Optimization-based algorithms for the rank-$(L_t,L_t,1)$ Block Term Decomposition and related decompositions

The canonical polyadic and rank-(Lt,Lt,1) block term decomposition are two closely related tensor decompositions. We present a decomposition that generalizes both of the former and examine which algorithms are suited for its computation. Accordingly, these algorithms are also directly applicable to the CPD and rank-(Lt,Lt,1) BTD. We compare the popular alternating least squares scheme with several general unconstrained optimization techniques, as well as inexact nonlinear least squares methods. In the latter, the structure of the Jacobian's Gramian is exploited by means of efficient expressions for its matrix-vector product. Combined with an effective preconditioner -- which is in fact equivalent to an ALS step -- these methods prove to be a promising alternative for ALS, often converging to a more accurate result using significantly less floating point operations, especially for difficult problems. This is joint work with Lieven De Lathauwer (K.U.Leuven Kortrijk, Belgium) and Marc Van Barel (K.U.Leuven, Belgium).status: publishe