# Partial Sum Minimization of Singular Values in Robust PCA: Algorithm and Applications

arXiv: https://arxiv.org/abs/1503.01444 · HTML: https://mlsanity.com/paper/1503.01444/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Mar 2015.

## Built on by (most cited first, 11 of 11)

- [Organic Priors in Non-Rigid Structure from Motion](https://mlsanity.com/paper/2207.06262/) (2022)
- [Developability Approximation for Neural Implicits through Rank Minimization](https://mlsanity.com/paper/2308.03900/) (2023)
- [ALPCAH: Sample-wise Heteroscedastic PCA with Tail Singular Value Regularization](https://mlsanity.com/paper/2307.02745/) (2023)
- [ALPCAH: Subspace Learning for Sample-wise Heteroscedastic Data](https://mlsanity.com/paper/2505.07272/) (2025)
- [Autoencoders for a manifold learning problem with a Jacobian rank constraint](https://mlsanity.com/paper/2306.14194/) (2023)
- [Efficient Differentiable Approximation of Generalized Low-rank Regularization](https://mlsanity.com/paper/2505.15407/) (2025)
- [Hyperspectral Image Denoising via Multi-modal and Double-weighted Tensor Nuclear Norm](https://mlsanity.com/paper/2101.07681/) (2021)
- ["Sparse + Low-Rank'' Tensor Completion Approach for Recovering Images and Videos](https://mlsanity.com/paper/2110.09298/) (2021)
- [Tensor Robust PCA with Nonconvex and Nonlocal Regularization](https://mlsanity.com/paper/2211.02404/) (2022)
- [A Novel Truncated Norm Regularization Method for Multi-channel Color Image Denoising](https://mlsanity.com/paper/2307.07932/) (2023)
- [Quaternion Infrared Visible Image Fusion](https://mlsanity.com/paper/2505.02364/) (2025)

## Abstract (arXiv)

Robust Principal Component Analysis (RPCA) via rank minimization is a powerful tool for recovering underlying low-rank structure of clean data corrupted with sparse noise/outliers. In many low-level vision problems, not only it is known that the underlying structure of clean data is low-rank, but the exact rank of clean data is also known. Yet, when applying conventional rank minimization for those problems, the objective function is formulated in a way that does not fully utilize a priori target rank information about the problems. This observation motivates us to investigate whether there is a better alternative solution when using rank minimization. In this paper, instead of minimizing the nuclear norm, we propose to minimize the partial sum of singular values, which implicitly encourages the target rank constraint. Our experimental analyses show that, when the number of samples is deficient, our approach leads to a higher success rate than conventional rank minimization, while the solutions obtained by the two approaches are almost identical when the number of samples is more than sufficient. We apply our approach to various low-level vision problems, e.g. high dynamic range imaging, motion edge detection, photometric stereo, image alignment and recovery, and show that our results outperform those obtained by the conventional nuclear norm rank minimization method.