Accurate Optimization of Weighted Nuclear Norm for Non-Rigid Structure from Motion
Paper in proceedings, 2020

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.

Author

José Pedro Lopes Iglesias

Chalmers, Electrical Engineering, Signal Processing and Biomedical Engineering, Imaging and Image Analysis

Carl Olsson

Lund University

Chalmers, Electrical Engineering, Signal Processing and Biomedical Engineering, Imaging and Image Analysis

Marcus Valtonen Örnhag

Lund University

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

03029743 (ISSN) 16113349 (eISSN)

Vol. 12372

European Conference on Computer Vision
, ,

Optimization Methods with Performance Guarantees for Subspace Learning

Swedish Research Council (VR), 2019-01-01 -- 2022-12-31.

Subject Categories

Computational Mathematics

Control Engineering

Mathematical Analysis

DOI

10.1007/978-3-030-58583-9_2

ISBN

9783030585822

More information

Latest update

12/22/2020