arXiv subjects

Nathan Krislock

Publications and source records attributed to Nathan Krislock.

5 recordsLinked to original sources

arXiv2020

On Inexact Accelerated Proximal Gradient Methods with Relative Error Rules

One of the most popular and important first-order iterations that provides optimal complexity of the classical proximal gradient method (PGM) is the "Fast Iterative Shrinkage/Thresholding Algorithm" (FISTA). In this paper, two inexact versions of FISTA for minimizing the sum of two convex functions are studied. The proposed schemes inexactly solve their subproblems by using relative error criteria instead of exogenous and diminishing error rules. When the evaluation of the proximal operator is difficult, inexact versions of FISTA are necessary and the relative error rules proposed here may have certain advantages over previous error rules. The same optimal convergence rate of FISTA is recovered for both proposed schemes. Some numerical experiments are reported to illustrate the numerical behavior of the new approaches.

math.OC↗

arXiv2015

Noisy Euclidean distance realization: robust facial reduction and the Pareto frontier

We present two algorithms for large-scale low-rank Euclidean distance matrix completion problems, based on semidefinite optimization. Our first method works by relating cliques in the graph of the known distances to faces of the positive semidefinite cone, yielding a combinatorial procedure that is provably robust and parallelizable. Our second algorithm is a first order method for maximizing the trace---a popular low-rank inducing regularizer---in the formulation of the problem with a constrained misfit. Both of the methods output a point configuration that can serve as a high-quality initialization for local optimization techniques. Numerical experiments on large-scale sensor localization problems illustrate the two approaches.

math.OC↗

arXiv2014

Resolving Histogram Binning Dilemmas with Binless and Binfull Algorithms

The histogram is an analysis tool in widespread use within many sciences, with high energy physics as a prime example. However, there exists an inherent bias in the choice of binning for the histogram, with different choices potentially leading to different interpretations. This paper aims to eliminate this bias using two "debinning" algorithms. Both algorithms generate an observed cumulative distribution function from the data, and use it to construct a representation of the underlying probability distribution function. The strengths and weaknesses of these two algorithms are compared and contrasted. The applicability and future prospects of these algorithms is also discussed.

physics.data-an↗

arXiv2010

Explicit Sensor Network Localization using Semidefinite Representations and Facial Reductions

The sensor network localization, SNL, problem in embedding dimension r, consists of locating the positions of wireless sensors, given only the distances between sensors that are within radio range and the positions of a subset of the sensors (called anchors). Current solution techniques relax this problem to a weighted, nearest, (positive) semidefinite programming, SDP, completion problem, by using the linear mapping between Euclidean distance matrices, EDM, and semidefinite matrices. The resulting SDP is solved using primal-dual interior point solvers, yielding an expensive and inexact solution. This relaxation is highly degenerate in the sense that the feasible set is restricted to a low dimensional face of the SDP cone, implying that the Slater constraint qualification fails. Cliques in the graph of the SNL problem give rise to this degeneracy in the SDP relaxation. In this paper, we take advantage of the absence of the Slater constraint qualification and derive a technique for the SNL problem, with exact data, that explicitly solves the corresponding rank restricted SDP problem. No SDP solvers are used. For randomly generated instances, we are able to efficiently solve many huge instances of this NP-hard problem to high accuracy, by finding a representation of the minimal face of the SDP cone that contains the SDP matrix representation of the EDM. The main work of our algorithm consists in repeatedly finding the intersection of subspaces that represent the faces of the SDP cone that correspond to cliques of the SNL problem.

math.OC↗

arXiv2006

Sensor Network Localization, Euclidean Distance Matrix Completions, and Graph Realization

We study Semidefinite Programming, \SDPc relaxations for Sensor Network Localization, \SNLc with anchors and with noisy distance information. The main point of the paper is to view \SNL as a (nearest) Euclidean Distance Matrix, \EDM, completion problem and to show the advantages for using this latter, well studied model. We first show that the current popular \SDP relaxation is equivalent to known relaxations in the literature for \EDM completions. The existence of anchors in the problem is {\em not} special. The set of anchors simply corresponds to a given fixed clique for the graph of the \EDM problem. We next propose a method of projection when a large clique or a dense subgraph is identified in the underlying graph. This projection reduces the size, and improves the stability, of the relaxation. In addition, viewing the problem as an \EDM completion problem yields better low rank approximations for the low dimensional realizations. And, the projection/reduction procedure can be repeated for other given cliques of sensors or for sets of sensors, where many distances are known. Thus, further size reduction can be obtained. Optimality/duality conditions and a primal-dual interior-exterior path following algorithm are derived for the \SDP relaxations We discuss the relative stability and strength of two formulations and the corresponding algorithms that are used. In particular, we show that the quadratic formulation arising from the \SDP relaxation is better conditioned than the linearized form, that is used in the literature and that arises from applying a Schur complement.

math.OC↗