Abstract
Tensor Networks are non-trivial representations of high-dimensional tensors, originally designed to describe quantum many-body systems. We show that Tensor Networks are ideal vehicles to connect quantum mechanical concepts to machine learning techniques, thereby facilitating an improved interpretability of neural networks. This study presents the discrimination of top quark signal over QCD background processes using a Matrix Product State classifier. We show that entanglement entropy can be used to interpret what a network learns, which can be used to reduce the complexity of the network and feature space without loss of generality or performance. For the optimisation of the network, we compare the Density Matrix Renormalization Group (DMRG) algorithm to stochastic gradient descent (SGD) and propose a joined training algorithm to harness the explainability of DMRG with the efficiency of SGD.
Highlights
JHEP08(2021)112 used in unsupervised learning [30], for anomaly detection [31] and has been shown that it can produce comparable results to recurrent neural networks [32].1 Beyond Matrix Product States (MPS), there have been various ML applications with MERA [34, 35] and in the 2D case with projected entangled pair states (PEPS) [36]
We show that Tensor Networks are ideal vehicles to connect quantum mechanical concepts to machine learning techniques, thereby facilitating an improved interpretability of neural networks
This study will show that MPS can be used to discriminate top jets over QCD jets with comparable precisions to state-of-the-art classifiers and that the tensor network learns the volume and correlations of the projected geometry of topological relations in the data, which is reflected by the entanglement entropy of the network
Summary
Tensors are general multidimensional objects which can describe the multilinear relationship between algebraic objects within a vector space. Notation (or tensor diagram notation) [46] to describe tensors as briefly shown in figure 1.2 In this notation, a node without any edge describes a scalar, and each edge represents a higher rank object, such as one edge for vectors, two for matrices, where tensors can be rank N objects. The Einstein summation has been applied to the connected edges where the leg between A and C indicates summation over index k. These are called auxiliary (or bond) dimensions of the network. The size of these connections indicates each tensor’s influence on each other, which will be further detailed . Such objects have been widely used in the theoretical description of quantum many-body systems [48], and in the design of quantum computing algorithms
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