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.

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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.

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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.

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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

The main purpose of the present paper is the study of computational aspects, and primarily the convergence rate, of genetic algorithms (GAs). Despite the fact that such algorithms are widely used in practice, little is known so far about their theoretical properties, and in particular about their long-term behavior. This situation is perhaps not too surprising, given the inherent hardness of analyzing nonlinear dynamical systems, and the complexity of the problems to which GAs are usually applied. In the present paper we concentrate on a number of very simple and natural systems of this sort, and show that at least for these systems the analysis can be properly carried out. Various properties and tight quantitative bounds on the long-term behavior of such systems are established. It is our hope that the techniques developed for analyzing these simple systems prove to be applicable to a wider range of genetic algorithms, and contribute to the development of the mathematical foundations of this promising optimization method. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 14, 111–138, 1999

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Techniques for bounding the convergence rate of genetic algorithms

In this paper, we study testing of sequence properties that are defined by forbidden order patterns. A sequence f : {1, . . . , n} → ℝ of length n contains a pattern π ∈ &#x1d516;k (&#x1d516;k is the group of permutations of k elements), iff there are indices i1 f(iy) whenever π(x) > π(y). If f does not contain π, we say f is π-free. For example, for π = (2, 1), the property of being π-free is equivalent to being non-decreasing, i.e. monotone. The property of being (k, k − 1, . . . , 1)-free is equivalent to the property of having a partition into at most k − 1 non-decreasing subsequences.Let π ∈ &#x1d516;k, k constant, be a (forbidden) pattern. Assuming f is stored in an array, we consider the property testing problem of distinguishing the case that f is π-free from the case that f differs in more than ϵn places from any π-free sequence. We show the following results: There is a clear dichotomy between the monotone patterns and the non-monotone ones:• For monotone patterns of length k, i.e., (k, k − 1, . . . , 1) and (1, 2, . . . , k), we design non-adaptive one-sided error ϵ-tests of (ϵ−1 log n)O(k2) query complexity.• For non-monotone patterns, we show that for any size-k non-monotone π, any non-adaptive one-sided error ϵ-test requires at least [EQUATION] queries. This general lower bound can be further strengthened for specific non-monotone k-length patterns to Ω(n1−2/(k+1)).On the other hand, there always exists a non-adaptive one-sided error ϵ-test for π ∈ &#x1d516;k with O(ϵ−1/kn1−1/k) query complexity. Again, this general upper bound can be further strengthened for specific non-monotone patterns. E.g., for π = (1, 3, 2), we describe an ϵ-test with (almost tight) query complexity of [EQUATION].Finally, we show that adaptivity can make a big difference in testing non-monotone patterns, and develop an adaptive algorithm that for any π ∈ &#x1d516;3, tests π-freeness by making (ϵ−1 log n)O(1) queries.For all algorithms presented here, the running times are linear in their query complexity.

Testing for forbidden order patterns in an array

We introduce and study the notion of the average distortion of a nonexpanding embedding of one metric space into another. Less sensitive than the multiplicative metric distortion, the average distortion captures well the global picture and, overall, is a quite interesting new measure of metric proximity, related to the concentration of measure phenomenon. The paper mostly deals with embeddings into the real line with a low (as much as it is possible) average distortion. Our main technical contribution is that the shortest-path metrics of special (e.g., planar, bounded treewidth, etc.) undirected weighted graphs can be embedded into the line with constant average distortion. This has implications, e.g., on the value of the MinCut–MaxFlow gap in uniform-demand multicommodity flows on such graphs.

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On Average Distortion of Embedding Metrics into the Line

Witness sets for families of binary vectors

Let $\mathcal{F}$ be a set system over an underlying finite set $X$, and let $\mu$ be a nonnegative measure over $X$; i.e., for every $S \subseteq X$, $\mu(S)=\sum_{x\in S} \mu(x)$. A measure $\mu^*$ on $X$ is called a multiplicative ${\lambda}$-approximation of $\mu$ on $(\mathcal{F},X)$ if for every $S\in \mathcal{F}$ it holds that $a\mu(S) \leq \mu^*(S) \leq b \mu(S)$, and $b/a = \lambda \geq 1$. The central question raised and partially answered in the present paper is about the existence of meaningful structural properties of $\mathcal{F}$ implying that for any $\mu$ on $X$ there exists an ${{1+\epsilon} \over {1-\epsilon}}$-approximation $\mu^*$ supported on a small subset of $X$. It turns out that the parameter that governs the support size of a multiplicative approximation is the triangular rank of $\mathcal{F}$, ${\rm trk}(\mathcal{F})$. It is defined as the maximal length of a sequence of sets $\{S_i\}_{i=1}^t $ in $\mathcal{F}$ such that for all $1<i\leq t$, $S_i \not\subseteq \cup_{j<i}\,S_j$....

Yuri Rabinovich

Papers

Techniques for bounding the convergence rate of genetic algorithms

Testing for forbidden order patterns in an array

On Average Distortion of Embedding Metrics into the Line

Witness sets for families of binary vectors

On Multiplicative $\lambda$-Approximations and Some Geometric Applications