Dimensionality-Dependent Generalization Bounds for $k$-Dimensional Coding Schemes

So what we call matrix factorization is really a coding scheme: Dimensionality-Dependent Generalization Bounds for $k$-Dimensional Coding Schemes by  Tongliang Liu, Dacheng Tao, Dong Xu

The $k$-dimensional coding schemes refer to a collection of methods that attempt to represent data using a set of representative $k$-dimensional vectors, and include non-negative matrix factorization, dictionary learning, sparse coding, $k$-means clustering and vector quantization as special cases. Previous generalization bounds for the reconstruction error of the $k$-dimensional coding schemes are mainly dimensionality independent. A major advantage of these bounds is that they can be used to analyze the generalization error when data is mapped into an infinite- or high-dimensional feature space. However, many applications use finite-dimensional data features. Can we obtain dimensionality-dependent generalization bounds for $k$-dimensional coding schemes that are tighter than dimensionality-independent bounds when data is in a finite-dimensional feature space? The answer is positive. In this paper, we address this problem and derive a dimensionality-dependent generalization bound for $k$-dimensional coding schemes by bounding the covering number of the loss function class induced by the reconstruction error. The bound is of order $\mathcal{O}\left(\left(mk\ln(mkn)/n\right)^{\lambda_n}\right)$, where $m$ is the dimension of features, $k$ is the number of the columns in the linear implementation of coding schemes, $n$ is the size of sample, $\lambda_n>0.5$ when $n$ is finite and $\lambda_n=0.5$ when $n$ is infinite. We show that our bound can be tighter than previous results, because it avoids inducing the worst-case upper bound on $k$ of the loss function and converges faster. The proposed generalization bound is also applied to some specific coding schemes to demonstrate that the dimensionality-dependent bound is an indispensable complement to these dimensionality-independent generalization bounds.

 
Image Credit: NASA/JPL-Caltech/Space Science Institute
 N00254238.jpg was taken on January 14, 2016 and received on Earth January 15, 2016. The camera was pointing toward ENCELADUS, and the image was taken using the CL1 and CL2 filters.

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