# Scalable Algorithms for Tractable Schatten Quasi-Norm Minimization

arXiv: https://arxiv.org/abs/1606.01245 · HTML: https://mlsanity.com/paper/1606.01245/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Jun 2016.

## Built on by (most cited first, 3 of 3)

- [Efficient Differentiable Approximation of Generalized Low-rank Regularization](https://mlsanity.com/paper/2505.15407/) (2025)
- [Dynamic Proximal Gradient Algorithms for Schatten-p Quasi-Norm Regularized Problems](https://mlsanity.com/paper/2603.00333/) (2026)
- [Fast and Effective Computation of Generalized Symmetric Matrix Factorization](https://mlsanity.com/paper/2603.19147/) (2026)

## Abstract (arXiv)

The Schatten-p quasi-norm (0<p<1) is usually used to replace the standard nuclear norm in order to approximate the rank function more accurately. However, existing Schatten-p quasi-norm minimization algorithms involve singular value decomposition (SVD) or eigenvalue decomposition (EVD) in each iteration, and thus may become very slow and impractical for large-scale problems. In this paper, we first define two tractable Schatten quasi-norms, i.e., the Frobenius/nuclear hybrid and bi-nuclear quasi-norms, and then prove that they are in essence the Schatten-2/3 and 1/2 quasi-norms, respectively, which lead to the design of very efficient algorithms that only need to update two much smaller factor matrices. We also design two efficient proximal alternating linearized minimization algorithms for solving representative matrix completion problems. Finally, we provide the global convergence and performance guarantees for our algorithms, which have better convergence properties than existing algorithms. Experimental results on synthetic and real-world data show that our algorithms are more accurate than the state-of-the-art methods, and are orders of magnitude faster.