Blessing of High-Order Dimensionality: from Non-Convex to Convex Optimization for Sensor Network Localization

TL;DR

This paper explores how increasing the problem's dimensionality can transform a non-convex sensor network localization problem into a convex one, explaining the success of semi-definite relaxation methods and their relation to other high-dimensional problems.

Contribution

It demonstrates that second-order dimension augmentation can convexify the SNL problem, providing theoretical insights into SDR's effectiveness and its connection to other high-dimensional optimization problems.

Findings

Second-order dimension augmentation convexifies the SNL loss function.

More edges do not necessarily improve convexity.

SDR+GD benefits from warm-starting with SDR solutions.

Abstract

This paper investigates the Sensor Network Localization (SNL) problem, which seeks to determine sensor locations based on known anchor locations and partially given anchors-sensors and sensors-sensors distances. Two primary methods for solving the SNL problem are analyzed: the low-dimensional method that directly minimizes a loss function, and the high-dimensional semi-definite relaxation (SDR) method that reformulates the SNL problem as an SDP (semi-definite programming) problem. The paper primarily focuses on the intrinsic non-convexity of the loss function of the low-dimensional method, which is shown in our main theorem. The SDR method, via second-order dimension augmentation, is discussed in the context of its ability to transform non-convex problems into convex ones; while the first-order direct dimension augmentation fails. Additionally, we will show that more edges don't necessarily contribute to the better convexity of the loss function. Moreover, we provide an explanation for the success of the SDR+GD (gradient descent) method which uses the SDR solution as a warm-start of the minimization of the loss function by gradient descent. The paper also explores the parallels among SNL, max-cut, and neural networks in terms of the blessing of high-order dimension augmentation.