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：我们研究了一种基于受限玻尔兹曼机（RBM）的二值图像去噪框架。该框架通过引入二次无约束二元优化（QUBO）形式的去噪目标函数，特别适用于量子退火。该去噪目标通过在训练好的RBM习得的分布与噪声图像偏离惩罚项之间取得平衡来实现。我们推导了在目标分布被良好近似条件下惩罚参数的统计最优选择，并进一步提出了一种基于经验支持的修正方案，使方法对该理想化假设具有鲁棒性。我们还证明了在额外假设下，通过该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。尽管我们将该模型定位为图像去噪模型，但其可应用于任何二值数据。由于QUBO形式天然适合在量子退火器上实现，我们在D-Wave Advantage量子计算机上测试了该模型，同时针对当前量子退火器无法处理的大规模数据，采用经典启发式算法近似求解QUBO进行了测试。