THESIS
2021
1 online resource (viii, 133 pages) : illustrations (some color)
Abstract
Neural networks are highly complex dynamical systems consisting of large numbers of neurons
interacting through synapses. How can we formulate adequate theoretical frameworks for understanding
such systems from statics to dynamics, and from macroscopic to microscopic? In this
thesis, we analyse two paradigms, Restricted Boltzmann Machine (RBM) and Hopfield model,
which have been studied by tools originating from disordered statistical mechanics. RBM, a two
layer neural network to learn the hidden features in datasets and generate data, is the cornerstone
of unsupervised learning. Hopfield model is fundamental to theoretical neuroscience, which also
can be regarded as a RBM with quadratic function in the hidden layer. In the first part, we analyse
the permutation symmetry of RBM with two...[
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Neural networks are highly complex dynamical systems consisting of large numbers of neurons
interacting through synapses. How can we formulate adequate theoretical frameworks for understanding
such systems from statics to dynamics, and from macroscopic to microscopic? In this
thesis, we analyse two paradigms, Restricted Boltzmann Machine (RBM) and Hopfield model,
which have been studied by tools originating from disordered statistical mechanics. RBM, a two
layer neural network to learn the hidden features in datasets and generate data, is the cornerstone
of unsupervised learning. Hopfield model is fundamental to theoretical neuroscience, which also
can be regarded as a RBM with quadratic function in the hidden layer. In the first part, we analyse
the permutation symmetry of RBM with two hidden units, revealing a series of continuous phase
transitions driven by data. In the second part, we study a popular correlated Hopfield model
which is used as a prototype of many neuroscience experiments and investigate the model from a
dynamics perspective using random matrices and its equilibrium properties by the replica theory.
From these two angles, we get more insights into the temporal and spatial correlations in neural
circuits.
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