Multiscale Analysis for Higher-order Tensors
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The widespread use of multisensor technology and the emergence of big data sets have created the necessity to develop more versatile tools to represent large and multimodal data such as higher-order tensors. Tensor decomposition based methods have been shown to be flexible in the choice of the constraints and to extract more general latent components in such data compared to matrix-based methods. For these reasons, tensor decompositions have found applications in many different signal processing problems including dimensionality reduction, signal separation, linear regression, feature extraction, and classification. However, most of the existing tensor decomposition methods are founded on the principle of finding a low-rank approximation in a linear subspace structure, where the definition of the rank may change depending on the particular decomposition. Since most data are not necessarily low-rank in a linear subspace, this often results in high approximation errors or low compression rates. In this paper, we introduce a new adaptive, multi-scale tensor decomposition method for higher order data inspired by hybrid linear modeling and subspace clustering techniques. In particular, we develop a multi-scale higher-order singular value decomposition (MS-HoSVD) approach where a given tensor is first permuted and then partitioned into several sub-tensors each of which can be represented as a low-rank tensor increasing the efficiency of the representation. The proposed approach is evaluated for two different signal processing applications: dimensionality reduction and classification.
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