  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.
