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Embedding Approximately Low-Dimensional l_2^2 Metrics into l_1

Foundations of Software Technology and Theoretical Computer Science Pub Date : 2016-12-13 DOI:10.4230/LIPIcs.FSTTCS.2016.10
A. Deshpande, P. Harsha, Rakesh Venkat
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Abstract

Goemans showed that any n points x_1,..., x_n in d-dimensions satisfying l_2^2 triangle inequalities can be embedded into l_{1}, with worst-case distortion at most sqrt{d}. We consider an extension of this theorem to the case when the points are approximately low-dimensional as opposed to exactly low-dimensional, and prove the following analogous theorem, albeit with average distortion guarantees: There exists an l_{2}^{2}-to-l_{1} embedding with average distortion at most the stable rank, sr(M), of the matrix M consisting of columns {x_i-x_j}_{i
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在l_1中嵌入近似低维l_2^2度量
Goemans证明任意n个点x_1，…， d维中满足l_2^2三角形不等式的x_n可以嵌入到l_{1}中，最坏情况失真最多为sqrt{d}。我们考虑将该定理推广到点是近似低维而不是完全低维的情况，并证明了以下类似定理，尽管有平均畸变保证:存在一个l_{2}^{2}到l_{1}的平均畸变嵌入，在由列{x_i-x_j}_{i
本文章由计算机程序翻译，如有差异，请以英文原文为准。
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Foundations of Software Technology and Theoretical Computer Science
Foundations of Software Technology and Theoretical Computer Science
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