Comparing Quantum Annealing and Spiking Neuromorphic Computing for Sampling Binary Sparse Coding QUBO Problems

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.

Sampling binary sparse coding QUBO models using a spiking neuromorphic processor

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.