Multilinear Subspace Clustering.
Kernfeld, Eric M.
2014
- Abstract: This thesis is on a topic from the mathematics of machine learning and statistical pattern recognition. In sections 1 to 4, we consider extensions of the subspace clustering problem, which has lately inspired algorithmic developments applicable to computer vision, genomics, music analysis, and electroencephalography. This problem is simple to state: given multivariate data grouped along ... read moremany low-dimensional subspaces, find the subspaces and the points’ memberships. Although some applications of subspace clustering algorithms involve data with matrix structure, recent algorithms disregard that, vectorizing data as part of the processing. We modify the subspace clustering problem using a construct from multilinear algebra, the tensor product of subspaces. This allows us to derive new algorithms that do not alter the original form of the data. We will argue that the new algorithms can carry out analysis more quickly and accurately than we could through subspace clustering. We mention applications where it may be possible to exploit this performance improvement. Our algorithms convert the multilinear subspace clustering problem into a graph clustering problem, where every point in a subspace is represented by a node on the graph and points are (ideally) connected if they belong to the same subspace. The algorithm we present to solve the modified subspace clustering problem is a randomized algorithm, and multiple trials are needed. Since every trial produces a new realization of the graph, we use sections 4 through 8 to consider various ways of combining graph realizations. We conduct tests on simulated data in order to evaluate them.read less
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