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This paper investigates the Sensor Network Localization (SNL) problem, whichseeks to determine sensor locations based on known anchor locations andpartially given anchors-sensors and sensors-sensors distances. Two primarymethods for solving the SNL problem are analyzed: the low-dimensional methodthat directly minimizes a loss function, and the high-dimensional semi-definiterelaxation (SDR) method that reformulates the SNL problem as an SDP(semi-definite programming) problem. The paper primarily focuses on theintrinsic non-convexity of the loss function of the low-dimensional method,which is shown in our main theorem. The SDR method, via second-order dimensionaugmentation, is discussed in the context of its ability to transformnon-convex problems into convex ones; while the first-order direct dimensionaugmentation fails. Additionally, we will show that more edges don'tnecessarily 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 theloss function by gradient descent. The paper also explores the parallels amongSNL, max-cut, and neural networks in terms of the blessing of high-orderdimension augmentation.

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