Multiple sequence alignments are fundamental to many sequence analysis methods. Most alignments are computed using the progressive alignment heuristic. These methods are starting to become a bottleneck in some analysis pipelines when faced with data sets of the size of many thousands of sequences. Some methods allow computation of larger data sets while sacrificing quality, and others produce high-quality alignments, but scale badly with the number of sequences. In this paper, we describe a new program called Clustal Omega, which can align virtually any number of protein sequences quickly and that delivers accurate alignments. The accuracy of the package on smaller test cases is similar to that of the high-quality aligners. On larger data sets, Clustal Omega outperforms other packages in terms of execution time and quality. Clustal Omega also has powerful features for adding sequences to and exploiting information in existing alignments, making use of the vast amount of precomputed information in public databases like Pfam.

/pdf/fast-scalable-generation-of-high-quality-protein-multiple-4ypo2ssntq.pdf

Fast, scalable generation of high‐quality protein multiple sequence alignments using Clustal Omega

We consider a problem of considerable practical interest: the recovery of a data matrix from a sampling of its entries. Suppose that we observe m entries selected uniformly at random from a matrix M. Can we complete the matrix and recover the entries that we have not seen?

We show that one can perfectly recover most low-rank matrices from what appears to be an incomplete set of entries. We prove that if the number m of sampled entries obeys $$m\ge C\,n^{1.2}r\log n$$ for some positive numerical constant C, then with very high probability, most n×n matrices of rank r can be perfectly recovered by solving a simple convex optimization program. This program finds the matrix with minimum nuclear norm that fits the data. The condition above assumes that the rank is not too large. However, if one replaces the 1.2 exponent with 1.25, then the result holds for all values of the rank. Similar results hold for arbitrary rectangular matrices as well. Our results are connected with the recent literature on compressed sensing, and show that objects other than signals and images can be perfectly reconstructed from very limited information.

/pdf/exact-matrix-completion-via-convex-optimization-3nmwb4bacp.pdf

Exact Matrix Completion via Convex Optimization

We analyze the performance of a Genetic Type Algorithm we call Culling and a variety of other algorithms on a problem we refer to as ASP. Culling is near optimal for this problem, highly noise tolerant, and the best known a~~roach . . in some regimes. We show that the problem of learning the Ising perception is reducible to noisy ASP. These results provide an example of a rigorous analysis of GA’s and give insight into when and how C,A’s can beat competing methods. To analyze the genetic algorithm, we view it as a special type of submartingale. We prove some new large deviation bounds on this submartingale w~ich enable us to determine the running time of the algorithm.

On genetic algorithms

We present two algorithms for the approximate nearest neighbor problem in high-dimensional spaces. For data sets of size n living in R d , the algorithms require space that is only polynomial in n and d, while achieving query times that are sub-linear in n and polynomial in d. We also show applications to other high-dimensional geometric problems, such as the approximate minimum spanning tree. The article is based on the material from the authors' STOC'98 and FOCS'01 papers. It unifies, generalizes and simplifies the results from those papers.

/pdf/approximate-nearest-neighbors-towards-removing-the-curse-of-27lmlz5prr.pdf

Approximate nearest neighbors: towards removing the curse of dimensionality

Covering the basic techniques used in the latest research work, the author consolidates progress made so far, including some very recent and promising results, and conveys the beauty and excitement of work in the field. He gives clear, lucid explanations of key results and ideas, with intuitive proofs, and provides critical examples and numerous illustrations to help elucidate the algorithms. Many of the results presented have been simplified and new insights provided. Of interest to theoretical computer scientists, operations researchers, and discrete mathematicians.

Approximation Algorithms

In this paper we explore some implications of viewing graphs asgeometric objects. This approach offers a new perspective on a number of graph-theoretic and algorithmic problems. There are several ways to model graphs geometrically and our main concern here is with geometric representations that respect themetric of the (possibly weighted) graph. Given a graphG we map its vertices to a normed space in an attempt to (i) keep down the dimension of the host space, and (ii) guarantee a smalldistortion, i.e., make sure that distances between vertices inG closely match the distances between their geometric images. In this paper we develop efficient algorithms for embedding graphs low-dimensionally with a small distortion. Further algorithmic applications include:
 Given faithful low-dimensional representations of statistical data, it is possible to obtain meaningful and efficientclustering. This is one of the most basic tasks in pattern-recognition. For the (mostly heuristic) methods used in the practice of pattern-recognition, see [20], especially chapter 6. Our studies of multicommodity flows also imply that every embedding of (the metric of) ann-vertex, constant-degree expander into a Euclidean space (of any dimension) has distortion Ω(logn). This result is tight, and closes a gap left open by Bourgain [12].

