Abstract
We introduce a machine learning model, the q-CNN model, sharing key features with convolutional neural networks and admitting a tensor network description. As examples, we apply q-CNN to the MNIST and Fashion MNIST classification tasks. We explain how the network associates a quantum state to each classification label, and study the entanglement structure of these network states. In both our experiments on the MNIST and Fashion-MNIST datasets, we observe a distinct increase in both the left/right as well as the up/down bipartition entanglement entropy (EE) during training as the network learns the fine features of the data. More generally, we observe a universal negative correlation between the value of the EE and the value of the cost function, suggesting that the network needs to learn the entanglement structure in order the perform the task accurately. This supports the possibility of exploiting the entanglement structure as a guide to design the machine learning algorithm suitable for given tasks.
Highlights
Convolutional neural networks (CNNs) have seen remarkable successes in various applications
In both our experiments on the MNIST and Fashion-MNIST datasets, we initialize the network in a random way which renders a quantum state with no particular entanglement structure
It can be read out that the entanglement needed for the (Fashion-) MNIST classification tasks is low, which could be viewed as a “justification” why a simple CNN is capable of performing these tasks
Summary
Convolutional neural networks (CNNs) have seen remarkable successes in various applications. Tensor network [2, 3] is one of the most popular tools utilised in many-body quantum physics to overcome this problem Speaking, they provide a way to approximate high-order tensors in terms of lower-order tensors, and by doing so greatly reduce the parameters needed to describe the relevant quantum states, circumventing the curse of dimensionality. We compute the entanglement entropies of the network quantum states, with respect to bipartitions of the configuration space corresponding to the left/right and up/down partitions of an image. In both our experiments on the MNIST and Fashion-MNIST datasets, we initialize the network in a random way which renders a quantum state with no particular entanglement structure. It can be read out that the entanglement needed for the (Fashion-) MNIST classification tasks is low, which could be viewed as a “justification” why a simple CNN is capable of performing these tasks
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