Outlier-Robust Multi-Model Fitting on Quantum Annealers

TL;DR

This work tackles the challenging problem of outlier-prone multi-model fitting by introducing Robust Quantum Multi-Model Fitting (R-QuMF), a formulation that explicitly models outliers within a maximum-set-coverage objective compatible with quantum annealers. By sampling provisional models to build a binary preference matrix $P$ and reformulating MSC as a QUBO with $ extbf{w}=[ extbf{y}; extbf{z}]$, the method jointly selects a small number of non-redundant models without requiring prior knowledge of the true model count $k$. A decomposition strategy, De-RQuMF, partitions large MSC problems into subproblems solvable on current quantum hardware, enabling scalable, robust fitting on real and synthetic data; experiments demonstrate superior robustness to outliers compared with prior quantum approaches and competitive performance against classical baselines. The results, including line, plane, and motion-segmentation tasks on synthetic and AdelaideRMF datasets, indicate the practical potential of quantum-accelerated MMF for noisy, real-world data, while also acknowledging hardware maturity as a current limitation. Overall, R-QuMF advances robust MMF in quantum settings and opens avenues for handling heterogeneous models and larger-scale vision tasks as quantum hardware continues to evolve.

Abstract

Multi-model fitting (MMF) presents a significant challenge in Computer Vision, particularly due to its combinatorial nature. While recent advancements in quantum computing offer promise for addressing NP-hard problems, existing quantum-based approaches for model fitting are either limited to a single model or consider multi-model scenarios within outlier-free datasets. This paper introduces a novel approach, the robust quantum multi-model fitting (R-QuMF) algorithm, designed to handle outliers effectively. Our method leverages the intrinsic capabilities of quantum hardware to tackle combinatorial challenges inherent in MMF tasks, and it does not require prior knowledge of the exact number of models, thereby enhancing its practical applicability. By formulating the problem as a maximum set coverage task for adiabatic quantum computers (AQC), R-QuMF outperforms existing quantum techniques, demonstrating superior performance across various synthetic and real-world 3D datasets. Our findings underscore the potential of quantum computing in addressing the complexities of MMF, especially in real-world scenarios with noisy and outlier-prone data.

Figures (17)

• Figure 1: Overview of our R-QuMF, a multi-model fitting approach that is robust to outliers and admissible to modern quantum annealers. We first sample models that along with the data define the preference matrix $P$. Next, a QUBO problem is prepared that can be minimised by quantum annealing (after a minor embedding of the logical problem on quantum hardware) or other solvers. Finally, the best solution is selected. R-QuMF outperforms previous quantum-admissible model fitting approaches.
• Figure 2: A sample visualization of the synthetic dataset (Ground Truth) and results for various methods for $50$ models and $33\%$ outliers (i.e. 10 outliers out of 30 points).
• Figure 3: Top: Misclassification Error [%] on synthetic data for $40$ sampled models with increasing outliers ($0$-$50$%); the problem size is fixed to $70$ ($30$ data points + $40$ models). SA is used for quantum methods. Bottom: Misclassification Error [%] for synthetic data on quantum hardware with increasing problem size; outlier percentage is fixed to $17\%$. Note that R-QuMF's $E_{mis}$ breaks starting from $120$ qubits. Non-robust quantum methods (i.e. QuMF and De-QuMF) are omitted because they fail in this scenario.
• Figure 4: Analysis of R-QuMF runs on our synthetic dataset. Left: The number of physical qubits as a function of the number of logical problem qubits for data points varying in the range $[2; 32]$; sampled models are $6$ times the data size. Right: The sparsity of $\widetilde{Q}$ in $\%$ as the function of the input data size.
• Figure 5: Sample results of fundamental matrix fitting on biscuitbook using SA. Our method performs as well as QuMF and De-QuMF which use the information about the number of ground-truth models.