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，给定一幅图像和一个过完备的非标准正交基，我们的目标是找到一个稀疏二进制向量，该向量指示一组最小的基向量，当它们相加时能够最佳地重构给定的输入。我们通过重构误差的$L_2$损失和施加稀疏性的二进制向量的$L_0$（或等价地$L_1$）损失来形式化该问题。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难问题。本工作的贡献有两方面。首先，提出了一种针对所需稀疏度的无监督且非归一化的字典特征学习方法，以最佳匹配数据。其次，通过使用随机神经元网络遍历非凸能量景观，在Loihi 1神经形态芯片上解决了二进制稀疏编码问题。这些解与经典启发式模拟退火进行了基准比较。我们证明，神经形态计算适用于对二进制稀疏编码QUBO模型的低能解进行采样，尽管Loihi 1能够对QUBO模型的非常稀疏解进行采样，但其实现仍需改进才能在竞争性上与模拟退火相媲美。