📝 SummarXiv

Abstract

The landmark multi-dimensional scaling (LMDS) is a leading method that embeds new points to an existing coordinate system based on observed distance information. It has long been known as a variant of Nystr\"{o}m algorithm. It was recently revealed that LMDS is Gower's method proposed in 1960s. However, the relationship with other range-based embedding methods including the least-squares (LS) has been unexplored, proposing the question of which method to use in practice. This paper provides a fresh look at those methods and explicitly differentiates them in terms of the objectives they try to minimize. For the first time for the case of single source localization, we show that both LMDS and LS are generated from a same family of objectives, which balance between length and angle preservation among the embedding points. Despite being nonconvex, the new objectives can be globally optimized through a trust-region method. An important result is that the LS solution can be thought as a regularized solution of LMDS. Extension to the case of multiple source localization is also explored. Comprehensive numerical results demonstrate the quality of the proposed objectives and the efficiency of the trust-region method.