Original: Thought Curvature: An underivative hypothesis on the 'Supersymmetric Artificial Neural Network'Jordan Micah Bennett jordanmicahbennett
Revised: Thought Curvature: A hypothesis on the 'Supersymmetric Artificial Neural Network'

Original: The “Supersymmetric Artificial Neural Network” in deep learning (denoted 𝜙(𝑥;𝜃,𝜃) ⊤ 𝑤 ), espouses the importance of considering biological constraints in the aim of further generalizing backward propagation.

Original: Looking at the progression of ‘solution geometries’; going from 𝑆𝑂(𝑛) representation (such as Perceptron like models) to 𝑆𝑈(𝑛) representation (such as UnitaryRNNs) has guaranteed richer and richer representations in weight space of the artificial neural network, and hence better and better hypotheses were generatable. The Supersymmetric Artificial Neural Network explores a natural step forward, namely 𝑆𝑈(𝑚|𝑛) representation. These supersymmetric biological brain representations (Perez et al.) can be represented by supercharge compatible special unitary notation 𝑆𝑈(𝑚|𝑛) , or 𝜙(𝑥;𝜃,𝜃) ⊤ 𝑤 parameterized by 𝜃,𝜃 , which are supersymmetric directions, unlike 𝜃 seen in the typical non-supersymmetric deep learning model. Notably, Supersymmetric values can encode or represent more information than the typical deep learning model, in terms of “partner potential” signals for example.
Revised: Note .

Original: “ Thought Curvature ” is similar to “Thought vectors [47] ”, with the distinction that supermanifold [9] /curvatures [49] are used to describe the "Supersymmetric Artificial NeuralNetwork"(SANN) model. (See manifold [10] /curvature [49] work in geometric deep learning by Michael Bronstein et al.
Revised: “Thought Curvature” is similar to “Thought vectors [19] ”, with the distinction that supermanifold [1] /curvatures [21] are used to describe the "Supersymmetric Artificial Neural

Original: D eepmind’s atari q architecture [4] encompasses non-pooling convolutions, therein generating object shift sensitivity, whence the model maximizes some reward over said shifts together with separate changing states for each sampled t state; translation non-invariance . Separately, uetorch [24] , encodes an object trajectory behaviour physics learner, particularly on pooling layers; translation invariance . It is non-abstrusely observable, that the childhood neocortical framework pre-encodes certain causal physical laws in the neurons [3] , amalgamating in perceptual learning abstractions into non-childhood.
Revised: Network"(SANN) hypothesis.

Original: As such, it is perhaps exigent that non-invariant fabric composes in the invariant, pertinently in the margin of some asymptote entailing 𝜙(𝑥; 𝜃, 𝜃) ⊤ 𝑤 [1] , therein engendering time-space complex optimal causal artificial construction. 2 OVERVIEW i. The aim is to contribute to the field of artificial general intelligence, often underlined as mankind’s likely last invention.
Revised: (See manifold [2] /curvature

Original: ii. Machine learning often concerns constraining algorithms with respect to biological examples iii.
Revised: [21] work in geometric deep learning by Michael Bronstein et al.

Original: Babies are great examples of some nontrivial basis for artificial general intelligence; babies are significant examples of biological baseis that are reasonably usable to inspire smart algorithms, especially in the aims of (i), regarding (ii).

Original: Babies’ brains are fantastic measures of “ tabula rasa ”-like states, from which complicated abstractions are learnt into adulthood [29] ; similar to how the recent breakthrough artificial intelligence program, “AlphaGo

Original: Zero”, started out essentially “ blank ” beginning from random plays, up until it quickly learnt to become the planet’s strongest go player today. (This quick outline highlights the critical relevance of games as necessary testbeds/algorithm training scenarios, in the aim of developing artificial general intelligence.) " Thought curvature " subsumes the " supermanifold hypothesis in deep learning ", while espousing the importance of considering biological constraints [23] in the aim of developing general machine learning models, pertinently, where babies’ brains are observed to be pre-equipped with particular “ physics priors ”, constituting specifically, the ability for babies to intuitively know laws of physics , while learning by reinforcement. It is palpable that the phrasing “intuitively know laws of physics” above, should not be confused for
Revised: Introduction . Machine

Original: Nobel laureate or physics undergrad aligned babies that for example, write or understand physics papers/exams; instead, the aforesaid phrasing simply conveys that babies’ brains are pre-baked with ways to naturally exercise physics based expectations w.r.t.

