Supersymmetric Artificial Neural Network

Thought Curvature or the "Supersymmetric Artificial Neural Network" hypothesis, (accepted to the 2019 String Theory and Cosmology Conference GRC ) is a Lie Superalgebra bound algorithmic learning model, on the horizon of evidence pertaining to Supersymmetry in the biological brain.

It was introduced by Jordan Micah Bennett on May 10, 2016.

"Thought Curvature" or the "Supersymmetric Artificial Neural Network (2016)" is reasonably observable as a new branch or field of Deep Learning in Artificial Intelligence, called Supersymmetric Deep Learning, by Bennett. Supersymmetric Artificial Intelligence (though not Deep Gradient Descent-like machine learning) can be traced back to work by Czachor et al, concerning a single section/four paragraph thought experiment via segment "Supersymmetry and dimensional Reduction" on a so named "Supersymmetric Latent Semantic Analysis (2004)" based thought experiment; i.e. supersymmetry based single value decomposition, absent neural/gradient descent. Most of that paper apparently otherwise focusses on comparisons between non supersymmetric LSA/Single Value Decomposition, traditional Deep Neural Networks and Quantum Information Theory. Biological science/Neuroscience saw application of supersymmetry, as far back as 2007 by Perez et al. (See reference 3 from Bennett's paper )

Method

Notation 1 - Manifold Learning: $\varphi (x;\theta {)}^{\mathrm{\top }}w$

Notation 2 - Supermanifold Learning: $\varphi (x;\theta ,\overline{\theta }{)}^{\mathrm{\top }}w$

Instead of some $\theta$ neural network representation as is typical in mean field theory or manifold learning models, the Supersymmetric Artificial Neural Network is parameterized by the Supersymmetric directions $\theta ,\overline{\theta }$.

An informal proof of the representation power gained by deeper abstractions of the “Supersymmetric Artificial Neural Network”

Machine learning non-trivially concerns the application of families of functions that guarantee more and more variations in weight space. This means that machine learning researchers study what functions are best to transform the weights of the artificial neural network, such that the weights learn to represent good values for which correct hypotheses or guesses can be produced by the artificial neural network.

The 'Supersymmetric Artificial Neural Network' is yet another way to represent richer values in the weights of the model; because supersymmetric values can allow for more information to be captured about the input space. For example, supersymmetric systems can capture potential-partner signals, which are beyond the feature space of magnitude and phase signals learnt in typical real valued neural nets and deep complex neural networks respectively. As such, a brief historical progression of geometric solution spaces for varying neural network architectures follows:

Naive Architecture for the “Supersymmetric Artificial Neural Network"

Following, is another view of “solution geometry” history, which may promote a clear way to view the reasoning behind the subsequent naive architecture sequence:

The “Edward Witten/String theory powered artificial neural network”, is simply an artificial neural network that learns supersymmetric weights.

Looking at the above progression of ‘solution geometries’; going from $SO(n)$ representation to $SU(n)$ representation has guaranteed richer and richer representations in weight space of the artificial neural network, and hence better and better hypotheses were generatable. It is only then somewhat natural to look to $SU(m|n)$ representation, i.e. the “Edward Witten/String theory powered artificial neural network” (“Supersymmetric Artificial Neural Network”).

To construct an “Edward Witten/String theory powered artificial neural network”, it may be feasible to compose a system, which includes a grassmann manifold artificial neural network then generate ‘charts’ until scenarios occur where the “Edward Witten/String theory powered artificial neural network” is achieved, in the following way:

See points 1 to 5 in this reference

It seems feasible that a $C^{\mathrm{\infty }}$ bound atlas-based learning model, where said $C^{\mathrm{\infty }}$ is in the family of supermanifolds from supersymmetry, may be obtained from a system, which includes charts ${(}_k^n)$of grassmann manifold networks $G{R}_{k,n}$ and stiefel manifolds $G{F}_{k,n}$, in $({\varphi }_I,U_I)$terms, where there exists some invertible submatrix $A\in {\varphi }_I(U_I\cap U_J)$ for $U_I=\pi (V_i)$ entailing matrix for where $\pi$is a submersion mapping on some stiefel manifold $G{F}_{k,n}$, thereafter enabling some differentiable grassmann manifold $G{R}_k({ℝ}^n)$, and $V_I=\{u\in {ℝ}^{n\times k}:det(u_I)\ne 0\}$.

Artificial Neural Network/Symmetry group landscape visualization

1. $O\left(n\right)$ structure – Orthogonal is not connected enough, therefore not amenable to gradient descent in machine learning. (Paper: See note 2 at end of page 2, in reference .)

2. $SO\left(n\right)$ structure – Special Orthogonal; is connected, gradient descent compatible, while preserving orthogonality, concerning normal space-time. (Paper: See paper in item 1).

3. $SU\left(n\right)$ structure – Special Unitary; is connected, gradient descent compatible; complex generalization of $O\left(n\right)$, but only a subspace of larger unitary space, concerning normal space-time. (The Unitary Evolution Recurrent Neural Network related to complex unit circle seen in $SU\left(1\right)$ in physics (See page 2 in (See page 7 in ).))

4. $U\left(n\right)$ structure – Unitary; is connected, gradient descent compatible; Larger unitary landscape than $SU\left(n\right)$, concerning normal space-time.

5. $SU(m|n)$ structure – Supersymmetric; is connected, thereafter reasonably gradient descent compatible and even larger than the $U\left(n\right)$ landscape, to permit sparticle invariance, being a Poincare group extension (See page 7 in ) containing both normal space-time and anti-commuting components, as seen in the Supersymmetric Artificial Neural Network which this page proposes.

Ending Remarks

Pertinently, the “Edward Witten/String theory powered supersymmetric artificial neural network”, is one wherein supersymmetric weights are sought. Many machine learning algorithms are not empirically shown to be exactly biologically plausible, i.e. Deep Neural Network algorithms, have not been observed to occur in the brain, but regardless, such algorithms work in practice in machine learning.

Likewise, regardless of Supersymmetry's elusiveness at the LHC, as seen above, it may be quite feasible to borrow formal methods from strategies in physics even if such strategies are yet to show related physical phenomena to exist; thus it may be pertinent/feasible to try to construct a model that learns supersymmetric weights, as I proposed throughout this paper, following the progression of solution geometries going from $SO(n)$ to $SU(n)$ and onwards to $SU(m|n)$.