تقرير
Clustering Mixtures of Bounded Covariance Distributions Under Optimal Separation
| العنوان: | Clustering Mixtures of Bounded Covariance Distributions Under Optimal Separation |
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| المؤلفون: | Diakonikolas, Ilias, Kane, Daniel M., Lee, Jasper C. H., Pittas, Thanasis |
| سنة النشر: | 2023 |
| المجموعة: | Computer Science Mathematics Statistics |
| مصطلحات موضوعية: | Computer Science - Machine Learning, Computer Science - Data Structures and Algorithms, Computer Science - Information Theory, Mathematics - Statistics Theory, Statistics - Machine Learning |
| الوصف: | We study the clustering problem for mixtures of bounded covariance distributions, under a fine-grained separation assumption. Specifically, given samples from a $k$-component mixture distribution $D = \sum_{i =1}^k w_i P_i$, where each $w_i \ge \alpha$ for some known parameter $\alpha$, and each $P_i$ has unknown covariance $\Sigma_i \preceq \sigma^2_i \cdot I_d$ for some unknown $\sigma_i$, the goal is to cluster the samples assuming a pairwise mean separation in the order of $(\sigma_i+\sigma_j)/\sqrt{\alpha}$ between every pair of components $P_i$ and $P_j$. Our contributions are as follows: For the special case of nearly uniform mixtures, we give the first poly-time algorithm for this clustering task. Prior work either required separation scaling with the maximum cluster standard deviation (i.e. $\max_i \sigma_i$) [DKK+22b] or required both additional structural assumptions and mean separation scaling as a large degree polynomial in $1/\alpha$ [BKK22]. For general-weight mixtures, we point out that accurate clustering is information-theoretically impossible under our fine-grained mean separation assumptions. We introduce the notion of a clustering refinement -- a list of not-too-small subsets satisfying a similar separation, and which can be merged into a clustering approximating the ground truth -- and show that it is possible to efficiently compute an accurate clustering refinement of the samples. Furthermore, under a variant of the "no large sub-cluster'' condition from in prior work [BKK22], we show that our algorithm outputs an accurate clustering, not just a refinement, even for general-weight mixtures. As a corollary, we obtain efficient clustering algorithms for mixtures of well-conditioned high-dimensional log-concave distributions. Moreover, our algorithm is robust to $\Omega(\alpha)$-fraction of adversarial outliers. |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2312.11769 |
| رقم الأكسشن: | edsarx.2312.11769 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |