arXiv subjects

Nadav Dym

Publications and source records attributed to Nadav Dym.

42 records · Page 3Linked to original sources

arXiv2018

Sinkhorn Algorithm for Lifted Assignment Problems

Recently, Sinkhorn's algorithm was applied for approximately solving linear programs emerging from optimal transport very efficiently. This was accomplished by formulating a regularized version of the linear program as Bregman projection problem onto the polytope of doubly-stochastic matrices, and then computing the projection using the efficient Sinkhorn algorithm, which is based on alternating closed-form Bregman projections on the larger polytopes of row-stochastic and column-stochastic matrices. In this paper we suggest a generalization of this algorithm for solving a well-known lifted linear program relaxations of the Quadratic Assignment Problem (QAP), which is known as the Johnson Adams (JA) Relaxation. First, an efficient algorithm for Bregman projection onto the JA polytope by alternating closed-form Bregman projections onto one-sided local polytopes is devised. The one-sided polytopes can be seen as a high-dimensional, generalized version of the row/column-stochastic polytopes. Second, a new method for solving the original linear programs using the Bregman projections onto the JA polytope is developed and shown to be more accurate and numerically stable than the standard approach of driving the regularizer to zero. The resulting algorithm is considerably more scalable than standard linear solvers and is able to solve significantly larger linear programs.

math.OC↗

arXiv2018

A Linear Variational Principle for Riemann Mappings and Discrete Conformality

We consider Riemann mappings from bounded Lipschitz domains in the plane to a triangle. We show that in this case the Riemann mapping has a linear variational principle: it is the minimizer of the Dirichlet energy over an appropriate affine space. By discretizing the variational principle in a natural way we obtain discrete conformal maps which can be computed by solving a sparse linear system. We show that these discrete conformal maps converge to the Riemann mapping in $H^1$, even for non-Delaunay triangulations. Additionally, for Delaunay triangulations the discrete conformal maps converge uniformly and are known to be bijective. As a consequence we show that the Riemann mapping between two bounded Lipschitz domains can be uniformly approximated by composing the Riemann mappings between each Lipschitz domain and the triangle.

cs.CG↗

arXiv2017

Exact Recovery with Symmetries for Procrustes Matching

The Procrustes matching (PM) problem is the problem of finding the optimal rigid motion and labeling of two point sets so that they are as close as possible. Both rigid and non-rigid shape matching problems can be formulated as PM problems. Recently [Maron et al.] presented a novel convex semi-definite programming relaxation (PM-SDP) for PM which achieves state of the art results on common shape matching benchmarks. In this paper we analyze the successfulness of PM-SDP in solving PM problems without noise (Exact PM problems). We begin by showing Exact PM to be computationally equivalent to the graph isomorphism problem. We demonstrate some natural theoretical properties of the relaxation, and use these properties together with the moment interpretation of [Lasserre] to show that for exact PM problems and for (generic) input shapes which are asymmetric or bilaterally symmetric, the relaxation returns a correct solution of PM. For symmetric shapes, PM has multiple solutions. The non-convex set of optimal solutions of PM is strictly contained in the convex set of optimal solutions of PM-SDP, so that `most' solutions of PM-SDP will not be solutions of PM. We deal with this by showing the solution set of PM to be the extreme points of the solution set of PM-SDP, and suggesting a random algorithm which returns a solution of PM with probability one, and returns all solutions of PM with equal probability. We also show these results can be extended to the almost-exact case. To the best of our knowledge, our work is the first to achieve exact recovery in the presence of multiple solutions.

math.OC↗

arXiv2017

Exact Recovery with Symmetries for the Doubly-Stochastic Relaxation

Graph matching or quadratic assignment, is the problem of labeling the vertices of two graphs so that they are as similar as possible. A common method for approximately solving the NP-hard graph matching problem is relaxing it to a convex optimization problem over the set of doubly stochastic (DS) matrices. Recent analysis has shown that for almost all pairs of isomorphic and asymmetric graphs, the DS relaxation succeeds in correctly retrieving the isomorphism between the graphs. Our goal in this paper is to analyze the case of symmetric isomorphic graphs. This goal is motivated by shape matching applications where the graphs of interest usually have reflective symmetry. For symmetric problems the graph matching problem has multiple isomorphisms and so convex relaxations admit all convex combinations of these isomorphisms as viable solutions. If the convex relaxation does not admit any additional superfluous solution we say that it is convex exact. In this case there are tractable algorithms to retrieve an isomorphism from the convex relaxation. We show that convex exactness depends strongly on the symmetry group of the graphs; For a fixed symmetry group $G$, either the DS relaxation will be convex exact for almost all pairs of isomorphic graphs with symmetry group $G$, or the DS relaxation will fail for all such pairs. We show that for reflective groups with at least one full orbit convex exactness holds almost everywhere, and provide some simple examples of non-reflective symmetry groups for which convex exactness always fails. When convex exactness holds, the isomorphisms of the graphs are the extreme points of the convex solution set. We suggest an efficient algorithm for retrieving an isomorphism in this case. We also show that the "convex to concave" projection method will also retrieve an isomorphism in this case.

math.OC↗

arXiv2017

DS++: A flexible, scalable and provably tight relaxation for matching problems

Correspondence problems are often modelled as quadratic optimization problems over permutations. Common scalable methods for approximating solutions of these NP-hard problems are the spectral relaxation for non-convex energies and the doubly stochastic (DS) relaxation for convex energies. Lately, it has been demonstrated that semidefinite programming relaxations can have considerably improved accuracy at the price of a much higher computational cost. We present a convex quadratic programming relaxation which is provably stronger than both DS and spectral relaxations, with the same scalability as the DS relaxation. The derivation of the relaxation also naturally suggests a projection method for achieving meaningful integer solutions which improves upon the standard closest-permutation projection. Our method can be easily extended to optimization over doubly stochastic matrices, partial or injective matching, and problems with additional linear constraints. We employ recent advances in optimization of linear-assignment type problems to achieve an efficient algorithm for solving the convex relaxation. We present experiments indicating that our method is more accurate than local minimization or competing relaxations for non-convex problems. We successfully apply our algorithm to shape matching and to the problem of ordering images in a grid, obtaining results which compare favorably with state of the art methods. We believe our results indicate that our method should be considered the method of choice for quadratic optimization over permutations.

cs.GR↗

arXiv2016

Spatial Recurrence for Ergodic Fractal Measures

We discuss an invertible version of Furstenberg's `Ergodic CP Shift Systems'. We show that the explicit regularity of these dynamical systems with respect to magnification of measures, implies certain regularity with respect to translation of measures; We show that the translation action on measures is non-singular, and prove pointwise discrete and continuous ergodic theorems for the translation action.

math.DS↗