期刊
APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS
卷 51, 期 -, 页码 333-373出版社
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.acha.2020.11.006
关键词
Multidimensional scaling; Euclidean embedding; Dimensionality reduction; Singular value thresholding; Optimal shrinkage; MDS
资金
- ICRC Blavatnik Interdisciplinary Research Center (Tel Aviv University)
- Federmann Research Center (Hebrew University)
- Israel Science Foundation [1523/16]
- German-Israeli Foundation for Scientific Research and Development(GIF) Program [I-1100-407.1-2015]
This paper investigates the performance of MDS in high dimensions and measurement noise, introducing MDS+ as an improved variant that offers better embedding quality and calculates the optimal embedding dimension. MDS+ is proven to be the unique, asymptotically optimal shrinkage function compared to traditional MDS.
Multidimensional Scaling (MDS) is a classical technique for embedding data in low dimensions, still in widespread use today. In this paper we study MDS in a modern setting specifically, high dimensions and ambient measurement noise. We show that as the ambient noise level increases, MDS suffers a sharp breakdown that depends on the data dimension and noise level, and derive an explicit formula for this breakdown point in the case of white noise. We then introduce MDS+, a simple variant of MDS, which applies a shrinkage nonlinearity to the eigenvalues of the MDS similarity matrix. Under a natural loss function measuring the embedding quality, we prove that MDS+ is the unique, asymptotically optimal shrinkage function. MDS+ offers improved embedding, sometimes significantly so, compared with MDS. Importantly, MDS+ calculates the optimal embedding dimension, into which the data should be embedded. (c) 2020 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
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