Low-Rank Tensor Completion Based on Fractional Regularization with Ky Fan p-k Norm

2026-06-17 · Source: Computer Vision and Pattern Recognition · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Computer Vision & Pattern Recognition · Depth: Expert, quick

Summary

This paper introduces a novel approach to low-rank tensor completion (LRTC) by proposing a nonconvex surrogate called the ratio of the tensor nuclear norm to the tensor Ky Fan p-k norm (TNPK). TNPK is designed to accurately approximate the tensor tubal rank and exhibits desirable properties such as scale invariance, parameter flexibility, and the existence of closed-form solutions for specific parameter choices of p and k. The method can reduce to the ratio of the tensor nuclear norm to the tensor Ky Fan k norm (TNK) or the tensor Frobenius norm (TNF) under specific settings. The authors construct an LRTC model and prove that low-rank tensors act as local minimizers under the tensor null space property (NSP). An efficient alternating direction method of multipliers (ADMM) algorithm is developed, deriving the proximal operator of the Ky Fan p-k inverse-norm, with guaranteed subsequential convergence. Extensive experiments on synthetic and real-world datasets demonstrate the superior performance of this method compared to existing state-of-the-art competitors.

Key takeaway

For AI Scientists and Machine Learning Engineers working with low-rank tensor completion, this research presents a robust new method that significantly outperforms current state-of-the-art techniques. You should consider evaluating the TNPK-based LRTC model for your applications requiring accurate tensor tubal rank approximation, especially where scale invariance and parameter flexibility are beneficial. Integrating this ADMM-driven approach could lead to improved model performance and more efficient handling of complex tensor datasets.

Key insights

A novel nonconvex surrogate, TNPK, improves low-rank tensor completion by accurately approximating tubal rank with flexible parameters and proven convergence.

Principles

• TNPK offers scale invariance and parameter flexibility.
• Low-rank tensors are local minimizers under NSP.
• ADMM ensures subsequential convergence for the model.

Method

The method constructs an LRTC model using the TNPK surrogate, derives the proximal operator of the Ky Fan p-k inverse-norm, and applies an ADMM algorithm for optimization.

In practice

• Applicable to synthetic and real-world datasets.
• Improves performance over state-of-the-art LRTC.
• Offers flexible parameter tuning (p, k).

Topics

• Low-Rank Tensor Completion
• Tensor Nuclear Norm
• Ky Fan p-k Norm
• Alternating Direction Method of Multipliers
• Nonconvex Optimization
• Computer Vision

Best for: Computer Vision Engineer, Research Scientist, AI Scientist, Machine Learning Engineer

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Editorial summary, takeaway, and curation by AIssential. Original article published by Computer Vision and Pattern Recognition.