A Fast Semidefinite Approach to Solving Binary Quadratic Problems

TL;DR

This paper introduces a new semidefinite programming formulation for binary quadratic problems that achieves a balance between tight bounds and computational efficiency, making it suitable for large-scale computer vision tasks.

Contribution

A novel SDP formulation for BQPs that maintains tight bounds while enabling a dual optimization approach with complexity comparable to spectral methods.

Findings

The new SDP approach is significantly more scalable than traditional SDP methods.

Experimental results show effectiveness in clustering, segmentation, and registration tasks.

The method achieves a good trade-off between solution quality and computational efficiency.

Abstract

Many computer vision problems can be formulated as binary quadratic programs (BQPs). Two classic relaxation methods are widely used for solving BQPs, namely, spectral methods and semidefinite programming (SDP), each with their own advantages and disadvantages. Spectral relaxation is simple and easy to implement, but its bound is loose. Semidefinite relaxation has a tighter bound, but its computational complexity is high for large scale problems. We present a new SDP formulation for BQPs, with two desirable properties. First, it has a similar relaxation bound to conventional SDP formulations. Second, compared with conventional SDP methods, the new SDP formulation leads to a significantly more efficient and scalable dual optimization approach, which has the same degree of complexity as spectral methods. Extensive experiments on various applications including clustering, image segmentation, co-segmentation and registration demonstrate the usefulness of our SDP formulation for solving large-scale BQPs.