Efficient Regression in Metric Spaces via Approximate Lipschitz Extension

Lee Ad Gottlieb, Aryeh Kontorovich, Robert Krauthgamer

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


We present a framework for performing efficient regression in general metric spaces. Roughly speaking, our regressor predicts the value at a new point by computing an approximate Lipschitz extension - the smoothest function consistent with the observed data - after performing structural risk minimization to avoid overfitting. We obtain finite-sample risk bounds with minimal structural and noise assumptions, and a natural runtime-precision tradeoff. The offline (learning) and online (prediction) stages can be solved by convex programming, but this naive approach has runtime complexity $O(n^{3})$, which is prohibitive for large data sets. We design instead a regression algorithm whose speed and generalization performance depend on the intrinsic dimension of the data, to which the algorithm adapts. While our main innovation is algorithmic, the statistical results may also be of independent interest.

Original languageEnglish
Article number7944658
Pages (from-to)4838-4849
Number of pages12
JournalIEEE Transactions on Information Theory
Issue number8
StatePublished - 1 Aug 2017


  • Regression analysis


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