The geometry of graphs and some of its algorithmic applications

Motivated by many recent algorithmic applications, this paper aims to promote a systematic study of the relationship between the topology of a graph and the metric distortion incurred when the graph is embedded into \math space. The main results are: 1. Explicit constant-distortion embeddings of all series-parallel graphs, and all graphs with bounded Euler number. These are thus the first natural families known to have constant distortion (strictly greater than 1). Using the above embeddings, we obtain algorithms to approximate the sparsest cut in such graphs to within a constant factor. 2. A constant-distortion embedding of outerplanar graphs into the restricted class of \math-metrics known as "dominating tree metrics". We also show a lower bound of \math on the distortion for embeddings of series-parallel graphs into (distributions over) dominating tree metrics. This shows, surprisingly, that such metrics approximate distances very poorly even for families of graphs with low treewidth, and excludes the possibility of using them to explore the finer structure of \math-embeddability.

Cuts, Trees and ℓ 1 -Embeddings of Graphs*

We explore some implications of viewing graphs as geometric objects. This approach offers a new perspective on a number of graph-theoretic and algorithmic problems. There are several ways to model graphs geometrically and our main concern here is with geometric representations that respect the metric of the (possibly weighted) graph. Given a graph G we map its vertices to a normed space in an attempt to (i) Keep down the dimension of the host space and (ii) Guarantee a small distortion, i.e., make sure that distances between vertices in G closely match the distances between their geometric images. We develop efficient algorithms for embedding graphs low-dimensionally with a small distortion. >

The main question discussed in this paper is how well a finite metric space of size n can be embedded into a graph with certain topological restrictions

/pdf/lower-bounds-on-the-distortion-of-embedding-finite-metric-1d3bnqqaqg.pdf

Lower Bounds on the Distortion of Embedding Finite Metric Spaces in Graphs

A Computational View of Population Genetics (preliminary version) Yuval Rabanit Yuri Rabinovicht Alistair Sinclair] This paper contributes to the study of nonlinear dynamical systems from a computational perspective. These systems are inherently more powerful than their linear counterparts (such as Markov chains), which have had a wide impact in Computer Science, and they seem likely to play an increasing role in future. However, there are as yet no general techniques available for handling the computational aspects of discrete nonlinear systems, and even the simplest examples seem very hard to analyze. We focus in this paper on a class of quadratic systems that are widely used as a model in population genetics and also in genetic algorithms. These systems describe a process where random matings occur between parental chromosomes via a mechanism known as “crossover”: i.e., children inherit pieces of genetic material from different parents according to some random rule. Our results concern two fundamental quantitative properties of crossover systems: 1. We develop a general technique for computing the rate of convergence to equilibrium. We apply this technique to obtain tight bounds on the rate of convergence in several cases of biological and computational interest. In general, we prove that these systems are “rapidly mixing”, in the sense that the convergence time is very small in comparison with the size of the state space. 2. We show that, for crossover systems, the classical quadratic system is a good model for the behavior of finite populations of small size. This stands in sharp contrast to recent results of Arora et al and Pudlak, who show that such a correspondence is unlikely to hold for general quadratic systems. tD~p~~t~ent Of Computer Science, University of TorontO, Toronto, Ontario M5S 1A4, Canada. Email: {rabani ,yuri}Q cs. toronto. edu. t Computer Science Division, University of California, Berkeley CA 94720-1776, U.S.A. Email: sinclair@cs .berkeley. edu. Perrmssion to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyri ht notice and the % title of the publication and Its dat~ appear, an notice IS gwen that copyin is by permission of tne Association of Computing ? Machinery o copy otherwise, or to republish, requires a fee andlor specific permission. STOC” 95, Las Vegas, Nevada, USA @ 1995 ACM 0-89791 -718-9/95/0005 .$3.50

Yuri Rabinovich

Papers

The geometry of graphs and some of its algorithmic applications

Cuts, Trees and ℓ 1 -Embeddings of Graphs*

The geometry of graphs and some of its algorithmic applications

Lower Bounds on the Distortion of Embedding Finite Metric Spaces in Graphs

A computational view of population genetics