Accelerated canonical polyadic decomposition using mode reduction.

Area of Science:

• Multilinear algebra
• Numerical analysis
• Data science

Background:

• Canonical Polyadic Decomposition (CPD), also known as PARAllel FACtor analysis (PARAFAC), is crucial for Nth-order tensor analysis (N ≥ 3).
• Existing CPD methods often rely on alternating least squares, requiring frequent tensor unfolding, which creates performance bottlenecks for large-scale, high-order tensors.

Purpose of the Study:

• To develop a more efficient and robust method for high-order tensor decomposition.
• To address the computational challenges and component collinearity issues in existing CPD algorithms.

Main Methods:

• Propose a novel CPD method that decomposes high-order tensors by first applying CPD to a mode-reduced (typically third-order) tensor.
• Incorporate a Khatri-Rao product projection procedure to complete the decomposition.
• Analyze the theoretical equivalence and uniqueness conditions between Nth-order and third-order CPD.

Main Results:

• The proposed method significantly reduces computational cost by avoiding frequent unfolding of high-order tensors.
• Demonstrate that Nth-order CPD can be converted to an equivalent third-order CPD without losing essential uniqueness.
• Show that the method is more efficient and less prone to local minima compared to state-of-the-art CPD techniques.

Conclusions:

• The novel CPD approach offers a computationally efficient solution for analyzing large-scale, high-order tensor data.
• This method provides a promising alternative for overcoming performance bottlenecks and improving the accuracy of tensor decomposition, especially in the presence of component collinearity.