A Fast Algorithm for Rank-(L, M, N) Block Term Decomposition of Multi-Dimensional Data

Attribute to its powerful representation ability, block term decomposition (BTD) has recently attracted many views of multi-dimensional data processing, e.g., hyperspectral image unmixing and blind source separation. However, the popular alternating least squares algorithm for rank-(L, M, N) BTD (BTD-ALS) suffers expensive time and space costs from Kronecker products and solving low-rank approximation subproblems, hindering the deployment of BTD for real applications, especially for large-scale data. In this paper, we propose a fast sketching-based Kronecker product-free algorithm for rank-(L, M, N) BTD (termed as KPF-BTD), which is suitable for real-world multi-dimensional data. Specifically, we first decompose the original optimization problem into several rank-(L, M, N) approximation subproblems, and then we design the bilateral sketching to obtain the approximate solutions of these subproblems instead of the exact solutions, which allows us to avoid Kronecker products and rapidly solve rank-(L, M, N) approximation subproblems. As compared with BTD-ALS, the time and space complexities \(\mathcal {O}{(2(p+1)(I^3LR+I^2L^2R+IL^3R)+I^3LR)}\) $\mathcal{O}(2(p+1)(I^{3}LR+I^2{L}^{2}R+I{L}^{3}R)+I^{3}LR)$ and \(\mathcal {O}{(I^3)}\) $\mathcal{O}(I^3)$ of KPF-BTD are significantly cheaper than \(\mathcal {O}{(I^3L^6R^2+I^3L^3R+I^3LR+I^2L^3R^2+I^2L^2R)}\) $\mathcal{O}(I^3{L}^6{R}^2+I^3{L}^{3}R+I^{3}LR+I^2{L}^3{R}^2+I^2{L}^{2}R)$ and \(\mathcal {O}{(I^3L^3R)}\) $\mathcal{O}(I^3{L}^{3}R)$ of BTD-ALS, where \(p \ll I\) $p\ll I$ . Moreover, we establish the theoretical error bound for KPF-BTD. Extensive synthetic and real experiments show KPF-BTD achieves substantial speedup and memory saving while maintaining accuracy (e.g., for a \(150\times 150\times 150\) $150\times 150\times 150$ synthetic tensor, the running time 0.2 seconds per iteration of KPF-BTD is significantly faster than 96.2 seconds per iteration of BTD-ALS while their accuracies are comparable).