Faster Algorithms for Testing under Conditional Sampling

There has been considerable recent interest in distribution-tests whose run-time and sample requirements are sublinear in the domain-size $k$. We study two of the most important tests under the conditional-sampling model where each query specifies a subset $S$ of the domain, and the response is a sample drawn from $S$ according to the underlying distribution. For identity testing, which asks whether the underlying distribution equals a specific given distribution or $\epsilon$-differs from it, we reduce the known time and sample complexities from $\tilde{\mathcal{O}}(\epsilon^{-4})$ to $\tilde{\mathcal{O}}(\epsilon^{-2})$, thereby matching the information theoretic lower bound. For closeness testing, which asks whether two distributions underlying observed data sets are equal or different, we reduce existing complexity from $\tilde{\mathcal{O}}(\epsilon^{-4} \log^5 k)$ to an even sub-logarithmic $\tilde{\mathcal{O}}(\epsilon^{-5} \log \log k)$ thus providing a better bound to an open problem in Bertinoro Workshop on Sublinear Algorithms [Fisher, 2004].

Universal Compression of Envelope Classes: Tight Characterization via Poisson Sampling

The Poisson-sampling technique eliminates dependencies among symbol appearances in a random sequence. It has been used to simplify the analysis and strengthen the performance guarantees of randomized algorithms. Applying this method to universal compression, we relate the redundancies of fixed-length and Poisson-sampled sequences, use the relation to derive a simple single-letter formula that approximates the redundancy of any envelope class to within an additive logarithmic term. As a first application, we consider i.i.d. distributions over a small alphabet as a step-envelope class, and provide a short proof that determines the redundancy of discrete distributions over a small al- phabet up to the first order terms. We then show the strength of our method by applying the formula to tighten the existing bounds on the redundancy of exponential and power-law classes, in particular answering a question posed by Boucheron, Garivier and Gassiat.

Near-optimal-sample estimators for spherical Gaussian mixtures

Statistical and machine-learning algorithms are frequently applied to high-dimensional data. In many of these applications data is scarce, and often much more costly than computation time. We provide the first sample-efficient polynomial-time estimator for high-dimensional spherical Gaussian mixtures. For mixtures of any $k$ $d$-dimensional spherical Gaussians, we derive an intuitive spectral-estimator that uses $\mathcal{O}_k\bigl(\frac{d\log^2d}{\epsilon^4}\bigr)$ samples and runs in time $\mathcal{O}_{k,\epsilon}(d^3\log^5 d)$, both significantly lower than previously known. The constant factor $\mathcal{O}_k$ is polynomial for sample complexity and is exponential for the time complexity, again much smaller than what was previously known. We also show that $\Omega_k\bigl(\frac{d}{\epsilon^2}\bigr)$ samples are needed for any algorithm. Hence the sample complexity is near-optimal in the number of dimensions. We also derive a simple estimator for one-dimensional mixtures that uses $\mathcal{O}\bigl(\frac{k \log \frac{k}{\epsilon} }{\epsilon^2} \bigr)$ samples and runs in time $\widetilde{\mathcal{O}}\left(\bigl(\frac{k}{\epsilon}\bigr)^{3k+1}\right)$. Our other technical contributions include a faster algorithm for choosing a density estimate from a set of distributions, that minimizes the $\ell_1$ distance to an unknown underlying distribution.