Tensor Robust Kernel PCA for Multidimensional Data.

Area of Science:

• Multidimensional data analysis
• Machine learning
• Signal processing

Background:

• Tensor robust principle component analysis (TRPCA) using tensor nuclear norm (TNN) is effective for multidimensional data.
• TNN assumes low-rankness of tensor slices, which is often violated by nonlinear structures in real-world data like videos and images.
• Exploiting intrinsic data structures efficiently remains a challenge.

Purpose of the Study:

• To propose a novel method that addresses the limitations of existing low-rank assumptions in multidimensional data processing.
• To introduce a kernelized tensor nuclear norm (KTNN) capable of capturing implicit low-rank structures.
• To develop an efficient algorithm for a new tensor robust kernel PCA (TRKPCA) model.

Main Methods:

• Kernelized Tensor Nuclear Norm (KTNN) is proposed by incorporating nonlinear kernel mapping in the transform domain.
• A tensor robust kernel PCA (TRKPCA) model is developed to decompose observed tensors into implicit low-rank and sparse components.
• An efficient alternating direction method of multipliers (ADMM)-based algorithm is designed to solve the nonlinear and nonconvex TRKPCA model.

Main Results:

• KTNN effectively captures the intrinsic nonlinear structure and implicit low-rankness of multidimensional data.
• The TRKPCA model successfully decomposes data into low-rank and sparse components, handling nonlinearities.
• Extensive experiments demonstrate TRKPCA's superior performance compared to state-of-the-art robust PCA methods on real-world applications.

Conclusions:

• The proposed KTNN provides a more faithful representation of multidimensional data structures than traditional TNN.
• TRKPCA offers a powerful and efficient approach for robust principal component analysis of data with complex nonlinear structures.
• The ADMM-based algorithm ensures efficient computation for the proposed TRKPCA model, validating its practical applicability.