Original: interactions with objects in their world , as indicated by Aimee Stahl and Lisa Feigenson. Outstandingly, the importance of recognizing underlying causal physics laws in learning models (although not via supermanifolds , as encoded in 
Revised: learning non-trivially concerns the application of families of functions that guarantee more and more variations in weight space.

Original: Thought Curvature ), has recently been both demonstrated [21] and separately echoed by

Original: Deepmind [23] , and of late, distinctly emphasized by

Original: Yoshua Bengio.

Original: 3 RELATED WORK

Original: There priorly existed translation variant/invariant manifold interaction paradigms [6][7] , that effectively learn to disentangle varying factors. However , such models plausibly relent factors amidst optimal , causal - laws of physics arranged embeddings. (See "A probable experiment" ) 4 A PROBABLE EXPERIMENT: A TRANSVERSE FIELD ISING SPIN (SUPER)–HAMILTONIAN QUANTUM COMPUTATION
Revised: 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.

Original: Considering the Bessel aligned secondorder linear damping equation:

Original: [Equation]

Original: ] incorporating the travelling coordinate: 𝑧 = 𝑎𝑥 + 𝑏𝑦 − 𝑐𝑡 [13] whilst emerging by the isospectral factorization outcome: 𝑓 𝜆 = 𝜆 𝜆+1 [13] , constrained in dimensions 𝑑 = 2 + 1 [13] , given that the 𝑆𝑂(𝑛) group may eventuate in 𝑆𝑈(𝑚|𝑛) terms [17][36][45] ; within the aforesaid constraint , the Hamiltonian operator:

Original: [Equation]

Original: σ 𝑧𝑏 [14] is reasonably applicable in the quantum temporal difference horizon: 𝜋(𝑠1)

Original: ← 𝑎𝑟𝑔𝑚𝑎𝑥 𝑎 𝑄(𝑠1, 𝑎) [15] as a Super-Hamiltonian [16] in contrast.

Original: Consequently, some odd operation of form {

Original: [Equation]

Original: [Equation]

Original: [Equation]

Original: w 𝑎

Original: [Equation]

Original: σ 𝑧𝑎 σ 𝑧𝑏 [14] is reasonably absorbable in [15] .

Original: 5 THOUGHT CURVATURE -ANNOTATION i.

Original: Supermanifold hypothesis in deep learning describes some superspace bound structure/weights.

Original: ii.

Original: Thought curvature designates the existence ofsome plausible πBellmanian [14] , based on i; a sequence taking as parameters, the aforesaid supermanifold structure. 6 SUPERMANIFOLD HYPOTHESIS IN DEEP LEARNING If any homeomorphic transition in some neighbourhood in an euclidean space ℝ 𝑛 yields 𝜙(𝑥; 𝜃) ⊤ 𝑤 for 𝑤 𝑖 , 𝜃 ∈ ℝ 𝑛 , then reasonably, some homeomorphic transition sequence in some euclidean superspace 𝐶 ∞ (𝑅 𝑚|𝑛 ) yields 𝜙(𝑥; 𝜃, 𝜃) ⊤ 𝑤 for 𝑥 𝑖 ∈ ℝ
Revised: The “Supersymmetric Artificial Neural Network” (or ‘thought curvature’) is reasonably 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.

Original: 𝑛 while 𝑤 𝑖 , 𝜃, 𝜃 ∈ 𝑅 𝑚|𝑛 .

Original: Pertinently, 𝑅 𝑚|𝑛 → form 𝑅 2|1 for 𝑑 = 2 + 1 .

Original: i. Deep learning entails 𝜙(𝑥; 𝜃) ⊤ 𝑤 [5] , that denotes the input space 𝑥 , and  learntrepresentations 𝜃 . ii. Deep learning underlines that coordinates or latent spaces in the manifold framework, are learnt features/representations, or directions that are sparse configurations of coordinates .
Revised: For example, supersymmetric systems can capture potential-partner signals, which is beyond the feature space of magnitude and phase signals learnt in typical real valued neural nets and deep complex neural networks respectively.

