Algorithms for Canonical Polyadic Decomposition With Block-Circulant Factors

Higher order tensors and their decompositions are well-known tools in signal processing. More specifically, tensors admitting decompositions with structured factor matrices arise in various applications, such as telecommunications or convolutive independent component analysis. These applications motivate the development of efficient algorithms for structured tensor decompositions. In this letter, we develop a method for canonical polyadic decompositions (CPDs) with block-circulant factor matrices by extending a method for circulant factors. Block-circulant factor matrices arise in the identification of homogeneous Wiener–Hammerstein models. By transforming the data tensor to the frequency domain, the available structure is exploited using simple element-wise divisions. This approach reformulates the structured CPD as an unstructured CPD of lower rank. The resulting CPD of a possibly incomplete tensor can then be solved algebraically or using optimization-based methods.