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Near-Optimal-Sample Estimators for Spherical Gaussian Mixtures
Ananda Theertha Suresh · Alon Orlitsky · Jayadev Acharya · Ashkan Jafarpour

Mon Dec 08 04:00 PM -- 08:59 PM (PST) @ Level 2, room 210D #None
Many important distributions are high dimensional, and often they can be modeled as Gaussian mixtures. We derive the first sample-efficient polynomial-time estimator for high-dimensional spherical Gaussian mixtures. Based on intuitive spectral reasoning, it approximates mixtures of $k$ spherical Gaussians in $d$-dimensions to within$\ell_1$ distance $\epsilon$ using $\mathcal{O}({dk^9(\log^2 d)}/{\epsilon^4})$ samples and $\mathcal{O}_{k,\epsilon}(d^3\log^5 d)$ computation time. Conversely, we show that any estimator requires $\Omega\bigl({dk}/{\epsilon^2}\bigr)$ samples, hence the algorithm's sample complexity is nearly optimal in the dimension. The implied time-complexity factor \mathcal{O}_{k,\epsilon}$ is exponential in $k$, but much smaller than previously known. We also construct a simple estimator for one-dimensional Gaussian mixtures that uses $\tilde\mathcal{O}(k /\epsilon^2)$ samples and $\tilde\mathcal{O}((k/\epsilon)^{3k+1})$ computation time.

Author Information

Ananda Theertha Suresh (University of California, San Diego)
Alon Orlitsky (University of California, San Diego)
Jayadev Acharya (Cornell University)
Ashkan Jafarpour (University of California, San Diego)

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