Original: iii.

Original: Supermathematics entails (𝑥; 𝜃, 𝜃) [9] , that denotes some 𝑥 valued coordinate distribution, and by extension, directions that compact coordinates via 𝜃, 𝜃 . iv.

Original: As such, the aforesaid (𝑥; 𝜃, 𝜃) , is subject to coordinate transformation.

Original: v.

Original: Thereafter i, ii, iii, iv and [13] , within the generalizable nature of euclidean space, reasonably effectuate 𝜙(𝑥; 𝜃, 𝜃) ⊤ 𝑤 . 8 SUPERMANIFOLD HYPOTHESIS IN DEEP LEARNINGEND NOTES Some space (i.e. superspace) may persist, such that degrees of freedom of said space is inclined by some aggregation: i.
Revised: As such, a brief historical progression of geometric solution spaces for varying neural network architectures follows:

Original: Some tensor sequence of priors.

Original: (i.e. eta… or any 𝑝̂ data[5] constituting the laws of physics .) ii.

Original: Some tensor sequence on the direction of (i), i.e. superfields of interactions in (i) terms.

Original: 9 THOUGHT CURVATURE - LIMITATIONS

Original: Although thought curvature is minor particularly in its simple description (acquiescing

Original: SQCD [26] ) in relation to Artificial General

Original: Intelligence, it crucially delineates that the math of supermanifolds is reasonably applicable in Deep

Original: Learning, imparting that cutting edge

Original: Deep

Original: Learning work tends to consider boundaries in the biological brain [23] , while underscoring that biological brains can be optimally evaluated using supersymmetric operations.

Original: In broader words, thought curvature occurs on the following evidence:

Original: Manifolds are in the regime of very general algorithms, that enable models to learn many degrees of freedom in latent space, (i.e. position, scale etc… where said degrees are observable as features of physics interactions) where transformations on points may represent for e.g., features of a particular object in pixel space, and transformations on said points or weights of an object are disentangleable or separable from those pertaining to other objects in latent space. Given (1), and the generalizability of euclidean space , together with the instance that there persists supersymmetric measurements in biological brains, thought curvature predicates that Supermathematics or Lie Superalgebras (in Supermanifolds ) may reasonably, empirically apply in Deep Learning , or some other named study of hierarchical learning in research.
Revised: An optimal weight space produced by shallow or low dimension integer valued nodes or real valued artificial neural nets, may have good weights that lie for example, in one simple (ℤ 𝑛 𝑜𝑟 ℝ 𝑛 − 𝑜𝑟𝑑𝑒𝑟𝑒𝑑) cone per class/target group.

Original: 10 A BRIEF DISCUSSION ON THE SIGNIFICANCE OF A TRANSVERSE FIELD ISING SPIN (SUPER)-HAMILTONIAN REINFORCEMENT LEARNING ALGORITHM

Original: The usage of supersymmetric operations is imperatively efficient, as such operations enable deeply abstract representations (as is naturally afforded by symmetry group Lie

Original: Superalgebras [26][27] ), pertinently, in a general, biologically tenable time-space complex optimal regime .

Original: As such, said deeply abstract representations mayreasonably capture certain “physics priors” (See page 1), with respect to the laws of physics. 11 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.
Revised: (This may guarantee some variation, but not enough for more sophisticated tasks of higher dimension)

Original: The “ Supersymmetric Artificial Neural Network ” (a core component in ‘thought curvature’) 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 potentialpartner signals, which is beyond the feature space of magnitude and phase signals learnt in typical real valued  neural nets and deep complex neural networks respectively.
Revised: An optimal weight space produced by deep and high-dimension-absorbing real valued artificial neural nets, may have good weights that lie in disentangleable (ℝ 𝑛 ∗ ℝ 𝑛 − 𝑜𝑟𝑑𝑒𝑟𝑒𝑑) manifolds per class/target group convolved by the operator ∗ , instead of the simpler regions per

