Fast and Structured Block-Term Tensor Decomposition for Hyperspectral Unmixing
Meng Ding, Xiao Fu, Xi-Le Zhao
Abstract
The block-term tensor decomposition model with multilinear rank- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(L_{r},L_{r},1)$</tex-math></inline-formula> terms (or the “ <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\mathsf{LL1}}$</tex-math></inline-formula> tensor decomposition” in short) offers a valuable alternative formulation for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">hyperspectral unmixing</i> (HU), which ensures the identifiability of the endmembers/abundances in cases where classic matrix factorization (MF) approaches cannot provide such guarantees. However, the existing <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\mathsf{LL1}}$</tex-math></inline-formula> -tensor-decomposition-based HU algorithms use a three-factor parameterization of the tensor (i.e., the hyperspectral image cube), which causes difficulties in incorporating structural prior information arising in HU. Consequently, their algorithms often exhibit high per-iteration complexity and slow convergence. This article focuses on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\mathsf{LL1}}$</tex-math></inline-formula> tensor decomposition under structural constraints and regularization terms in HU. Our algorithm uses a two-factor reparameterization of the tensor model. Like in the MF-based approaches, the factors correspond to the endmembers and abundances in the context of HU. Thus, the proposed framework is natural to incorporate physics-motivated priors in HU. To tackle the formulated optimization problem, a two-block alternating <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">gradient projection</i> (GP)-based algorithm is proposed. Carefully designed projection solvers are proposed to implement the GP algorithm with a relatively low per-iteration complexity. An extrapolation-based acceleration strategy is proposed to expedite the GP algorithm. Such an extrapolated multiblock algorithm only had asymptotic convergence assurances in the literature. Our analysis shows that the algorithm converges to the vicinity of a stationary point within finite iterations, under reasonable conditions. Empirical study shows that the proposed algorithm often attains orders-of-magnitude speedup and substantial HU performance gains compared with the existing <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\mathsf{LL1}}$</tex-math></inline-formula> -decomposition-based HU algorithms.