Testing k-Modal Distributions: Optimal Algorithms via Reductions.
C. Daskalakis and I. Diakonikolas and R. Servedio and G. Valiant and P. Valiant.
In ACM-SIAM Symposium on Discrete Algorithms (SODA), 2013, pp. 1833--1852.

Abstract:

We give highly efficient algorithms, and almost matching lower bounds, for a range of basic statistical problems that involve testing and estimating the $L_1$ (total variation) distance between two $k$-modal distributions $p$ and $q$ over the discrete domain $\{1,\dots,n\}$. 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:

• Determine whether $p = q$ (for an explicitly given $k$-modal distribution $q$) versus $p$ is $\eps$-far from $q$;
• Determine whether $p=q$ (where $q$ is available via sample access) versus $p$ is $\eps$-far from $q$;
• Estimating $L_1$ distance (tolerant testing'') against a known or unknown distribution:

• Approximate $d_{TV}(p,q)$ to within additive $\epsilon$ where $q$ is an explicitly given $k$-modal distribution $q$;
• Approximate $d_{TV}(p,q)$ to within additive $\epsilon$ 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 $\polylog\log n+ \polylog k$ factors. Our algorithms significantly improve the previous results of \cite{BKR:04}, 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 $O((\log n)^3)$ 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 $\{1,\dots,n\}$ to the corresponding distribution testing problems for unrestricted distributions over a much smaller domain $\{1,\dots,\ell\}$ where $\ell = O(k \log n).$