Near-Optimal Bounds for Testing Histogram Distributions
This work addresses a fundamental problem in distribution testing for data approximation, with incremental improvements in sample complexity bounds.
The paper tackled the problem of testing whether a discrete probability distribution is a histogram with a specified number of bins, providing a near-optimal sample complexity bound of ϕ(√(nk)/ε + k/ε² + √n/ε²) and a computationally efficient algorithm.
We investigate the problem of testing whether a discrete probability distribution over an ordered domain is a histogram on a specified number of bins. One of the most common tools for the succinct approximation of data, $k$-histograms over $[n]$, are probability distributions that are piecewise constant over a set of $k$ intervals. The histogram testing problem is the following: Given samples from an unknown distribution $\mathbf{p}$ on $[n]$, we want to distinguish between the cases that $\mathbf{p}$ is a $k$-histogram versus $\varepsilon$-far from any $k$-histogram, in total variation distance. Our main result is a sample near-optimal and computationally efficient algorithm for this testing problem, and a nearly-matching (within logarithmic factors) sample complexity lower bound. Specifically, we show that the histogram testing problem has sample complexity $\widetilde Θ(\sqrt{nk} / \varepsilon + k / \varepsilon^2 + \sqrt{n} / \varepsilon^2)$.