Kernel $$K$$-Means Clustering of Distributional Data
Abstract
<jats:p> We consider the problem of clustering a sample of probability distributions from a random distribution on <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$\mathbb R^d$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>d</mml:mi> </mml:msup> </mml:math> </jats:alternatives> </jats:inline-formula> . Our proposed partitioning method makes use of a symmetric, positive-definite kernel <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$k$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> and its associated reproducing kernel Hilbert space <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$\mathscr {H}$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>H</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> . By mapping each distribution to its corresponding kernel mean embedding in <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$\mathscr {H}$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>H</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> , we obtain a sample in this space where we carry out the <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$K$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> -means clustering procedure, which provides an unsupervised classification of the original sample. The procedure is simple and computationally feasible even for dimension <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$d>1$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> . The simulation studies provide insight into the choice of the kernel and its tuning parameter. The performance of the proposed clustering procedure is illustrated on a collection of synthetic aperture radar images and on meteorological data. </jats:p>