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 so-called Quadratic Unconstrained Binary Optimization (QUBO) problem, whose solution is generally NP-hard to find. The contribution of this work is twofold. First, the method of unsupervised and unnormalized dictionary feature learning for a desired sparsity level to best match the data is presented. Second, the binary sparse coding problem is then solved on the Loihi 1 neuromorphic chip by the use of stochastic networks of neurons to traverse the non-convex energy landscape. The solutions are benchmarked against the classical heuristic simulated annealing. We demonstrate neuromorphic computing is suitable for sampling low energy solutions of binary sparse coding QUBO models, and although Loihi 1 is capable of sampling very sparse solutions of the QUBO models, there needs to be improvement in the implementation in order to be competitive with simulated annealing.

翻译：我们考虑计算图像稀疏二进制表示的问题。具体而言，给定一张图像和一个过完备、非正交基，我们旨在找到一个稀疏二进制向量，指示一组最小基向量，这些基向量相加后能够最好地重建给定输入。我们用重建误差的$L_2$损失和二进制向量上强制稀疏性的$L_0$（或等价地，$L_1$）损失来表述该问题。这产生了一个所谓的二次无约束二进制优化（QUBO）问题，其解通常为NP难求解。本工作的贡献有两点。首先，提出了一种非监督且非归一化的字典特征学习方法，用于在期望的稀疏度水平下最佳匹配数据。其次，利用神经元随机网络遍历非凸能量景观，在Loihi 1神经形态芯片上求解二进制稀疏编码问题。以经典启发式模拟退火为基准对解进行评测。我们证明神经形态计算适用于采样二进制稀疏编码QUBO模型的低能量解，尽管Loihi 1能够采样QUBO模型的极稀疏解，但其实现仍需改进以与模拟退火竞争。