Abstract

The population recovery problem is a basic problem in noisy unsupervised learning that has attracted significant research attention in recent years [WY12,DRWY12, MS13, BIMP13, LZ15,DST16]. A number of different variants of this problem have been studied, often under assumptions on the unknown distribution (such as that it has restricted support size). In this work we study the sample complexity and algorithmic complexity of the most general version of the problem, under both bit-flip noise and erasure noise model. We give essentially matching upper and lower sample complexity bounds for both noise models, and efficient algorithms matching these sample complexity bounds up to polynomial factors.

Highlights

  • For the sake of a compact representation, we assume the learner only lists the nonzero values of D; this means that a successful learner need only list O(1/ε) nonzero values

  • For the bit-flip noise population recovery problem, our main result is a lower bound on the sample complexity of estimation, as well as a full noisy population recovery (NPR) algorithm whose running time matches it up to polynomial factors

  • An earlier paper by Moitra and Saks [11] gave an algorithm with sample complexity and running time (n/ε)O(log(1/ν)/ν)

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Summary

The erasure noise and bit-flip noise population recovery problems

The noisy population recovery (NPR) problem is to learn an unknown probability distribution D on {0, 1}n, under ν-noise, to ∞-accuracy ε.1 In this problem the learner gets access to independent samples y,. The noisy population recovery (NPR) problem is to learn an unknown probability distribution D on {0, 1}n, under ν-noise, to ∞-accuracy ε.1. In this problem the learner gets access to independent samples y,. Each coordinate of x is retained with probability ν (as in erasure noise), and is otherwise replaced with a uniformly random bit. This is the model of noise associated with the so-called “Bonami noise operator” Tν (see [13] for the precise description and many applications of this operator)

Our results
Preliminaries
Well-known preliminary reductions
Reduction to an analytic problem
Circle bounds for erasure noise
Circle bounds for bit-flip noise
Reduction between the restricted and the general lower bound
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