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arXiv:2105.02878 · IEEE/CVF International Conference on Computer Vision (ICCV)

Q-Match: Iterative Shape Matching via Quantum Annealing

Marcel Seelbach Benkner, Zorah Lähner, Vladislav Golyanik, Christof Wunderlich, Christian Theobalt, Michael Moeller

Published Oct 1, 2021• 28 citations
arXivHTMLPDF

Abstract

Finding shape correspondences can be formulated as an ${\mathcal{N}}{\mathcal{P}} - hard$ quadratic assignment problem (QAP) that becomes infeasible for shapes with high sampling density. A promising research direction is to tackle such quadratic optimization problems over binary variables with quantum annealing, which allows for some problems a more efficient search in the solution space. Unfortunately, enforcing the linear equality constraints in QAPs via a penalty significantly limits the success probability of such methods on currently available quantum hardware. To address this limitation, this paper proposes Q-Match, i.e., a new iterative quantum method for QAPs inspired by the α-expansion algorithm, which allows solving problems of an order of magnitude larger than current quantum methods. It implicitly enforces the QAP constraints by updating the current estimates in a cyclic fashion. Further, Q-Match can be applied iteratively, on a subset of well-chosen correspondences, al-lowing us to scale to real-world problems. Using the latest quantum annealer, the D-Wave Advantage, we evaluate the proposed method on a subset of QAPLIB as well as on isometric shape matching problems from the FAUST dataset.

