Robust Low-Rank Tensor Minimization via a New Tensor Spectral k-Support Norm

Area of Science:

• Multilinear Algebra
• Numerical Analysis
• Computer Vision
• Signal Processing

Background:

• Tensor singular value decomposition (t-SVD) provides a framework for low-rank tensor modeling.
• Existing tensor norms like tensor nuclear norm (TNN) and tensor Frobenius norm (TFN) have limitations.
• t-SVD excels at modeling cross-channel/frame information in image and video data.

Purpose of the Study:

• To introduce a novel tensor norm, the tensor spectral k-support norm (TSP-k), within the t-SVD framework.
• To develop an alternative convex relaxation for TSP-k that interpolates between TNN and TFN.
• To enhance low-rank tensor approximation by simultaneously driving minor singular values to zero and preserving global structure.

Main Methods:

• Proposed the tensor spectral k-support norm (TSP-k) via convex relaxation.
• Developed proximal and polar operators for the TSP-k norm as essential optimization components.
• Designed two optimization algorithms tailored for medium- and large-scale tensor computations.

Main Results:

• The TSP-k norm effectively induces low-rankness while capturing more global information than TNN or TFN.
• Experimental results on synthetic, image, and video datasets demonstrate superior performance of TSP-k.
• The proposed optimization algorithms proved effective for medium- and large-size tensor problems.

Conclusions:

• The tensor spectral k-support norm (TSP-k) represents a significant advancement in low-rank tensor approximation.
• TSP-k offers a more robust approach to tensor modeling, improving data recovery and structure preservation.
• The developed optimization techniques provide efficient solutions for applying TSP-k to real-world problems.