We investigate a framework for binary image denoising via restricted Boltzmann machines (RBMs) that introduces a denoising objective in quadratic unconstrained binary optimization (QUBO) form and is well-suited for quantum annealing. The denoising objective is attained by balancing the distribution learned by a trained RBM with a penalty term for derivations from the noisy image. We derive the statistically optimal choice of the penalty parameter assuming the target distribution has been well-approximated, and further suggest an empirically supported modification to make the method robust to that idealistic assumption. We also show under additional assumptions that the denoised images attained by our method are, in expectation, strictly closer to the noise-free images than the noisy images are. While we frame the model as an image denoising model, it can be applied to any binary data. As the QUBO formulation is well-suited for implementation on quantum annealers, we test the model on a D-Wave Advantage machine, and also test on data too large for current quantum annealers by approximating QUBO solutions through classical heuristics.

翻译：本文研究了一种通过受限玻尔兹曼机（RBM）进行二值图像去噪的框架，该框架将去噪目标引入二次无约束二元优化（QUBO）形式，非常适合量子退火。去噪目标通过平衡训练好的RBM所学到的分布与对偏离含噪图像的惩罚项来实现。我们推导了在目标分布已得到良好近似假设下的统计最优惩罚参数选择，并进一步提出了一种基于经验支持的修正方法，使该方法对该理想化假设具有鲁棒性。我们还在额外假设下证明，通过我们的方法获得的去噪图像在期望意义上严格比含噪图像更接近无噪图像。虽然我们将该模型构建为图像去噪模型，但它可应用于任何二值数据。由于QUBO形式非常适合在量子退火器上实现，我们在D-Wave Advantage机器上测试了该模型，并通过经典启发式方法近似QUBO解，对超出当前量子退火器处理能力的大规模数据进行了测试。