We consider the problem of computing a sparse binary representation of an image. To be precise, given an image and an overcomplete, non-orthonormal basis, we aim to find a sparse binary vector indicating the minimal set of basis vectors that when added together best reconstruct the given input. We formulate this problem with an $L_2$ loss on the reconstruction error, and an $L_0$ (or, equivalently, an $L_1$) loss on the binary vector enforcing sparsity. This yields a quadratic binary optimization problem (QUBO), whose optimal solution(s) in general is NP-hard to find. The method of unsupervised and unnormalized dictionary feature learning for a desired sparsity level to best match the data is presented. Next, we solve the sparse representation QUBO by implementing it both on a D-Wave quantum annealer with Pegasus chip connectivity via minor embedding, as well as on the Intel Loihi 2 spiking neuromorphic processor. On the quantum annealer, we sample from the sparse representation QUBO using parallel quantum annealing combined with quantum evolution Monte Carlo, also known as iterated reverse annealing. On Loihi 2, we use a stochastic winner take all network of neurons. The solutions are benchmarked against simulated annealing, a classical heuristic, and the optimal solutions are computed using CPLEX. Iterated reverse quantum annealing performs similarly to simulated annealing, although simulated annealing is always able to sample the optimal solution whereas quantum annealing was not always able to. The Loihi 2 solutions that are sampled are on average more sparse than the solutions from any of the other methods. Loihi 2 outperforms a D-Wave quantum annealer standard linear-schedule anneal, while iterated reverse quantum annealing performs much better than both unmodified linear-schedule quantum annealing and iterated warm starting on Loihi 2.

翻译：我们研究图像稀疏二元表示的计算问题。具体而言，给定图像和一组过完备非正交基，我们的目标是找到一个稀疏二元向量，该向量指示能够最佳重构给定输入的最小基向量集合。我们采用重构误差的$L_2$损失和强制稀疏性的二元向量$L_0$（等价于$L_1$）损失来构建该问题，从而得到一个二次二元优化问题（QUBO），其最优解在一般情况下属于NP难问题。本文提出了一种针对特定稀疏度水平的无监督非标准化字典特征学习方法，以实现对数据的最佳匹配。随后，我们通过两种方式求解稀疏表示QUBO问题：一是在采用Pegasus芯片连接架构的D-Wave量子退火机上通过小图嵌入实现；二是在英特尔Loihi 2脉冲神经形态处理器上实现。在量子退火机上，我们采用并行量子退火结合量子演化蒙特卡洛（亦称迭代反向退火）的方法对稀疏表示QUBO进行采样。在Loihi 2上，我们使用随机胜者通吃神经元网络。我们将这些解与经典启发式算法模拟退火进行基准测试，并使用CPLEX计算最优解。迭代反向量子退火的表现与模拟退火相近，但模拟退火始终能采样到最优解，而量子退火未能始终实现。Loihi 2采样的解平均比其他任何方法得到的解更具稀疏性。Loihi 2的性能优于采用标准线性退火方案的D-Wave量子退火机，而迭代反向量子退火的表现则显著优于未修改的线性退火方案量子退火以及在Loihi 2上进行的迭代热启动方法。