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 unconstrained binary optimization problem (QUBO), whose optimal solution(s) in general is NP-hard to find. The contribution of this work is twofold. First, we solve the sparse representation QUBOs by solving them 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 using a stochastic Non-equilibrium Boltzmann Machine (NEBM). Second, we deploy Quantum Evolution Monte Carlo with Reverse Annealing and iterated warm starting on Loihi 2 to evolve the solution quality from the respective machines. 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. We demonstrate that both quantum annealing and neuromorphic computing are suitable for binary sparse coding QUBOs, and that 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，给定图像及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，该向量指示能够通过叠加最佳重构输入图像的最小基向量集合。我们采用重构误差的$L_2$损失函数与强制稀疏性的二元向量$L_0$（等价于$L_1$）损失函数来构建该问题，从而得到一个二次无约束二元优化问题（QUBO），其最优解在一般情况下属于NP难问题。本研究的贡献包含两个方面：首先，我们通过两种方式求解稀疏表示QUBO问题——在具有Pegasus芯片连接架构的D-Wave量子退火器上采用小图嵌入技术求解，以及在英特尔Loihi 2脉冲神经形态处理器上使用随机非平衡玻尔兹曼机求解。其次，我们在Loihi 2处理器上部署结合逆向退火与迭代热启动的量子演化蒙特卡洛方法，以提升各计算平台所得解的质量。我们将所得解与经典启发式算法模拟退火进行基准测试，并使用CPLEX计算最优解作为参照。迭代逆向量子退火的表现与模拟退火相当，但模拟退火始终能采样到最优解，而量子退火未能始终保持此能力。Loihi 2采样所得解的平均稀疏度优于其他所有方法。实验表明，量子退火与神经形态计算均适用于二元稀疏编码QUBO问题，且Loihi 2的表现优于采用标准线性退火方案的D-Wave量子退火器，而迭代逆向量子退火的性能则显著优于未改进的线性退火方案量子退火与Loihi 2的迭代热启动方法。