深度学习和量子纠缠：网络设计的基本联系
Deep Learning and Quantum Entanglement: Fundamental Connections with Implications to Network Design
摘要：
深度卷积网络在各种机器学习应用中取得了前所未有的成功。关于是什么让这些网络如此成功的正式理解正在逐步展开，但在大多数情况下，仍然有重大的谜团有待解开。归纳偏差就是其中之一，它反映了嵌入在网络结构中的先验知识。在这项工作中，我们建立了量子物理学和深度学习领域之间的基本联系。我们使用这种联系来断言关于卷积网络每层中通道数量在整体归纳偏差中所起作用的新理论观察。具体来说，我们展示了由深度卷积算术电路（ConvAC）实现的函数与量子多体波函数之间的等价性，这依赖于它们共同的底层张量结构。这有助于使用量子纠缠度量作为深度网络表达能力的定义良好的量化器，以模拟其输入的复杂相关结构。最重要的是，可以使用张量网络构建深度ConvAC。这种描述使我们能够对卷积网络进行图论分析，通过这种分析，我们证明了通过其通道数直接控制深度网络的感应偏置，这些通道数与底层图中的最小割相关。这一结果对于任何为特定任务设计网络的从业者都是相关的。我们从理论上分析了ConvAC，并在涉及ReLU激活和最大池化的更常见的ConvNets上实证验证了我们的发现。除了上述结果之外，在定义明确的图论工具中描述深度卷积网络以及与量子纠缠的正式联系是这项工作带来的两个跨学科桥梁。
英文摘要：
Deep convolutional networks have witnessed unprecedented success in various machine learning applications. Formal understanding on what makes these networks so successful is gradually unfolding, but for the most part there are still significant mysteries to unravel. The inductive bias, which reflects prior knowledge embedded in the network architecture, is one of them. In this work, we establish a fundamental connection between the fields of quantum physics and deep learning. We use this connection for asserting novel theoretical observations regarding the role that the number of channels in each layer of the convolutional network fulfills in the overall inductive bias. Specifically, we show an equivalence between the function realized by a deep convolutional arithmetic circuit (ConvAC) and a quantum many-body wave function, which relies on their common underlying tensorial structure. This facilitates the use of quantum entanglement measures as well-defined quantifiers of a deep network's expressive ability to model intricate correlation structures of its inputs. Most importantly, the construction of a deep ConvAC in terms of a Tensor Network is made available. This description enables us to carry a graph-theoretic analysis of a convolutional network, with which we demonstrate a direct control over the inductive bias of the deep network via its channel numbers, that are related to the min-cut in the underlying graph. This result is relevant to any practitioner designing a network for a specific task. We theoretically analyze ConvACs, and empirically validate our findings on more common ConvNets which involve ReLU activations and max pooling. Beyond the results described above, the description of a deep convolutional network in well-defined graph-theoretic tools and the formal connection to quantum entanglement, are two interdisciplinary bridges that are brought forth by this work.
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