期刊
PATTERN RECOGNITION LETTERS
卷 125, 期 -, 页码 349-356出版社
ELSEVIER
DOI: 10.1016/j.patrec.2019.04.001
关键词
k-means; Gaussian mixture models; Expectation maximization; Variational methods; Free energy
资金
- DFG [SFB 1330, 352015383, EXC 2177/1, 390895286]
We show that k-means (Lloyd's algorithm) is obtained as a special case when truncated variational EM approximations are applied to Gaussian mixture models (GMM) with isotropic Gaussians. In contrast to the standard way to relate k-means and GMMs, the provided derivation shows that it is not required to consider Gaussians with small variances or the limit case of zero variances. There are a number of consequences that directly follow from our approach: (A) k-means can be shown to increase a free energy (a.k.a. ELBO) associated with truncated distributions and this free energy can directly be reformulated in terms of the k-means objective; (B) k-means generalizations can directly be derived by considering the 2nd closest, 3rd closest etc. cluster in addition to just the closest one; and (C) the embedding of k-means into a free energy framework allows for theoretical interpretations of other k-means generalizations in the literature. In general, truncated variational EM provides a natural and rigorous quantitative link between k-means-like clustering and GMM clustering algorithms which may be very relevant for future theoretical and empirical studies. (C) 2019 Elsevier B.V. All rights reserved.
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