[Paper Review] Tensor and Its Tucker Core: the Invariance Relationships

Bo Jiang, Fan Yang+1 more|arXiv (Cornell University)|Jan 7, 2016

Tensor decomposition and applications32 references3 citations

TL;DR

This paper establishes that key tensor properties—such as CP rank, border rank, tensor Schatten quasi-norms, and Z-eigenvalues—are invariant under Tucker decomposition, enabling significant computational savings by performing analyses on the smaller core tensor instead of the original high-dimensional tensor. The method achieves up to 99% reduction in computation time, as demonstrated in numerical experiments with tensors of order 3–5 and sizes up to 8×8×8.

ABSTRACT

In [13], Hillar and Lim famously demonstrated that "multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NP-hard". Despite many recent advancements, the state-of-the-art methods for computing such `tensor analogues' still suffer severely from the curse of dimensionality. In this paper we show that the Tucker core of a tensor however, retains many properties of the original tensor, including the CP rank, the border rank, the tensor Schatten quasi norms, and the Z-eigenvalues. When the core tensor is smaller than the original tensor, this property leads to considerable computational advantages as confirmed by our numerical experiments. In our analysis, we in fact work with a generalized Tucker-like decomposition that can accommodate any full column-rank factor matrices.

Motivation & Objective

• To systematically establish the invariance of fundamental tensor properties under Tucker decomposition.
• To reduce computational complexity for NP-hard tensor problems by leveraging the smaller Tucker core instead of the original tensor.
• To provide a unified framework for computing tensor ranks, norms, and eigenvalues via the core tensor.
• To demonstrate substantial computational savings in practice through numerical experiments on structured tensors.
• To generalize Tucker decomposition to include any full column-rank factor matrices, broadening applicability.

Proposed method

• Propose a generalized Tucker decomposition with full column-rank factor matrices, extending beyond orthonormal HOSVD.
• Prove that CP rank, border rank, tensor Schatten quasi-norms, and Z-eigenvalues are preserved from original tensor to its Tucker core.
• Use the Tucker core as a computational surrogate: perform tensor analysis (e.g., eigenvalue computation) on the smaller core and map results back.
• Apply the SOS (sum of squares) approach to compute Z-eigenvalues on the core tensor, reducing problem size and solving time.
• Utilize numerical experiments to validate the computational advantage across various tensor orders and structures.
• Implement a three-step scheme: Tucker decomposition → core analysis → result transformation to original tensor.

Experimental results

Research questions

• RQ1 Does the CP rank of a tensor remain unchanged in its Tucker core?
• RQ2 Are tensor Schatten quasi-norms invariant under Tucker decomposition?
• RQ3 Can Z-eigenvalues and eigenvectors of a high-order tensor be computed more efficiently via its Tucker core?
• RQ4 To what extent does the Tucker core reduce computational time for NP-hard tensor problems?
• RQ5 Is the invariance of tensor properties preserved under general Tucker decompositions with full column-rank factors?

Key findings

• The CP rank of a tensor is equal to that of its Tucker core, as confirmed by theoretical analysis and numerical validation.
• The tensor Schatten quasi-norm is invariant under Tucker decomposition, preserving this norm in the core tensor.
• Z-eigenvalues and their corresponding eigenvectors of the original tensor are fully recoverable from the Tucker core, with exact correspondence observed in all test cases.
• For Example 2 (3rd-order tensor of size 8×8×8), computation on the core (size 2×2×2) took only 2.66 seconds versus 17,670.71 seconds on the original tensor.
• In Example 3 (5th-order tensor of size 4×4×4×4×4), core computation took 5.23 seconds compared to 186.58 seconds on the original tensor, achieving a 97% time reduction.
• In all examples, the Tucker core size was reduced to 2, regardless of original tensor size, enabling dramatic speedups in eigenvalue computation.