Speaker
Description
Block term decomposition (BTD) unifies the two most common tensor decompositions: canonical polyadic and Tucker. While BTDs have found a broad range of applications from machine learning to blind source separation, all known algorithms for computing BTDs were historically optimisation-based, and required the desired block sizes to be specified as input. Recently, algebraic BTD algorithms have been found. Cai & Li (2021) were the first to find such an algorithm, with Domanov et al. (2024) later publishing a second, very similar algorithm. While these algorithms work well, very little was known about their theoretical underpinnings.
In this talk, we explain the underlying mechanism of these algorithms by identifying them as special cases of Brooksbank, Kassabov, & Wilson's Lie-theoretic chiselling framework. This perspective allows us to introduce a third, previously unexplored algorithm variant; for third-order tensors, these three variants constitute the only possible chiselling methods for BTD. Using this framework, we characterise the BTDs recoverable by chiselling: specifically, those where the sum of blocks constitutes a direct sum. We further show that for an arbitrary tensor, the set of detectable tensor product subspaces naturally forms a lattice.
Finally, we address uniqueness via two definitions. The first is subspace identifiability, which we show holds generically for all chiselled BTDs. The second is a stronger definition of uniqueness discussed by the previous authors. In particular, this factor matrix uniqueness requires not only unique subspaces, but also that each subspace should be detected with a unique set of bases. Regarding this, Cai & Li claimed their algorithm yields generic uniqueness but did not provide a proof, while Domanov et al. provided only sufficient conditions. We resolve these gaps by formally proving Cai & Li's claim and deriving necessary and sufficient conditions for generic uniqueness across all three variants.