# Biconvex Relaxation for Semidefinite Programming in Computer Vision

@inproceedings{Shah2016BiconvexRF, title={Biconvex Relaxation for Semidefinite Programming in Computer Vision}, author={Sohil Shah and Abhay Kumar Yadav and Carlos D. Castillo and David W. Jacobs and Christoph Studer and Tom Goldstein}, booktitle={ECCV}, year={2016} }

Semidefinite programming is an indispensable tool in computer vision, but general-purpose solvers for semidefinite programs are often too slow and memory intensive for large-scale problems. We propose a general framework to approximately solve large-scale semidefinite problems (SDPs) at low complexity. Our approach, referred to as biconvex relaxation (BCR), transforms a general SDP into a specific biconvex optimization problem, which can then be solved in the original, low-dimensional variable…

## 22 Citations

### The non-convex Burer-Monteiro approach works on smooth semidefinite programs

- Computer ScienceNIPS
- 2016

It is shown that the low-rank Burer--Monteiro formulation of SDPs in that class almost never has any spurious local optima, including applications such as max-cut, community detection in the stochastic block model, robust PCA, phase retrieval and synchronization of rotations.

### Exploiting low-rank structure in semidefinite programming by approximate operator splitting

- Computer ScienceOptimization
- 2020

This work aims to reduce this scalability gap by proposing a novel proximal algorithm for solving generalSemidefinite programming problems by exploiting the low-rank property inherent to several semidefinitely programming problems.

### A Decomposition Augmented Lagrangian Method for Low-rank Semidefinite Programming

- Computer Science, Mathematics
- 2021

A decomposition method based on the augmented Lagrangian framework to solve a broad family of semidefinite programming problems possibly with nonlinear objective functions, nonsmooth regularization, and general linear equality/inequality constraints, showing the strength of the proposed method compared to other state-of-the-art methods.

### low-rank structure in semideﬁnite programming by approximate operator splitting.

- Computer Science
- 2022

The key characteristic of the proposed algorithm is to be able to exploit the low-rank property inherent to several semideﬁnite programming problems, which provides a substantial speedup and allows the operator splitting method to eﬃciently scale to larger instances.

### Projection-free Graph-based Classifier Learning using Gershgorin Disc Perfect Alignment

- Computer ScienceArXiv
- 2021

This paper proposes a fast projection-free method by solving a sequence of linear programs (LP) by leveraging a recent linear algebraic theory called Gershgorin disc perfect alignment (GDPA), and extracts predicted labels from converged solution ¯ H.

### Partial Exactness for the Penalty Function of Biconvex Programming

- Computer ScienceEntropy
- 2021

Based on the penalty function, an algorithm is presented for finding a partial optimum solution to an inequality constrained biconvex optimization, and its convergence is proven under some conditions.

### DS*: Tighter Lifting-Free Convex Relaxations for Quadratic Matching Problems

- Computer Science2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition
- 2018

This work presents a lifting-free convex relaxation that is provably at least as tight as existing (lifting-free) convex relaxations and demonstrates experimentally that this approach is superior to existing convex and non-convex methods for various problems, including image arrangement and multi-graph matching.

### Bayesian Low-rank Matrix Completion with Dual-graph Embedding: Prior Analysis and Tuning-free Inference

- Computer ScienceSignal Processing
- 2022

A novel Bayesian learning algorithm is proposed that automatically learns the hyper-parameters associated with dual-graph regularization, and at the same time, guarantees the low-rankness of matrix completion.

### Smoothing Partially Exact Penalty Function of Biconvex Programming

- Computer Science, MathematicsAsia Pac. J. Oper. Res.
- 2020

It is proved that the partial KKT point is equal to the partial optimum point under the condition of partial Slater constraint qualification and the penalty function of biconvex programming is partially exact if partial K KT condition holds.

### Unfolding Projection-free SDP Relaxation of Binary Graph Classifier via GDPA Linearization

- Computer ScienceArXiv
- 2021

This paper unrolls a projection-free algorithm for semi-definite programming relaxation (SDR) of a binary graph classifier, where the PSD cone constraint is replaced by a set of “tightest possible” linear constraints per iteration.

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