Results 1  10
of
21
Computing Simplicial Homology Based on Efficient Smith Normal Form Algorithms
, 2002
"... We recall that the calculation of homology with integer coecients of a simplicial complex reduces to the calculation of the Smith Normal Form of the boundary matrices which in general are sparse. We provide a review of several algorithms for the calculation of Smith Normal Form of sparse matrices an ..."
Abstract

Cited by 38 (2 self)
 Add to MetaCart
We recall that the calculation of homology with integer coecients of a simplicial complex reduces to the calculation of the Smith Normal Form of the boundary matrices which in general are sparse. We provide a review of several algorithms for the calculation of Smith Normal Form of sparse matrices and compare their running times for actual boundary matrices. Then we describe alternative approaches to the calculation of simplicial homology. The last section then describes motivating examples and actual experiments with the GAP package that was implemented by the authors. These examples also include as an example of other homology theories some calculations of Lie algebra homology.
Mixedup Trees: the Structure of Phylogenetic Mixtures
 BULLETIN OF MATHEMATICAL BIOLOGY (2008)
, 2008
"... In this paper, we apply new geometric and combinatorial methods to the study of phylogenetic mixtures. The focus of the geometric approach is to describe the geometry of phylogenetic mixture distributions for the two state random cluster model, which is a generalization of the two state symmetric ( ..."
Abstract

Cited by 20 (4 self)
 Add to MetaCart
(Show Context)
In this paper, we apply new geometric and combinatorial methods to the study of phylogenetic mixtures. The focus of the geometric approach is to describe the geometry of phylogenetic mixture distributions for the two state random cluster model, which is a generalization of the two state symmetric (CFN) model. In particular, we show that the set of mixture distributions forms a convex polytope and we calculate its dimension; corollaries include a simple criterion for when a mixture of branch lengths on the star tree can mimic the site pattern frequency vector of a resolved quartet tree. Furthermore, by computing volumes of polytopes we can clarify how “common” nonidentifiable mixtures are under the CFN model. We also present a new combinatorial result which extends any identifiability result for a specific pair of trees of size six to arbitrary pairs of trees. Next we present a positive result showing identifiability of ratesacrosssites models. Finally, we answer a question raised in a previous paper concerning “mixed branch repulsion” on trees larger than quartet trees under the CFN model.
Computation and relaxation of conditions for equivalence between ℓ 1 and ℓ 0 minimization,” CSL
, 2007
"... Abstract—In this paper, we investigate the exact conditions under which the ℓ 1 and ℓ 0 minimizations arising in the context of sparse error correction or sparse signal reconstruction are equivalent. We present a much simplified condition for verifying equivalence, which leads to a provably correct ..."
Abstract

Cited by 11 (3 self)
 Add to MetaCart
(Show Context)
Abstract—In this paper, we investigate the exact conditions under which the ℓ 1 and ℓ 0 minimizations arising in the context of sparse error correction or sparse signal reconstruction are equivalent. We present a much simplified condition for verifying equivalence, which leads to a provably correct algorithm that computes the exact sparsity of the error or the signal needed to ensure equivalence. In the case when the encoding matrix is imbalanced, we show how an optimal diagonal rescaling matrix can be computed via linear programming, so that the rescaled system enjoys the widest possible equivalence. I.
Convex Hulls, Oracles, and Homology
 J. SYMBOLIC COMPUT
, 2004
"... This paper presents a new algorithm for the convex hull problem, which is based on a reduction to a combinatorial decision problem CompletenessC, which in turn can be solved by a simplicial homology computation. Like other convex hull algorithms, our algorithm is polynomial (in the size of input plu ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
This paper presents a new algorithm for the convex hull problem, which is based on a reduction to a combinatorial decision problem CompletenessC, which in turn can be solved by a simplicial homology computation. Like other convex hull algorithms, our algorithm is polynomial (in the size of input plus output) for simplicial or simple input. We show that the “no”case of CompletenessC has a certificate that can be checked in polynomial time (if integrity of the input is guaranteed).
Why Initialization Matters for IBM Model 1: Multiple Optima and NonStrict Convexity
"... Contrary to popular belief, we show that the optimal parameters for IBM Model 1 are not unique. We demonstrate that, for a large class of words, IBM Model 1 is indifferent among a continuum of ways to allocate probability mass to their translations. We study the magnitude of the variance in optimal ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
(Show Context)
Contrary to popular belief, we show that the optimal parameters for IBM Model 1 are not unique. We demonstrate that, for a large class of words, IBM Model 1 is indifferent among a continuum of ways to allocate probability mass to their translations. We study the magnitude of the variance in optimal model parameters using a linear programming approach as well as multiple random trials, and demonstrate that it results in variance in test set loglikelihood and alignment error rate. 1
Beneathandbeyond revisited
 Algebra, Geometry, and Software Systems
, 2003
"... Abstract. It is shown how the BeneathandBeyond algorithm can be used to yield another proof of the equivalence of V and Hrepresentations of convex polytopes. In this sense this paper serves as the sketch of an introduction to polytope theory with a focus on algorithmic aspects. Moreover, computa ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
Abstract. It is shown how the BeneathandBeyond algorithm can be used to yield another proof of the equivalence of V and Hrepresentations of convex polytopes. In this sense this paper serves as the sketch of an introduction to polytope theory with a focus on algorithmic aspects. Moreover, computational results are presented to compare BeneathandBeyond to other convex hull implementations. 1
POLYHEDRAL COMBINATORICS
"... Polyhedral combinatorics is a rich mathematical subject motivated by integer and linear programming. While not exhaustive, this survey covers a variety of interesting topics, so let’s get right to it! ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
Polyhedral combinatorics is a rich mathematical subject motivated by integer and linear programming. While not exhaustive, this survey covers a variety of interesting topics, so let’s get right to it!
Which nonnegative matrices are slack matrices?
, 2013
"... In this paper we characterize the slack matrices of cones and polytopes among all nonnegative matrices. This leads to an algorithm for deciding whether a given matrix is a slack matrix. The underlying decision problem is equivalent to the polyhedral verification problem whose complexity is unknown. ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
(Show Context)
In this paper we characterize the slack matrices of cones and polytopes among all nonnegative matrices. This leads to an algorithm for deciding whether a given matrix is a slack matrix. The underlying decision problem is equivalent to the polyhedral verification problem whose complexity is unknown.
MultiObjective Parametric Query Optimization
"... Classical query optimization compares query plans according to one cost metric and associates each plan with a constant cost value. In this paper, we introduce the MultiObjective Parametric Query Optimization (MPQ) problem where query plans are compared according to multiple cost metrics and the ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
Classical query optimization compares query plans according to one cost metric and associates each plan with a constant cost value. In this paper, we introduce the MultiObjective Parametric Query Optimization (MPQ) problem where query plans are compared according to multiple cost metrics and the cost of a given plan according to a given metric is modeled as a function that depends on multiple parameters. The cost metrics may for instance include execution time or monetary fees; a parameter may represent the selectivity of a query predicate that is unspecied at optimization time. MPQ generalizes parametric query optimization (which allows multiple parameters but only one cost metric) and multiobjective query optimization (which allows multiple cost metrics but no parameters). We formally analyze the novel MPQ problem and show why existing algorithms are inapplicable. We present a generic algorithm for MPQ and a specialized version for MPQ with piecewiselinear plan cost functions. We prove that both algorithms nd all relevant query plans and experimentally evaluate the performance of our second algorithm in a Cloud computing scenario. 1.