Robust Euclidean embedding via EDM optimization

Abstract

This paper aims to propose an efficient numerical method for the most challenging problem known as the robust Euclidean embedding (REE) in the family of multidimensional scaling (MDS). The problem is notoriously known to be nonsmooth, nonconvex and its objective is non-Lipschitzian. We first explain that the semidefinite programming (SDP) relaxations and Euclidean distance matrix (EDM) approach, popular for other types of problems in the MDS family, failed to provide a viable method for this problem. We then propose a penalized REE (PREE), which can be economically majorized. We show that the majorized problem is convex provided that the penalty parameter is above certain threshold. Moreover, it has a closed-form solution, resulting in an efficient algorithm dubbed as PREEEDM (for Penalized REE via EDM optimization). We prove among others that PREEEDM converges to a stationary point of PREE, which is also an approximate critical point of REE. Finally, the efficiency of PREEEDM is compared with several state-of-the-art methods including SDP and EDM solvers on a large number of test problems from sensor network localization and molecular conformation.

A penalized method of alternating projections for weighted low-rank hankel matrix optimization

“…Penalty approaches have long been used to develop globally convergent algorithms for problems with rank constraints, see [10,12,13,21,22,27,32,33]. For example, Gao [12] proposed the penalty function p(X ) based on the following observation:…”

“…Hence, we define the complex gradient to be ∇ f (z) := 2∂ f /∂ z, when it exists. When f is not differentiable, we can extend the subdifferential of f from the real case to the complex case by generalizing (33).…”

Feasibility and a fast algorithm for Euclidean distance matrix optimization with ordinal constraints

“…where nonnegative and nonincreasing sequencesŷ ∈ IR m and λ ∈ IR m are given. Problem (37) can be reformulated as typical isotonic regression (36) with y =ŷ− λ. Consequently, we reach the following algorithm (denoted as mFastProxSL1) for solving isotonic regression (36). S1 While x is not decreasing, do Identify strictly increasing subsequences, i.e.…”

“…We refer to [5,21,15,31] just to name a few of outstanding SDP based approaches for EDM optimization in SNL and MC. The characterization (4) has fundamental differences from (3) [25] as it describes an EDM via the conditional positive semidefinite cone, based on which great progress has been made on numerical algorithms for EDM optimization [25,28,27,26,13,22,35,36], as we will detail below.…”

“…Based on such technique, the resulting majorization-minimization approach demonstrates superior numerical performance on MC and SNL. Similar technique is used in [36] where a robust loss function…”