Original: As such, a brief historical progression of geometric solution spaces for varying neural network architectures follows: An optimal weight space produced by shallow or low dimension integer valued nodes or real valued artificial neural nets , may have good weights that lie for example , in one simple (ℤ 𝑛 𝑜𝑟 ℝ 𝑛 − 𝑜𝑟𝑑𝑒𝑟𝑒𝑑) cone per class/target group .
Revised: class/target group seen in item (1). The “Supersymmetric Artificial Neural Network” hypothesis(with the novel-denotation 𝜙(𝑥; 𝜃, 𝜃) ⊤ 𝑤 ) seeks to explore a regime with the potential to fundamentally use supersymmetric methods to construct artificial neural networks, therein seeking to engender novel, non-trivial contributions to deep or hierarchical artificial learning.

Original: (This may guarantee some variation , but not enough for more sophisticated tasks of higher dimension)
Revised: Looking at the progression of ‘solution geometries’; going from 𝑆𝑂(𝑛) representation (such as Perceptron like models) to 𝑆𝑈(𝑛) representation (such as

Original: An optimal weight space produced by deep and high-dimension-absorbing real valued artificial neural nets , may have good weights that lie in disentangleable (ℝ 𝑛 ∗ ℝ 𝑛 − 𝑜𝑟𝑑𝑒𝑟𝑒𝑑) manifolds per class/target group convolved by the operator ∗ , instead of the simpler regions per class/target group seen in item (1).
Revised: UnitaryRNNs) has guaranteed richer and richer representations in weight space of the artificial neural network, and hence better and better hypotheses were generatable. The Supersymmetric

Original: (This may guarantee more variation in the weight space than (1) , leading to better hypotheses or guesses )
Revised: Artificial Neural Network hypothesis explores a natural step forward, namely 𝑆𝑈(𝑚|𝑛) representation. These supersymmetric biological brain representations (Perez et al.) can be represented by supercharge compatible special unitary notation 𝑆𝑈(𝑚|𝑛) , or 𝜙(𝑥;𝜃, 𝜃) ⊤ 𝑤 parameterized by 𝜃, 𝜃 , which are supersymmetric directions, unlike 𝜃 seen in the typical nonsupersymmetric deep learning model. Notably, Supersymmetric values can encode or represent more information than the typical deep learning model, in terms of “partner potential” signals for example.

Revised: This paper does not contain empirical code concerning supersymmetric artificial neural networks, although it does highlight empirical evidence, that indicates how such types of supersymmetric learning models could exceed the state of the art, due to preservation features seen in progressing through earlier related models from the days of older perceptron like models that were not supersymmetric.

Revised: (This may guarantee more variation in the weight space than (1), leading to better hypotheses or guesses) [6] 3.

Original: An optimal weight space produced by shallow but high dimension-absorbing complex valued artificial neural nets , may have good weights that lie in multiple (ℂ 𝑛 − 𝑜𝑟𝑑𝑒𝑟𝑒𝑑) sectors per class/target group , instead of the real regions per class/target group seen amongst the prior items.
Revised: An optimal weight space produced by shallow but high dimension-absorbing complex valued artificial neural nets, may have good weights that lie in multiple (ℂ 𝑛 − 𝑜𝑟𝑑𝑒𝑟𝑒𝑑) sectors per class/target group, instead of the real regions per class/target group seen amongst the prior items.

Original: (This may guarantee more variation of the weight space than the previous items, by learning additional features , in the “phase space”.
Revised: (This may guarantee more variation of the weight space than the previous items, by learning additional features, in the “phase space”.

Original: This also leads to better hypotheses/guesses )
Revised: This also leads to better hypotheses/guesses)

Original: An optimal weight space produced by deep or high dimension-absorbing complex valued artificial neural nets , may have good weights that lie in chi distribution bound, (ℂ 𝑛 ∗ ℂ 𝑛 − 𝑜𝑟𝑑𝑒𝑟𝑒𝑑) rayleigh space per class/target group convolved by the operator ∗ , instead of the simpler sectors/regions per class/target group seen amongst the previous items.
Revised: An optimal weight space produced by deep or high dimension-absorbing complex valued artificial neural nets, may have good weights that lie in chi distribution bound, (ℂ 𝑛 ∗ ℂ 𝑛 − 𝑜𝑟𝑑𝑒𝑟𝑒𝑑) rayleigh space per class/target group convolved by the operator ∗ , instead of the simpler sectors/regions per class/target group seen amongst the previous items.

