this paper, we focus on an important class of parallel applications, which repeatedly perform computation, followed by a collective operation such as a barrier. We model this theoretically and demonstrate, in a rigorous way, the effect of noise on the scalability of such applications. We study three natural and important classes of noise distributions: The exponential distribution, the heavy-tailed distribution, and the Bernoulli distribution. We show that the systems scale well in the presence of exponential noise, but the performance goes down drastically in the presence of heavy-tailed or Bernoulli noise.

Deterministically testing sparse polynomial identities of unbounded degree

We present two deterministic algorithms for the arithmetic circuit identity testing problem. The running time of our algorithms is polynomially bounded in s and m, where s is the size of the circuit and m is an upper bound on the number monomials with non-zero coefficients in its standard representation. The running time of our algorithms also has a logarithmic dependence on the degree of the polynomial but, since a circuit of size s can only compute polynomials of degree at most 2^s, the running time does not depend on its degree. Before this work, all such deterministic algorithms had a polynomial dependence on the degree and therefore an exponential dependence on s. Our first algorithm works over the integers and it requires only black-box access to the given circuit. Though this algorithm is quite simple, the analysis of why it works relies on Linniks Theorem, a deep result from number theory about the size of the smallest prime in an arithmetic progression. Our second algorithm, unlike the first, uses elementary arguments and works over any integral domains, but it uses the circuit in a less restricted manner. In both cases the running time has a logarithmic dependence on the largest coefficient of the polynomial.

Unique games on expanding constraint graphs are easy

We present an efficient algorithm to find a good solution to the Unique Games problem when the constraint graph is an expander. We introduce a new analysis of the standard SDP in this case that involves correlations among distant vertices. It also leads to a parallel repetition theorem for unique games when the graph is an expander.

A Finite Population Model of Molecular Evolution: Theory and Computation

This article is concerned with the evolution of haploid organisms that reproduce asexually. In a seminal piece of work, Eigen and coauthors proposed the quasispecies model in an attempt to understand such an evolutionary process. Their work has impacted antiviral treatment and vaccine design strategies. Yet, predictions of the quasispecies model are at best viewed as a guideline, primarily because it assumes an infinite population size, whereas realistic population sizes can be quite small. In this paper we consider a population genetics-based model aimed at understanding the evolution of such organisms with finite population sizes and present a rigorous study of the convergence and computational issues that arise therein. Our first result is structural and shows that, at any time during the evolution, as the population size tends to infinity, the distribution of genomes predicted by our model converges to that predicted by the quasispecies model. This justifies the continued use of the quasispecies model to derive guidelines for intervention. While the stationary state in the quasispecies model is readily obtained, due to the explosion of the state space in our model, exact computations are prohibitive. Our second set of results are computational in nature and address this issue. We derive conditions on the parameters of evolution under which our stochastic model mixes rapidly. Further, for a class of widely used fitness landscapes we give a fast deterministic algorithm which computes the stationary distribution of our model. These computational tools are expected to serve as a framework for the modeling of strategies for the deployment of mutagenic drugs.

Entropy, optimization and counting

We study the problem of computing max-entropy distributions over a discrete set of objects subject to observed marginals. There has been a tremendous amount of interest in such distributions due to their applicability in areas such as statistical physics, economics, biology, information theory, machine learning, combinatorics and algorithms. However, a rigorous and systematic study of how to compute such distributions has been lacking. Since the underlying set of discrete objects can be exponential in the input size, the first question in such a study is if max-entropy distributions have polynomially-sized descriptions. We start by giving a structural result which shows that such succinct descriptions exist under very general conditions. Subsequently, we use techniques from convex programming to give a meta-algorithm that can efficiently (approximately) compute max-entropy distributions provided one can efficiently (approximately) count the underlying discrete set. Thus, we can translate a host of existing counting algorithms, developed in an unrelated context, into algorithms that compute max-entropy distributions. Conversely, we prove that counting oracles are necessary for computing max-entropy distributions: we show how algorithms that compute max-entropy distributions can be converted into counting algorithms.

Stochastic simulations suggest that HIV-1 survives close to its error threshold.

The use of mutagenic drugs to drive HIV-1 past its error threshold presents a novel intervention strategy, as suggested by the quasispecies theory, that may be less susceptible to failure via viral mutation-induced emergence of drug resistance than current strategies. The error threshold of HIV-1, , however, is not known. Application of the quasispecies theory to determine poses significant challenges: Whereas the quasispecies theory considers the asexual reproduction of an infinitely large population of haploid individuals, HIV-1 is diploid, undergoes recombination, and is estimated to have a small effective population size in vivo. We performed population genetics-based stochastic simulations of the within-host evolution of HIV-1 and estimated the structure of the HIV-1 quasispecies and . We found that with small mutation rates, the quasispecies was dominated by genomes with few mutations. Upon increasing the mutation rate, a sharp error catastrophe occurred where the quasispecies became delocalized in sequence space. Using parameter values that quantitatively captured data of viral diversification in HIV-1 patients, we estimated to be substitutions/site/replication, ∼2–6 fold higher than the natural mutation rate of HIV-1, suggesting that HIV-1 survives close to its error threshold and may be readily susceptible to mutagenic drugs. The latter estimate was weakly dependent on the within-host effective population size of HIV-1. With large population sizes and in the absence of recombination, our simulations converged to the quasispecies theory, bridging the gap between quasispecies theory and population genetics-based approaches to describing HIV-1 evolution. Further, increased with the recombination rate, rendering HIV-1 less susceptible to error catastrophe, thus elucidating an added benefit of recombination to HIV-1. Our estimate of may serve as a quantitative guideline for the use of mutagenic drugs against HIV-1.