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Near-Optimal Density Estimation in Near-Linear Time Using Variable-Width Histograms
The "approximation factor" $C$ in our result is inherent in the problem, as we prove that no algorithm with sample size bounded in terms of $k$ and $\epsilon$ can achieve $C<2$ regardless of what kind of hypothesis distribution it uses.
Optimal Algorithms for Testing Closeness of Discrete Distributions
We study the question of closeness testing for two discrete distributions.
Efficient Density Estimation via Piecewise Polynomial Approximation
We give an algorithm that draws $\tilde{O}(t\new{(d+1)}/\eps^2)$ samples from $p$, runs in time $\poly(t, d, 1/\eps)$, and with high probability outputs a piecewise polynomial hypothesis distribution $h$ that is $(O(\tau)+\eps)$-close (in total variation distance) to $p$.
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