Abstract
We use a quantum annealing D-Wave 2X computer to obtain solutions to NP-hard sparse coding problems. To reduce the dimensionality of the sparse coding problem to fit on the quantum D-Wave 2X hardware, we passed downsampled MNIST images through a bottleneck autoencoder. To establish a benchmark for classification performance on this reduced dimensional data set, we used an AlexNet-like architecture implemented in TensorFlow, obtaining a classification score of $94.54 \pm 0.7 \%$. As a control, we showed that the same AlexNet-like architecture produced near-state-of-the-art classification performance $(\sim 99\%)$ on the original MNIST images. To obtain a set of optimized features for inferring sparse representations of the reduced dimensional MNIST dataset, we imprinted on a random set of $47$ image patches followed by an off-line unsupervised learning algorithm using stochastic gradient descent to optimize for sparse coding. Our single-layer of sparse coding matched the stride and patch size of the first convolutional layer of the AlexNet-like deep neural network and contained $47$ fully-connected features, $47$ being the maximum number of dictionary elements that could be embedded onto the D-Wave $2$X hardware. Recent work suggests that the optimal level of sparsity corresponds to a critical value of the trade-off parameter associated with a putative second order phase transition, an observation supported by a free energy analysis of D-Wave energy states. When the sparse representations inferred by the D-Wave $2$X were passed to a linear support vector machine, we obtained a classification score of $95.68\%$. Thus, on this problem, we find that a single-layer of quantum inference is able to outperform a standard deep neural network architecture.
Abstract (translated)
我们使用量子退火D波2X计算机来求解NP硬稀疏编码问题。为了降低稀疏编码问题的维数以适应量子D波2X硬件，我们通过瓶颈自动编码器传递了低采样mnist图像。为了在这个缩减的维度数据集上建立分类性能的基准，我们使用了TensorFlow中实现的类似Alexnet的架构，获得了94.54美元/pm 0.7%$的分类分数。作为对照，我们发现相同的类似Alexnet的架构在原始mnist图像上产生了接近最先进的分类性能$（sim 99 \%）。为了获得一组用于推断减少维mnist数据集稀疏表示的优化特征，我们在一组$47$的随机图像补丁上刻印，然后使用随机梯度下降的离线无监督学习算法来优化稀疏编码。我们的单层稀疏编码匹配了Alexnet类深神经网络的第一个卷积层的步幅和补丁大小，包含47美元的完全连接功能，47美元是可以嵌入到d-wave$2$x硬件上的字典元素的最大数目。最近的研究表明，最佳稀疏度对应于与假定的二阶相变相关的权衡参数的临界值，这一观测得到了D波能量状态自由能分析的支持。当D波$2$x推断出的稀疏表示被传递到线性支持向量机时，我们得到了$95.68\%$的分类分数。因此，在这个问题上，我们发现单层的量子推理比标准的深神经网络结构更好。
URL
https://arxiv.org/abs/1905.13215
