AdS/CFT String Theory Correspondence and Boltzmann Machines-4

In the previous articles, we discussed the AdS/CFT and Boltzmann Machine as separate entities. Now we will demonstrate how the AdS/CFT solution is correlated to the Boltzmann Machine and some possible applications for this correlation. As stated earlier, a Boltzmann Machine is a statistical physics model used in a wide variety of areas such as energy, quantum dynamics, fluids, and is currently used in machine learning-AI models as a neural network [6][13][26].

Let us begin with identifying the analogous elements between AdS-CFT and the Deep Boltzmann Machine (DBM). This is taken primary from the results of [10]. The left column is the AdS-CFT variable, and the right column is the corresponding DBM parameter.. As can be seen the hidden DBM layers are equated to the d+1 AdS gravity bulk dimensions.?

This relationship can also be shown graphically, where the hidden layers of the DBM are analogous to the AdS (d+1) holographic dimensions. This is a simple representation of the DBM, where there are normally many hidden layers, not just the one shown here.?

The bottom diagram also demonstrates the holographic dimensional progression with the AdS (d+1) to the d-dimensional CFT.

A primary reason for this is to continue defining and discretizing the AdS/CFT Duality with further rigor in quantum gravity models ?[5][18][24]. This is the method we used previously to define the chaotic boundary conditions for the Feynman-Kac stochastic string solution [15][16][17]. The boundaries are Ising networks, the basis for the DBM. A further analogy is that tensor networks have been used as the training models for quantum field theory deep networks, and as a holographic feed forward network.

Some constraints

One major discrepancy in this analogy is that there is no large numerical limit for a DBM but there is in the AdS/CFT Duality [3][10][20][21]. The deeper a DBM is, the more hidden layers exist, with the non-degenerate weights as space time variables and the data errors as CFT partitions [6]. With the AdS/CFT there is an upper limit of dimensionality based on the physical constraints.?

References

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