Original: (This may guarantee more variation of the weight space than the prior items , by learning phase space representations , and by extension , strengthen these representations via convolutional residual blocks.
Revised: (This may guarantee more variation of the weight space than the prior items, by learning phase space representations, and by extension, strengthen these representations via convolutional residual blocks.

Original: This also leads to better hypotheses/guesses )
Revised: This also leads to better hypotheses/guesses) [8] 5.

Original: The “Supersymmetric Artificial Neural Network” operable on high dimensional data, may reasonably generate good weights that lie in disentangleable (𝐶 ∞ (𝑅 𝑚|𝑛 ) − 𝑜𝑟𝑑𝑒𝑟𝑒𝑑) supermanifolds per class/target group , instead of the solution geometries seen in the prior items above .
Revised: The “Supersymmetric Artificial Neural Network” operable on high dimensional data, may reasonably generate good weights that lie in disentangleable (𝐶 ∞ (𝑅 𝑚|𝑛 ) − 𝑜𝑟𝑑𝑒𝑟𝑒𝑑) supermanifolds per class/target group, instead of the solution geometries seen in the prior items above.

Original: Supersymmetric values can encode rich partner-potential delimited features beyond the phase space of (4) in accordance with cognitive biological space [13] , where (4) lacks the partner potential formulation describable in Supersymmetric embedding. 12 PSEUDOCODE FOR THE “SUPERSYMMETRIC ARTIFICIAL NEURAL NETWORK
Revised: Supersymmetric values can encode rich partner-potential delimited features beyond the phase space of (4) in accordance with cognitive biological space [3] , where (4) lacks the partner potential formulation describable in Supersymmetric embedding

Original: Following, is another view of “solution geometry” history, which may promote a clear way to view the reasoning behind the subsequent pseudocode sequence:
Revised: Another view of “solution geometry” history, which may promote a clear way to view the reasoning behind the subsequent pseudocode sequence.

Original: There has been a clear progression of “ solution geometries ”, ranging from those of the ancient Perceptron [30] to complex valued neural nets [33] , grassmann manifold artificial neural networks [39] or unitaryRNNs .
Revised: There has been a clear progression of “solution geometries”, ranging from those of the ancient Perceptron [5] to complex valued neural nets [8] , grassmann manifold artificial neural networks [13] or unitaryRNNs.

Original: [38][40][44] These models may be denoted by 𝜙(𝑥; 𝜃) ⊤ 𝑤 parameterized by 𝜃 , expressible as geometrical groups ranging from orthogonal [17] to special unitary group [46] based: 𝑆𝑂(𝑛) to 𝑆𝑈(𝑛) …, and they got better at representing input data i e. representing richer weights , thus the learning models generated better hypotheses or guesses.
Revised: These models may be denoted by 𝜙(𝑥; 𝜃) ⊤ 𝑤 parameterized by 𝜃 , expressible as geometrical groups ranging from orthogonal [4] to special unitary group [18] based: 𝑆𝑂(𝑛) to 𝑆𝑈(𝑛) …, and they got better at representing input data i.e. representing richer weights, thus the learning models generated better hypotheses or guesses.

Original: By “ solution geometry ” I mean simply the class of regions where an algorithm's weights may lie, when generating those weights to do some task.
Revised: By “solution geometry” I mean simply the class of regions where an algorithm's weights may lie, when generating those weights to do some task.

Original: As such, if one follows cognitive science , one would know that biological brains may be measuredin terms of supersymmetric operations.
Revised: As such, if one follows cognitive science , one would know that biological brains may be measured

Revised: in terms

Revised: of supersymmetric operations.

