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Learning Arithmetic Formulas in the Presence of Noise: A General Framework and Applications to Unsupervised Learning

Authors Pritam Chandra, Ankit Garg, Neeraj Kayal, Kunal Mittal, Tanmay Sinha

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Pritam Chandra
  • Microsoft Research, Bangalore, India
Ankit Garg
  • Microsoft Research, Bangalore, India
Neeraj Kayal
  • Microsoft Research, Bangalore, India
Kunal Mittal
  • Princeton University, NJ, USA
Tanmay Sinha
  • Microsoft Research, Bangalore, India

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Pritam Chandra, Ankit Garg, Neeraj Kayal, Kunal Mittal, and Tanmay Sinha. Learning Arithmetic Formulas in the Presence of Noise: A General Framework and Applications to Unsupervised Learning. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 25:1-25:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ITCS.2024.25

Abstract

We present a general framework for designing efficient algorithms for unsupervised learning problems, such as mixtures of Gaussians and subspace clustering. Our framework is based on a meta algorithm that learns arithmetic formulas in the presence of noise, using lower bounds. This builds upon the recent work of Garg, Kayal and Saha (FOCS '20), who designed such a framework for learning arithmetic formulas without any noise. A key ingredient of our meta algorithm is an efficient algorithm for a novel problem called Robust Vector Space Decomposition. We show that our meta algorithm works well when certain matrices have sufficiently large smallest non-zero singular values. We conjecture that this condition holds for smoothed instances of our problems, and thus our framework would yield efficient algorithms for these problems in the smoothed setting.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Arithmetic Circuits
  • Robust Vector Space Decomposition
  • Subspace Clustering
  • Mixtures of Gaussians

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