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• W3209102090 abstract "The <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$ktext {-means}$ </tex-math></inline-formula> clustering problem concerns finding a partition of the data points into <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$k$ </tex-math></inline-formula> clusters such that the total within-cluster squared distance is minimized. This optimization objective is non-convex, and not everywhere differentiable. In general, there exist spurious local solutions other than the global optimum. Moreover, the simplest and most popular algorithm for <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$ktext {-means}$ </tex-math></inline-formula> , namely Lloyd’s algorithm, generally converges to such spurious local solutions both in theory and in practice. In this paper, we investigate the <italic xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>structures</i> of these spurious local solutions under a probabilistic generative model with <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$k$ </tex-math></inline-formula> ground truth clusters. As soon as <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$k=3$ </tex-math></inline-formula> , spurious local minima provably exist, even for well-separated clusters. One such local minimum puts two centers at one true cluster, and the third center in the middle of the other two true clusters. We prove that this is essentially the <italic xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>only</i> type of spurious local minima under a separation condition. In particular, any local minimum solution only involves a configuration that puts multiple centers at a true cluster, and one center in the middle of multiple true clusters. Our results pertain to the <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$ktext {-means}$ </tex-math></inline-formula> formulation for mixtures of Gaussians or bounded distributions, and hold in the over- and under-parametrization regimes where the number of centers in <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$ktext {-means}$ </tex-math></inline-formula> may not equal to the number of true clusters. Our theoretical results corroborate existing empirical observations and provide justification for popular heuristics for <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$ktext {-means}$ </tex-math></inline-formula> clustering." @default.
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• W3209102090 date "2022-01-01" @default.
• W3209102090 modified "2023-10-17" @default.
• W3209102090 title "Structures of Spurious Local Minima in <i>k</i>-Means" @default.
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• W3209102090 doi "https://doi.org/10.1109/tit.2021.3122465" @default.
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