arXiv 2007.060822020-07-12quant-phcs.LGstat.ML

基于纠缠和张量网络的监督图像分类

Entanglement and Tensor Networks for Supervised Image Classification

X, The Moonshot Factory（X，登月工厂）

John Martyn, Guifre Vidal, Chase Roberts, Stefan Leichenauer

摘要：

张量网络最初是为了解决量子多体物理学中的计算问题而设计的，最近已被应用于机器学习任务。然而，与量子物理学相比，在过去30年中张量网络方法成功的原因已经很好地理解了，但对于这些技术为什么适用于机器学习却知之甚少。本文的目标是研究当前机器学习应用中张量网络模型的纠缠特性，以揭示可能指导未来发展的一般原则。我们重新使用张量网络进行监督图像分类，使用手写数字的MNIST数据集，由Stoudenmire和Schwab [Adv.通知。过程系统29，4799（2016）]。首先，我们假设张量网络在训练过程中可能正在学习哪个状态。为此，我们提出了一个合理的候选状态$|Σ_{\ell}\rangle$（建立作为对应于图像的训练集中的产品状态的叠加），并调查其纠缠特性。我们的结论是，$|Σ_{\ell}\rangle$是如此强大的纠缠，它不能近似的张量网络中使用的工作，因此必须代表一个非常不同的状态。其次，我们使用具有块积结构的张量网络，其中纠缠被限制在$n \times n$像素/量子比特的小块内。我们发现，这些状态是非常有表现力的（例如，$99.97 \%$的训练精度已经为$n=2$），这表明远程纠缠可能不是必不可少的图像分类。然而，在我们目前的实现中，优化会导致过度拟合，导致测试精度无法与其他当前方法竞争。

英文摘要：

Tensor networks, originally designed to address computational problems in quantum many-body physics, have recently been applied to machine learning tasks. However, compared to quantum physics, where the reasons for the success of tensor network approaches over the last 30 years is well understood, very little is yet known about why these techniques work for machine learning. The goal of this paper is to investigate entanglement properties of tensor network models in a current machine learning application, in order to uncover general principles that may guide future developments. We revisit the use of tensor networks for supervised image classification using the MNIST data set of handwritten digits, as pioneered by Stoudenmire and Schwab [Adv. in Neur. Inform. Proc. Sys. 29, 4799 (2016)]. Firstly we hypothesize about which state the tensor network might be learning during training. For that purpose, we propose a plausible candidate state $|Σ_{\ell}\rangle$ (built as a superposition of product states corresponding to images in the training set) and investigate its entanglement properties. We conclude that $|Σ_{\ell}\rangle$ is so robustly entangled that it cannot be approximated by the tensor network used in that work, which must therefore be representing a very different state. Secondly, we use tensor networks with a block product structure, in which entanglement is restricted within small blocks of $n \times n$ pixels/qubits. We find that these states are extremely expressive (e.g. training accuracy of $99.97 \%$ already for $n=2$), suggesting that long-range entanglement may not be essential for image classification. However, in our current implementation, optimization leads to over-fitting, resulting in test accuracies that are not competitive with other current approaches.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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