Testing $k$-Modal Distributions: Optimal Algorithms via Reductions
  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 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:   1. Determine whether $p = q$ (for an explicitly given k-modal distribution $q$) versus $p$ is $\eps$-far from $q$;   2. 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:   3. Approximate $d_{TV}(p,q)$ to within additive $\eps$ where $q$ is an explicitly given k-modal distribution $q$;   4. Approximate $d_{TV}(p,q)$ to within additive $\eps$ where $q$ is available via sample access.   For each of these four problems we give sub-logarithmic sample algorithms, that we show are tight up to additive $\poly(k)$ and multiplicative $\polylog\log n+\polylog k$ factors. Thus our bounds 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 $\ell = O(k \log n).$ 
Constantinos Daskalakis, Ilias Diakonikolas, Rocco A. Servedio,   Gregory Valiant, Paul Valiant
cs.DS math.PR math.ST stat.TH