MR-NTD: Manifold Regularization Nonnegative Tucker Decomposition for Tensor Data Dimension Reduction and...

Area of Science:

• Multidimensional Data Analysis
• Machine Learning
• Signal Processing

Background:

• Increasing volume of high-dimensional, nonnegative tensor data (e.g., EEG, images, video) necessitates efficient dimension reduction.
• Geometric properties of high-dimensional data often reside in lower-dimensional submanifolds.
• Existing tensor decomposition methods may not fully preserve geometric information or handle nonnegative constraints effectively.

Purpose of the Study:

• To develop a novel dimension reduction technique for nonnegative tensor objects.
• To preserve geometric information within tensor data during decomposition.
• To propose an efficient and effective algorithm for nonnegative tensor dimension reduction.

Main Methods:

• Nonnegative Tucker Decomposition (NTD) to obtain smaller core tensors.
• Incorporation of a manifold regularization term to preserve geometric structure.
• Development of the Manifold Regularization NTD (MR-NTD) algorithm using alternating least squares.
• Analysis of algorithm convergence and computational complexity.

Main Results:

• The MR-NTD algorithm effectively performs dimension reduction on nonnegative tensor objects.
• The method successfully preserves geometric information present in the original tensor data.
• Convergence of the MR-NTD algorithm is theoretically demonstrated.
• Linear scaling of computational complexity with data size and number of tensors indicates efficiency.

Conclusions:

• MR-NTD offers an effective and efficient solution for dimension reduction of nonnegative tensor data.
• The proposed method enhances the preservation of geometric structures in complex datasets.
• MR-NTD shows significant promise for applications involving large-scale tensor analysis.