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.

通过应用无监督词典特征学习方法，我们研究了稀疏二进制表示问题，并使用量子退火和 Loihi 2 神经形态处理器求解了该问题。我们的结果表明，量子退火在大多数情况下与模拟退火相似，但模拟退火总能找到最优解，而量子退火不能。Loihi 2 采样得到的解比其他方法更加稀疏，而且 Loihi 2 在性能上优于标准的 D-Wave 量子退火处理器。

比较量子退火与脉冲神经形态计算在二进制稀疏编码 QUBO 问题采样中的应用

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 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.

本研究探讨了计算图像稀疏二进制表示的问题，并使用 Loihi 1 神经形态芯片解决了二元稀疏编码问题，对比了其与经典的模拟退火方法的优劣。