Manifold Vector Machine

Manifold Vector Machine or MVM is a data points superpositioning process. It build data points into relative vector beams that can be accessed by quantum pointers to a set of superpositions with manifold transformation. It is a class of machine learning which works directly on the manifold of the data points from current dimension. This is distinctively different from Nonlinear dimensionality reduction and Support vector machine, the later category, which can be generalised as Dimension Augmentation methods (DIB-N: Dimension Increase By - N, DDB-N: Dimension Decrease By - N ). The concept was first introduced as such by Hao in 2018,. Notably, it is also distinctively different from the concept introduced by Geoffrey Hinton in his Capsule neural network. Although the philosophy of Capsule did pointed out the impact of integrity on data context through a learning process, it is a DIB-N.

Overview

A Manifold Vector Machine or MVM assumes that (1) distance within a set of data points can be relatively governed throughout manifold transformation from its current dimensional context; (2) $\exists$ discrete continuity that is iterable for the entire vector chain. Then we can count from one end to the other throughout these data points as we perform non-repetitive iteration algorithms. Suggest that one complete travel throughout the data points is one oscillation denotable as ${\overset {.}{O}}$, then we have: 

A set of ETE(End-To-End) vectors: ${\vec {A}}{_{0}},{\vec {A}}{_{1}},{\vec {A}}{_{2}}$,...${\vec {A}}{_{n}}$,…,${\vec {A}}{_{\infty }}$$:\equiv$${\underset {n=0}{\overset {\infty }{\circleddash }}}{\vec {A}}{_{n}}$$(\lVert {\vec {A}}{_{n}}\rVert \neq 0,\forall n\in [0,*\infty ])$ such that:

$Iteration$ ($d\delta$) = $Floor$$\left({\frac {N\times {\overset {.}{o}}}{\lVert concat({\underset {n=0}{\overset {\infty }{\circleddash }}}{\vec {A}}{_{n}})\rVert }}\right)$ .

$N$: number of oscillation through one marching.
$Floor$: guarantees each iteration is complete.

So that, ${\frac {dM_{T}}{d\delta }}={\frac {\lVert {\vec {\lambda }}\cdot M_{T}\rVert }{d_{H}}}$, where Hausdorff definable:

$\exists d_{H}{_{total}}:={max}\lbrace {min}\lbrace d({\vec {A}}{_{0}},{\vec {A}}{_{\infty }}),d({\vec {A}}{_{\infty }},{\vec {A}}{_{0}})\rbrace \rbrace$,
$\exists d_{H}{_{local}}:={max}\lbrace {min}\lbrace d({\vec {A}}{_{\infty }{_{-}{_{1}}}},{\vec {A}}{_{\infty }}),d({\vec {A}}{_{\infty }},{\vec {A}}{_{\infty }{_{-}{_{1}}}})\rbrace \rbrace =\lVert {\vec {A}}{_{n}}\rVert$

With a useful special case:

$\exists d_{H}{_{total}}:=\lVert {\vec {A_{0}}}-{\vec {A_{\infty }}}\rVert$, iff ${\underset {n=0}{\overset {\infty }{\circleddash }}}{\vec {A}}{_{n}}={\underset {n=0}{\overset {\infty }{\circledcirc }}}{\vec {A}}{_{n}}$.

Manifold Types

Depending on the type of the data, one manifold work better than the other on a variety of reasons.

Manifolds:
Dot $d_{H}{_{total}}:=\lVert {\vec {0}}\rVert$, (see Gravitational singularity)
Non-closed segments $d_{H}{_{total}}:=\sum _{n=0}^{\infty }\lVert {\vec {A}}{_{n}}\rVert$, (see Array data structure)
Closed Orient/Non-orient Manifolds $d_{H}{_{total}}:=\lVert {\vec {A_{0}}}-{\vec {A_{\infty }}}\rVert$, (see definition of Closed manifold, as well as Amplitude amplification in Quantum computing), as:
${\underset {n=0}{\overset {\infty }{\circleddash }}}{\vec {A}}{_{n}}\equiv {\underset {n=0}{\overset {\infty }{\circledcirc }}}{\vec {A}}{_{n}}$, where locally $\mid \varphi _{i}\rangle$ = $\sum _{}^{}\lVert {\vec {A}}{_{k}}\rVert \times \mid \circledast _{k}\rangle$

Applications

MVM is applicable on its own or embeddable to general machine learning strategies from Regression analysis, Support vector machine, and Deep learning.
A list of applications of MVM include:

• Image processing
• Neural Network
• Cryptography
• Prime Prediction

References

This article "Manifold Vector Machine" is from Wikipedia. The list of its authors can be seen in its historical and/or the page Edithistory:Manifold Vector Machine. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.

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