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：我们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，该框架以二次无约束二值优化（QUBO）形式引入去噪目标函数，并适用于量子退火。该去噪目标函数通过在训练后的RBM学习到的分布与噪声图像偏差惩罚项之间取得平衡来实现。我们推导了在目标分布已被良好近似假设下的统计最优惩罚参数选择，并进一步提出一种基于经验支持的修正，使方法对该理想化假设具有鲁棒性。我们在额外假设下证明，通过我们的方法获得的去噪图像在期望意义上严格比噪声图像更接近无噪声图像。尽管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形式天然适用于量子退火器实现，我们在一台D-Wave Advantage机器上测试该模型，并通过对当前量子退火器而言规模过大的数据，采用经典启发式算法近似QUBO解进行测试。