# Accurate Optimization of Weighted Nuclear Norm for Non-Rigid Structure from Motion

arXiv: https://arxiv.org/abs/2003.10281 · HTML: https://mlsanity.com/paper/2003.10281/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Mar 2020.

## Built on by (most cited first, 5 of 5)

- [Organic Priors in Non-Rigid Structure from Motion](https://mlsanity.com/paper/2207.06262/) (2022)
- [Fast Encoder-Based 3D from Casual Videos via Point Track Processing](https://mlsanity.com/paper/2404.07097/) (2024)
- [MHR-Net: Multiple-Hypothesis Reconstruction of Non-Rigid Shapes from 2D Views](https://mlsanity.com/paper/2207.09086/) (2022)
- [Non-rigid Structure-from-Motion: Temporally-smooth Procrustean Alignment and Spatially-variant Deformation Modeling](https://mlsanity.com/paper/2405.04309/) (2024)
- [On the Regularization Landscape for the Linear Recommendation Models](https://mlsanity.com/paper/2609.11876/) (2026)

## Abstract (arXiv)

Fitting a matrix of a given rank to data in a least squares sense can be done very effectively using 2nd order methods such as Levenberg-Marquardt by explicitly optimizing over a bilinear parameterization of the matrix. In contrast, when applying more general singular value penalties, such as weighted nuclear norm priors, direct optimization over the elements of the matrix is typically used. Due to non-differentiability of the resulting objective function, first order sub-gradient or splitting methods are predominantly used. While these offer rapid iterations it is well known that they become inefficent near the minimum due to zig-zagging and in practice one is therefore often forced to settle for an approximate solution. In this paper we show that more accurate results can in many cases be achieved with 2nd order methods. Our main result shows how to construct bilinear formulations, for a general class of regularizers including weighted nuclear norm penalties, that are provably equivalent to the original problems. With these formulations the regularizing function becomes twice differentiable and 2nd order methods can be applied. We show experimentally, on a number of structure from motion problems, that our approach outperforms state-of-the-art methods.