In this paper we present Collaborative Low-Rank Subspace Clustering. Given multiple observations of a phenomenon we learn a unified representation matrix. This unified matrix incorporates the features from all the observations, thus increasing the discriminative power compared with learning the representation matrix on each observation separately. Experimental evaluation shows that our method outperforms subspace clustering on separate observations and the state of the art collaborative learning algorithm.
The work most closely related and which inspires our work is the so called Multi-Task Low-rank Affinity Pursuit (MLAP) @cite_41 . MLAP adopts the low-rank representation model for each individual observation and fuses them together by seeking sparsity consistent low-rank coefficients. The MLAP objective is the following where @math , @math and @math are observation data, the fitting error and coefficient matrices for the @math th observation respectively, @math is the number of observations, the @math norm is defined as @math , @math and @math is the vectorisation operator such that @math .
@cite_41
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Recovering a low-rank tensor from incomplete information is a recurring problem in signal processing and machine learning. The most popular convex relaxation of this problem minimizes the sum of the nuclear norms of the unfoldings of the tensor. We show that this approach can be substantially suboptimal: reliably recovering a @math -way tensor of length @math and Tucker rank @math from Gaussian measurements requires @math observations. In contrast, a certain (intractable) nonconvex formulation needs only @math observations. We introduce a very simple, new convex relaxation, which partially bridges this gap. Our new formulation succeeds with @math observations. While these results pertain to Gaussian measurements, simulations strongly suggest that the new norm also outperforms the sum of nuclear norms for tensor completion from a random subset of entries. Our lower bound for the sum-of-nuclear-norms model follows from a new result on recovering signals with multiple sparse structures (e.g. sparse, low rank), which perhaps surprisingly demonstrates the significant suboptimality of the commonly used recovery approach via minimizing the sum of individual sparsity inducing norms (e.g. @math , nuclear norm). Our new formulation for low-rank tensor recovery however opens the possibility in reducing the sample complexity by exploiting several structures jointly.