AdS/CFT String Theory Correspondence and Boltzmann Machines

The first article was an introduction to the AdS-CFT Correspondence and its holographic solution. The focus of this second article will be to apply this holographic solution to the Boltzmann Machine. The Boltzmann Machine is a Machine Learning Neural Network derived from Statistical Mechanics [10][13]. This neural net has energy values to each bi-directional Markovian node. The Boltzmann distribution is represented by the energy states of the system, which are the weighted nodes in the network [25][26].

There is a possible relation between physics and machine learning by correlating the field energy to a deep neural network [6][13]. Hidden neural network variables are bulk gravitational field, variable inputs are quantum field vector source, quantum generating function is Boltzmann probability distribution, and bulk action is the Stochastic Hopfield energy function [25].

In the Boltzmann machine, the Hamiltonian is a network of energy units combined in a global lattice of binary values, similar to the Ising Model [10]. There is previous research on the Ising Model in Stochastic String Theory as an even self-dual lattice of electromagnetic energy values. These values are weighted based on the probabilistic energy scores for each combination of network nodes at that point in the lattice. The states are symmetric including zero eigenvalues. The Ising Model is a canonical Boltzmann Distribution of probabilistic energy values. The Ising Model is only detectable at the visible non-hidden layers [6][26].

The Boltzmann Machine is known as the connection between Deep Learning and Physics, is an energy model that uses weighted omni-directional energy level nodes to build a neural network with hidden and visible nodes [13][26]. They are primarily used in Classification and Regression (CART) Models, for features importance, dimensional reduction, and image pattern recognition. In the case of the Boltzmann Machine the energy distribution is the weighted deviation of the loss function, with the target being the lowest possible energy value.

The possible energy states are binary (0,1) or on/off (+/-). These states are determined by the weighted values and the connections. If the connection between nodes is positive, then the connection stays in the ‘on’ or ‘1’ position. The states are considered the hypotheses of the model and are used to calculate the loss or error functions, which is the difference between actual and predicted values. The binary values lead to a Stochastic Models such as the Logistic Regression Model with a binary Dependent Variable and S- Curve non-linear output [13].

Here is a simple diagram showing hidden (blue) and visible (red) nodes. The connections are non-directional, and the energy can flow in all possible directions between nodes or energy states.?

Figure 01 courtesy of Scott Little.

The local minima issue is solved by setting higher energies that converge from a local to a global energy state. The model will continue to train until the lowest possible energy state is reached [6].?

References?

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[15] S. Little. AdS-CFT Stochastic Feynman-Kac Mellin Transform with Chaotic Boundaries. Academia.edu. December 28, 2021. https://www.academia.edu/66244508/AdS_CFT_Stochastic_Feynman_Kac_Mellin_Transform_with_Chaotic_Boundaries?source=swp_share.

[16] S. Little. Chaotic Boundaries of AdS-CFT Stochastic Feynman-Kac Mellin Transform. Academia.edu. December 28, 2021. https://www.academia.edu/66245245/Chaotic_Boundaries_of_AdS_CFT_Stochastic_Feynman_Kac_Mellin_Transform?source=swp_share.

[17] S. Little. Feynman-Kac Formulation of Stochastic String DBI Helmholtz Action. Academia.edu. July 13, 2021. https://www.academia.edu/49860679/Feynman_Kac_Formulation_of_Stochastic_String_DBI_Helmholtz_Action.

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[19] S. Little. Session Chair and Contributed Speaker (CP10): Stochastic Helmholtz Finite Volume Method for DBI String-Brane Theory Simulations. SIAM Conference on Analysis of Partial Differential Equations (PD19),?December 11 – 14, 2019, La Quinta Resort & Club, La Quinta, California, Session:?https://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=67694

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[27] Git Hub Python boltzmannclean https://github.com/facultyai/boltzmannclean.