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We give a general unified method that can be used for $L_1$ {\em closenesstesting} of a wide range of univariate structured distribution families. Morespecifically, we design a sample optimal and computationally efficientalgorithm for testing the equivalence of two unknown (potentially arbitrary)univariate distributions under the $\mathcal{A}_k$-distance metric: Givensample access to distributions with density functions $p, q: I \to \mathbb{R}$,we want to distinguish between the cases that $p=q$ and$\|p-q\|_{\mathcal{A}_k} \ge \epsilon$ with probability at least $2/3$. We showthat for any $k \ge 2, \epsilon>0$, the {\em optimal} sample complexity of the$\mathcal{A}_k$-closeness testing problem is $\Theta(\max\{k^{4/5}/\epsilon^{6/5}, k^{1/2}/\epsilon^2 \})$. This is the first $o(k)$sample algorithm for this problem, and yields new, simple $L_1$ closenesstesters, in most cases with optimal sample complexity, for broad classes ofstructured distributions.

Optimal Algorithms and Lower Bounds for Testing Closeness of Structured Distributions