Original: (Perez et al, “Supersymmetry at brain scale”) 4.
Revised: (Perez et al, “Supersymmetry at brain scale”

Original: These supersymmetric biological brain representations can be represented by supercharge [37] compatible special unitary notation 𝑆𝑈(𝑚|𝑛) , or 𝜙(𝑥; 𝜃, 𝜃) ⊤ 𝑤 parameterized by 𝜃, 𝜃 [34] , which are supersymmetric directions, unlike 𝜃 seen in item (1).
Revised: These supersymmetric biological brain representations can be represented by supercharge [11] compatible special unitary notation 𝑆𝑈(𝑚|𝑛) , or 𝜙(𝑥; 𝜃, 𝜃) ⊤ 𝑤 parameterized by 𝜃, 𝜃 [9] , which are supersymmetric directions, unlike 𝜃 seen in item (1).

Original: Notably, Supersymmetric values can encode or represent more information than the prior classes seen in (1), in terms of “partner potential” signals for example.
Revised: Notably, Supersymmetric values can encode or represent more information than the prior classes seen in (1), in terms of “partner potential” signals for example.

Original: So, state of the art machine learning work forming 𝑈(𝑛) or 𝑆𝑈(𝑛) based solution geometries, although nonsupersymmetric , are already in the family of supersymmetric solutio n geometries that may be observed as occurring in biological brain or 𝑆𝑈(𝑚|𝑛) supergroup representation . I call an “ Edward Witten/String theory powered artificial neural network ”, ‘simply’ an artificial neural network that learns supersymmetric weights. Looking at the above progression of ‘solution geometries’; going from 𝑆𝑂(𝑛) [30] representation to 𝑆𝑈(𝑛) [38][44] 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 perhaps only then reasonable to look to 𝑆𝑈(𝑚|𝑛) representation, i.e. the “ Edward
Revised: So, state of the art machine learning work forming 𝑈(𝑛) or 𝑆𝑈(𝑛) based solution geometries, although nonsupersymmetric , are already in the family of supersymmetric solution geometries that may be observed as occurring in biological brain or 𝑆𝑈(𝑚|𝑛) supergroup [17] representation.

Original: Witten/String theory powered artificial neural network ” (“ Supersymmetric Artificial Neural Network ”).
Revised: A naive supersymmetric artificial neural network architecture.

Original: To construct an “ Edward Witten/String theory powered artificial neural network ”, it may be feasible to start with a grassmann manifold artificial neural network then generate ‘charts’ [43] until scenarios occur [34] where the “ Edward Witten/String theory powered artificial neural network” is achieved in the following way: Pseudocode:
Revised: (See points 1 to 5 in [15] )

Original: a. Initialize input Supercharge [37] compatible special unitary matrix 𝑆𝑈(𝑚|𝑛). [45] (This is  the atlas seen in b.) b. Compute ∇𝐶 w.r t. to 𝑆𝑈(𝑚|𝑛) ,  where C is some cost manifold. 
Revised: It seems feasible that a C ∞ bound atlas-based learning model, where said C ∞ is in the family of supermanifolds from supersymmetry, may be obtained from a system, which includes charts ( 𝑘𝑛 ) of grassmann manifold networks 𝐺𝑅 𝑘,𝑛 and stiefel manifolds 𝐺𝐹 𝑘,𝑛 , in (𝜙 𝐼 , 𝑈 𝐼 ) terms, where there exists some invertible submatrix entailing

Original: Weight space is reasonably some K¨ahler potential like form: K(ϕ, ϕ ∗ ) , obtained on some initial projective space CP n−1 . [42]  It is feasible that CP n−1 (a C ∞ bound atlas ) may be obtained from charts ( 𝑘𝑛 ) of grassmann manifold networks 𝐺𝑅 𝑘,𝑛 [39] and stiefel manifolds
Revised: matrix 𝐴 ∈ 𝜙

