Regularized Tensor Factorizations and Higher-Order Principal Components Analysis
This work addresses the challenge of handling high-dimensional tensors with irrelevant features in fields like biomedical imaging and chemometrics, offering an incremental improvement over existing tensor decomposition methods.
The paper tackles the problem of high-dimensional tensor data analysis by introducing sparse and regularized tensor factorization frameworks, which improve dimension reduction, feature selection, and signal recovery, as demonstrated on simulated data, microarrays, and functional MRIs.
High-dimensional tensors or multi-way data are becoming prevalent in areas such as biomedical imaging, chemometrics, networking and bibliometrics. Traditional approaches to finding lower dimensional representations of tensor data include flattening the data and applying matrix factorizations such as principal components analysis (PCA) or employing tensor decompositions such as the CANDECOMP / PARAFAC (CP) and Tucker decompositions. The former can lose important structure in the data, while the latter Higher-Order PCA (HOPCA) methods can be problematic in high-dimensions with many irrelevant features. We introduce frameworks for sparse tensor factorizations or Sparse HOPCA based on heuristic algorithmic approaches and by solving penalized optimization problems related to the CP decomposition. Extensions of these approaches lead to methods for general regularized tensor factorizations, multi-way Functional HOPCA and generalizations of HOPCA for structured data. We illustrate the utility of our methods for dimension reduction, feature selection, and signal recovery on simulated data and multi-dimensional microarrays and functional MRIs.