## Testing k-Modal Distributions: Optimal Algorithms via Reductions

##### Author(s)

Diakonikolas, Ilias; Servedio, Rocco A.; Valiant, Gregory; Valiant, Paul; Daskalakis, Konstantinos
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Show full item record##### Abstract

We give highly efficient algorithms, and almost matching lower bounds, for a range of basic statistical problems that involve testing and estimating the L[subscript 1] (total variation) distance between two k-modal distributions p and q over the discrete domain {1, …, n}. More precisely, we consider the following four problems: given sample access to an unknown k-modal distribution p,
Testing identity to a known or unknown distribution:
1. Determine whether p = q (for an explicitly given k-modal distribution q) versus p is e-far from q;
2. Determine whether p = q (where q is available via sample access) versus p is ε-far from q;
Estimating L[subscript 1] distance (“tolerant testing”) against a known or unknown distribution:
3. Approximate d[subscript TV](p, q) to within additive ε where q is an explicitly given k-modal distribution q;
4. Approximate d[subscript TV] (p, q) to within additive ε where q is available via sample access.
For each of these four problems we give sub-logarithmic sample algorithms, and show that our algorithms have optimal sample complexity up to additive poly (k) and multiplicative polylog log n + polylogk factors. Our algorithms significantly improve the previous results of [BKR04], which were for testing identity of distributions (items (1) and (2) above) in the special cases k = 0 (monotone distributions) and k = 1 (unimodal distributions) and required O((log n)[superscript 3]) samples.
As our main conceptual contribution, we introduce a new reduction-based approach for distribution-testing problems that lets us obtain all the above results in a unified way. Roughly speaking, this approach enables us to transform various distribution testing problems for k-modal distributions over {1, …, n} to the corresponding distribution testing problems for unrestricted distributions over a much smaller domain {1, …, ℓ} where ℓ = O(k log n).

##### Date issued

2013##### Department

Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science##### Journal

Proceedings of the Twenty-fourth Annual ACM-SIAM Symposium on Discrete Algorithms

##### Publisher

Society for Industrial and Applied Mathematics

##### Citation

Daskalakis, Constantinos, Ilias Diakonikolas, Rocco A. Servedio, Gregory Valiant, and Paul Valiant. “Testing k -Modal Distributions: Optimal Algorithms via Reductions.” Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms (January 6, 2013): 1833–1852.

Version: Original manuscript

##### ISBN

978-1-61197-251-1

978-1-61197-310-5