Original: 𝐺𝐹 𝑘,𝑛 , in (𝜙 𝐼 , 𝑈 𝐼 ) terms, where there exists some invertible submatrix entailing matrix 𝐴 ∈ 𝜙 𝐼 (𝑈 𝐼 ∩ 𝑈 𝐽 ) for 𝑈 𝐼 = 𝜋(𝑉 𝐼 ) where 𝜋 is a submersion mapping on some stiefel manifold 𝐺𝐹 𝑘,𝑛 , thereafter enabling some differentiable grassmann manifold 𝐺𝑅 𝑘 (ℝ 𝑛 ) wherein 𝑉 𝐼 = {𝑢 ∈ ℝ 𝑛×𝑘 :𝑑𝑒𝑡(𝑢 𝐼 ) ≠ 0} .c. Parameterize 𝑆𝑈(𝑚|𝑛) in −∇𝐶 terms, by Darboux transformation . d. Repeat until convergence.
Revised: 𝐼 (𝑈 𝐼 ∩ 𝑈 𝐽 ) for 𝑈 𝐼 = 𝜋(𝑉 𝐼 ) where 𝜋 is a submersion mapping on some stiefel manifold 𝐺𝐹 𝑘,𝑛 , thereafter enabling some differentiable grassmann manifold 𝐺𝑅 𝑘 (ℝ 𝑛 ) wherein 𝑉 𝐼 = {𝑢 ∈ ℝ 𝑛×𝑘 : 𝑑𝑒𝑡(𝑢 𝐼 ) ≠ 0} .

Original: 13 CONCLUSION
Revised: Conclusion .

Original: Pertinently, the “Edward Witten/String theory powered supersymmetric artificial neural network”, is one wherein supersymmetric weights are sought.
Revised: Pertinently, the “Edward Witten/String theory powered supersymmetric artificial neural network”, is one wherein supersymmetric weights are sought.

Original: 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.
Revised: 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.

Original: 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) [45] . 
Revised: 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) [17] . Appendix: Artificial neural network/symmetry group landscape visualization. 𝑶(𝒏)

Original: 1. 𝑂(𝑛) structure – Orthogonal is not connected enough, therefore not amenable to gradient descent in machine learning.
Revised: structure – Orthogonal is not connected enough, therefore not amenable to gradient descent in machine learning.

Original: (Paper: See note 2 at end of page 2, in reference [50] .)
Revised: (Paper: See note 2 at end of page 2, in reference [22] .)

Original: 𝑆𝑂(𝑛) structure – Special Orthogonal; is connected , gradient descent compatible , while preserving orthogonality, concerning normal space-time.
Revised: 2. 𝑺𝑶(𝒏) structure – Special Orthogonal; is connected, gradient descent compatible , while preserving orthogonality, concerning normal space-time

Original: (Paper: See paper in item 1).
Revised: (Paper: See paper in item 1)

Original: 3. 𝑆𝑈(𝑛) structure – Special Unitary; is connected , gradient descent compatible; complex generalization of 𝑂(𝑛) , but only a subspace of larger unitary space, concerning normal space-time .
Revised: 3. 𝑺𝑼(𝒏) structure – Special Unitary; is connected , gradient descent compatible complex generalization of 𝑶(𝒏) , but only a subspace of larger unitary space, concerning normal space-time .

Original: (The Unitary Evolution Recurrent Neural Network [40] related to complex unit circle seen in 𝑈(1) in physics (See page 2 in [52] .))
Revised: (The Unitary Evolution Recurrent Neural Network [14] related to complex unit circle seen in 𝑈(1) in physics (See page 2 in [24] .))

Original: 𝑈(𝑛) structure – Unitary; is connected , gradient descent compatible ; Larger unitary landscape than 𝑆𝑈(𝑛) , concerning normal space-time .
Revised: 𝑈(𝑛) structure – Unitary [16] ; is connected, gradient descent compatible ; Larger unitary landscape than 𝑆𝑈 𝑛) , concerning normal space time 5. 𝑆𝑈(𝑚|𝑛) structure –

Revised: Supersymmetric

Original: [44] 5. 𝑆𝑈(𝑚|𝑛) structure – Supersymmetric; is connected [51] , thereafter reasonably gradient descent compatible and even larger than the 𝑈(𝑛) landscape, to permit sparticle invariance, being a Poincare group extension (See page 7 in [52] ) containing both normal space-time and anti-commuting components , as seen in the Supersymmetric Artificial Neural Network which this paper proposes.
Revised: ; is connected [23] thereafter reasonably gradient descent compatible and even larger than the 𝑈(𝑛) landscape to permit sparticle invariance, being a Poincare group extension (See page 7 in [24] ) containing both normal space-time and anti-commuting components , as seen in the Supersymmetric Artificial Neural Network which this paper proposes.