Testing k-Modal Distributions: Optimal Algorithms via Reductions

• Main Authors: Diakonikolas, Ilias (Author), Servedio, Rocco A. (Author), Valiant, Gregory (Author), Valiant, Paul (Author), Daskalakis, Konstantinos (Contributor)
• Other Authors: Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science (Contributor)
• Format: Article
• Language: English
• Published: Society for Industrial and Applied Mathematics, 2015-11-20T17:19:02Z.
• Subjects: Article
• Online Access: Get fulltext

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).
National Science Foundation (U.S.) (CAREER Award CCF-0953960)
Alfred P. Sloan Foundation (Fellowship)