arXiv subjects

Eric Vigoda

Publications and source records attributed to Eric Vigoda.

60 records · Page 4Linked to original sources

arXiv2007

Analysis of top-swap shuffling for genome rearrangements

We study Markov chains which model genome rearrangements. These models are useful for studying the equilibrium distribution of chromosomal lengths, and are used in methods for estimating genomic distances. The primary Markov chain studied in this paper is the top-swap Markov chain. The top-swap chain is a card-shuffling process with $n$ cards divided over $k$ decks, where the cards are ordered within each deck. A transition consists of choosing a random pair of cards, and if the cards lie in different decks, we cut each deck at the chosen card and exchange the tops of the two decks. We prove precise bounds on the relaxation time (inverse spectral gap) of the top-swap chain. In particular, we prove the relaxation time is $Θ(n+k)$. This resolves an open question of Durrett.

math.PR↗

arXiv2007

Limitations of Markov chain Monte Carlo algorithms for Bayesian Inference of phylogeny

Markov chain Monte Carlo algorithms play a key role in the Bayesian approach to phylogenetic inference. In this paper, we present the first theoretical work analyzing the rate of convergence of several Markov chains widely used in phylogenetic inference. We analyze simple, realistic examples where these Markov chains fail to converge quickly. In particular, the data studied are generated from a pair of trees, under a standard evolutionary model. We prove that many of the popular Markov chains take exponentially long to reach their stationary distribution. Our construction is pertinent since it is well known that phylogenetic trees for genes may differ within a single organism. Our results shed a cautionary light on phylogenetic analysis using Bayesian inference and highlight future directions for potential theoretical work.

q-bio.PE↗

arXiv2006

Adaptive Simulated Annealing: A Near-optimal Connection between Sampling and Counting

We present a near-optimal reduction from approximately counting the cardinality of a discrete set to approximately sampling elements of the set. An important application of our work is to approximating the partition function $Z$ of a discrete system, such as the Ising model, matchings or colorings of a graph. The typical approach to estimating the partition function $Z(β^*)$ at some desired inverse temperature $β^*$ is to define a sequence, which we call a {\em cooling schedule}, $β_0=0<β_1<...<β_\ell=β^*$ where Z(0) is trivial to compute and the ratios $Z(β_{i+1})/Z(β_i)$ are easy to estimate by sampling from the distribution corresponding to $Z(β_i)$. Previous approaches required a cooling schedule of length $O^*(\ln{A})$ where $A=Z(0)$, thereby ensuring that each ratio $Z(β_{i+1})/Z(β_i)$ is bounded. We present a cooling schedule of length $\ell=O^*(\sqrt{\ln{A}})$. For well-studied problems such as estimating the partition function of the Ising model, or approximating the number of colorings or matchings of a graph, our cooling schedule is of length $O^*(\sqrt{n})$, which implies an overall savings of $O^*(n)$ in the running time of the approximate counting algorithm (since roughly $\ell$ samples are needed to estimate each ratio).

cs.DS↗

arXiv2006

Coupling with the stationary distribution and improved sampling for colorings and independent sets

We present an improved coupling technique for analyzing the mixing time of Markov chains. Using our technique, we simplify and extend previous results for sampling colorings and independent sets. Our approach uses properties of the stationary distribution to avoid worst-case configurations which arise in the traditional approach. As an application, we show that for $k/Δ>1.764$, the Glauber dynamics on $k$-colorings of a graph on $n$ vertices with maximum degree $Δ$ converges in $O(n\log n)$ steps, assuming $Δ=Ω(\log n)$ and that the graph is triangle-free. Previously, girth $\ge 5$ was needed. As a second application, we give a polynomial-time algorithm for sampling weighted independent sets from the Gibbs distribution of the hard-core lattice gas model at fugacity $λ<(1-ε)e/Δ$, on a regular graph $G$ on $n$ vertices of degree $Δ=Ω(\log n)$ and girth $\ge 6$. The best known algorithm for general graphs currently assumes $λ<2/(Δ-2)$.

math.PR↗

arXiv2006

Phylogeny of Mixture Models: Robustness of Maximum Likelihood and Non-identifiable Distributions

We address phylogenetic reconstruction when the data is generated from a mixture distribution. Such topics have gained considerable attention in the biological community with the clear evidence of heterogeneity of mutation rates. In our work, we consider data coming from a mixture of trees which share a common topology, but differ in their edge weights (i.e., branch lengths). We first show the pitfalls of popular methods, including maximum likelihood and Markov chain Monte Carlo algorithms. We then determine in which evolutionary models, reconstructing the tree topology, under a mixture distribution, is (im)possible. We prove that every model whose transition matrices can be parameterized by an open set of multi-linear polynomials, either has non-identifiable mixture distributions, in which case reconstruction is impossible in general, or there exist linear tests which identify the topology. This duality theorem, relies on our notion of linear tests and uses ideas from convex programming duality. Linear tests are closely related to linear invariants, which were first introduced by Lake, and are natural from an algebraic geometry perspective.

q-bio.PE↗

arXiv2005

Elementary bounds on Poincare and log-Sobolev constants for decomposable Markov chains

We consider finite-state Markov chains that can be naturally decomposed into smaller ``projection'' and ``restriction'' chains. Possibly this decomposition will be inductive, in that the restriction chains will be smaller copies of the initial chain. We provide expressions for Poincare (resp. log-Sobolev) constants of the initial Markov chain in terms of Poincare (resp. log-Sobolev) constants of the projection and restriction chains, together with further a parameter. In the case of the Poincare constant, our bound is always at least as good as existing ones and, depending on the value of the extra parameter, may be much better. There appears to be no previously published decomposition result for the log-Sobolev constant. Our proofs are elementary and self-contained.

math.PR↗