Multi-subspace power method for decomposing all tensors

TL;DR

The paper addresses the challenge of decomposing low-rank tensors with arbitrary partial symmetry by introducing the Multi-Subspace Power Method (MSPM), which reformulates a tensor to have orthonormal last slices. Decomposition components are shown to correspond to partially symmetric singular vector tuples (pSVTs) with singular value 1, computable via the Partially Symmetric Higher-Order Power Method (PS-HOPM) that enjoys global and local convergence guarantees; remaining summands are recovered through deflation and completion. A flattening-based framework is developed to ensure uniqueness bounds and to select optimal flattenings for different symmetry patterns, enabling practical recovery up to the rank. Empirical results across (2,1), (4,1), and order-three CP problems show MSPM achieves higher accuracy and faster runtimes than existing approaches, highlighting its effectiveness for tensors with symmetry constraints. The work provides both theoretical foundations and scalable algorithms for symmetry-aware tensor decomposition and suggests directions for extending convergence analyses and structure exploitation to broader settings.

Abstract

We present an algorithm for decomposing low rank tensors of any symmetry type, from fully asymmetric to fully symmetric. It generalizes the recent subspace power method from symmetric tensors to all tensors. The algorithm transforms an input tensor into a tensor with orthonormal slices. We show that for tensors with orthonormal slices and low rank, the summands of their decomposition are in one-to-one correspondence with the partially symmetric singular vector tuples (pSVTs) with singular value one. We use this to show correctness of the algorithm. We introduce a shifted power method for computing pSVTs and establish its global convergence. Numerical experiments demonstrate that our decomposition algorithm achieves higher accuracy and faster runtime than existing methods.

Figures (9)

• Figure 1: Building a tensor with last slices orthonormal from a general tensor. A fourth-order tensor $\mathscr{T}$ is represented as a 4D hypercube. Flattening with $f=(1,1,0,0)$ combines the first two directions to be columns in the flattening matrix $\mathbf{T}^{(f)}$. The SVD yields the left singular vectors $U$. The columns of $U$ are reshaped into tensors and stacked to form the tensor $\mathscr{T}^{(f)}$.
• Figure 2: The maximal rank $r(f)$ for $S^4(\mathbb{R}^{20}) \otimes \mathbb{R}^m$ with $m \in [2,120]$ for all flattenings $f$. The quantity $r(f)$ is piecewise linear in $m$ in this case.
• Figure 3: Symmetry–breaking case $d=(1,2,1)$, $f=(1,1,0)$: use shared vectors and the row space of $\mathbf{T}^{(f)}$ to recover the full summand.
• Figure 4: Steps to recover the rank-one tensor in the non-symmetry breaking case where $d = (1,1,1,1), f = (1,1,0,0), f' = (1,0,1,0)$.
• Figure 5: Basin of attraction for pSVTs of $\mathscr{T}^{(f)}$ under random initialization. Each point is an initialization vector $\mathbf{c} = [1, x, y]^\mathsf{T}$. It is colored according to which rank-one component $(\mathbf{a}_i, \mathbf{b}_i)$ PS-HOPM converges to. The three pSVTs are all attractors, converged to with comparable frequency.

Theorems & Definitions (39)

• Definition 2.1: Contraction
• Definition 2.2: Partially symmetric singular vector tuple (pSVT), see friedland2014number
• Proposition 2.3: See friedland2014number
• Definition 2.4: Flattenings of a partially symmetric tensor
• Definition 3.1: Orthonormal slices
• Example 3.2: A $2 \times 2 \times 2$ tensor
• Example 3.3
• Lemma 3.4
• proof
• Theorem 3.5