DOI
10.1109/iccv48922.2021.00749

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Finding shape correspondences can be formulated as an NP−hard{\mathcal{N}}{\mathcal{P}} - hardNP−hard quadratic assignment problem (QAP) that becomes infeasible for shapes with high sampling density. A promising research direction is to tackle such quadratic optimization problems over binary variables with quantum annealing, which allows for some problems a more efficient search in the solution space. Unfortunately, enforcing the linear equality constraints in QAPs via a penalty significantly limits the success probability of such methods on currently available quantum hardware. To address this limitation, this paper proposes Q-Match, i.e., a new iterative quantum method for QAPs inspired by the α-expansion algorithm, which allows solving problems of an order of magnitude larger than current quantum methods. It implicitly enforces the QAP constraints by updating the current estimates in a cyclic fashion. Further, Q-Match can be applied iteratively, on a subset of well-chosen correspondences, al-lowing us to scale to real-world problems. Using the latest quantum annealer, the D-Wave Advantage, we evaluate the proposed method on a subset of QAPLIB as well as on isometric shape matching problems from the FAUST dataset.
Matching problems on 3D shapes and images are challenging as they are frequently formulated as combinatorial quadratic assignment problems (QAPs) with permutation matrix constraints, which are NP-hard. In this work, we address such problems with emerging quantum computing technology and propose several reformulations of QAPs as unconstrained problems suitable for efficient execution on quantum hardware. We investigate several ways to inject permutation matrix constraints in a quadratic unconstrained binary optimization problem which can be mapped to quantum hardware. We focus on obtaining a sufficient spectral gap, which further increases the probability to measure optimal solutions and valid permutation matrices in a single run. We perform our experiments on the quantum computer D-Wave 2000Q(211qubits, adiabatic). Despite the observed discrepancy between simulated adiabatic quantum computing and execution on real quantum hardware, our reformulation of permutation matrix constraints increases the robustness of the numerical computations over other penalty approaches in our experiments. The proposed algorithm has the potential to scale to higher dimensions on future quantum computing architectures, which opens up multiple new directions for solving matching problems in 3D computer vision and graphics.
Matching problems on 3D shapes and images are challenging as they are frequently formulated as combinatorial quadratic assignment problems (QAPs) with permutation matrix constraints, which are NP-hard. In this work, we address such problems with emerging quantum computing technology and propose several reformulations of QAPs as unconstrained problems suitable for efficient execution on quantum hardware. We investigate several ways to inject permutation matrix constraints in a quadratic unconstrained binary optimization problem which can be mapped to quantum hardware. We focus on obtaining a sufficient spectral gap, which further increases the probability to measure optimal solutions and valid permutation matrices in a single run. We perform our experiments on the quantum computer D-Wave 2000Q (2^11 qubits, adiabatic). Despite the observed discrepancy between simulated adiabatic quantum computing and execution on real quantum hardware, our reformulation of permutation matrix constraints increases the robustness of the numerical computations over other penalty approaches in our experiments. The proposed algorithm has the potential to scale to higher dimensions on future quantum computing architectures, which opens up multiple new directions for solving matching problems in 3D computer vision and graphics.
Quantum annealing and D-Wave quantum annealer attracted considerable attention for their ability to solve combinatorial optimization problems. In order to solve other type of optimization problems, it is necessary to apply certain kinds of mathematical transformations. However, linear constraints reduce the size of problems that can be represented in quantum annealers, owing to the sparseness of connections between qubits. For example, the quadratic assignment problem (QAP) with linear equality constraints can be solved only up to size 12 in the quantum annealer D-Wave Advantage, which has 5640 qubits. To overcome this obstacle, Ohzeki developed a method for relaxing the linear equality constraints and numerically verified the effectiveness of this method with some target problems, but others remain unsolvable. In particular, it is difficult to obtain feasible solutions to problems with hard constraints, such as the QAP. We therefore propose a method for solving the QAP with quantum annealing by applying a post-processing bit-flip heuristic algorithm to solutions obtained by the Ohzeki method. In a numerical experiment, we solved a sparse QAP by the proposed method. This sparse QAP has been used in areas such as item listing on an E-commerce website. We successfully solved a QAP of size 19 with high accuracy for the first time using D-Wave Advantage. We also confirmed that the bit-flip heuristic algorithm moves infeasible solutions to nearby feasible solutions in terms of Hamming distance with good computational efficiency.
Since the ability of quantum computing to speed up certain, specific computations has a solid theoretical underpinning, it is a natural question to ask if problems in computer vision also benefit from it. This doctoral thesis is in particular concerned with the question if the recently emerging optimization method of quantum annealing is useful for solving combinatorial optimization problems that arise in computer vision. The computer vision problem that is mainly investigated here is the problem of shape matching. The task of shape matching is to obtain the correspondences of two objects in a scene that are described with point clouds. Most often, one is concerned with finding the correspondences for a person who is recorded in 3D two times but with a different pose. The reason why this computer vision problem was picked as an interesting candidate for an application of quantum annealing is because it can be formulated as a difficult combinatorial optimization problem with a quadratic objective. This is close to the type of optimization problem that recent quantum annealing devices try to solve. The biggest difference is that in quantum annealing one a priori has unconstrained problems so that one cannot include equality constraints in the optimization. In the first work of this thesis, multiple methods are presented to overcome this difference and for really small shape matching problems experiments with simulated quantum annealing and with quantum annealing on D-Wave quantum computers were presented. In the second work of this thesis an iterative method was developed so that it was even possible to solve shape matching problems with over 500 vertices. Finally, a third publication looked at problems where one needs to find the correspondences between multiple shapes. Besides applications of quantum annealing for shape matching, this thesis also contains work on solving the motion segmentation problem with quantum annealing and work about integrating a quantum annealer in a neural network to solve computer vision problems. The integration of the quantum annealer in a neural network can also be viewed as the task to learn an appropriate formulation of a computer vision problem in a way that is solvable with a current quantum annealer. Finally, we also present work about an iterative method for compensating limited connectivity in quantum annealing devices. This work has no particular computer vision application in mind, but investigates the performance of the different approaches on Max-Cut problems. The thesis is structured in a cumulative way, so that the individual publications are presented separately. On a broader picture, the works are motivated by the following questions: What could be the impact of quantum annealing on computer vision and how could computer vision algorithms that use quantum annealing look like? And from another perspective: How would the field of computer vision be changed if there were massive advances in building quadratic, unconstrained, binary optimization problem solvers with or even without utilizing quantum computing?
There is growing interest in solving computer vision problems such as mesh or point set alignment using Adiabatic Quantum Computing (AQC). Unfortunately, modern experimental AQC devices such as D-Wave only support Quadratic Unconstrained Binary Optimisation (QUBO) problems, which severely limits their applicability. This paper proposes a new way to overcome this limitation and introduces QuCOOP, an optimisation framework extending the scope of AQC to composite and binary-parametrised, possibly non-quadratic problems. The key idea of QuCOOP is to iteratively approximate the original objective function by a sequel of local (intermediate) QUBO forms, whose binary parameters can be sampled on AQC devices. We experiment with quadratic assignment problems, shape matching and point set registration without knowing the correspondences in advance. Our approach achieves state-of-the-art results across multiple instances of tested problems1.
Modern quantum annealers can find high-quality solutions to combinatorial optimisation objectives given as quadratic unconstrained binary optimisation (QUBO) problems. Unfortunately, obtaining suitable QUBO forms in computer vision remains challenging and currently requires problem-specific analytical derivations. Moreover, such explicit formulations impose tangible constraints on solution encodings. In stark contrast to prior work, this paper proposes to learn QUBO forms from data through gradient backpropagation instead of deriving them. As a result, the solution encodings can be chosen flexibly and compactly. Furthermore, our methodology is general and virtually independent of the specifics of the target problem type. We demonstrate the advantages of learnt QUBOs on the diverse problem types of graph matching, 2D point cloud alignment and 3D rotation estimation. Our results are competitive with the previous quantum state of the art while requiring much fewer logical and physical qubits, enabling our method to scale to larger problems. The code and the new dataset will be open-sourced.