(MDS) and (t-SNE) of Higher Dimensional Black Holes

 Multidimensional Scaling (MDS) and T-distributed Stochastic Neighbor Embedding (t-SNE) are techniques used for dimensionality reduction and embedding high-dimensional data into lower-dimensional spaces while preserving certain relationships. In the context of black hole information conservation, these techniques can be utilized to embed complex data related to the black hole's properties into lower-dimensional spaces. Here are two equations representing the application of MDS and t-SNE for black hole information conservation:

1. Equation using Multidimensional Scaling (MDS):

   Let $�$ be the high-dimensional data matrix representing the properties of a black hole (such as mass, spin, charge, etc.) for $�$ black holes. The goal is to embed this high-dimensional data into a lower-dimensional space ($�$ dimensions) while preserving pairwise distances.

   The pairwise Euclidean distance matrix in the high-dimensional space is represented as $�$. The embedded points in $�$ dimensions are represented by $�$. The objective of MDS is to minimize the stress function, which measures the discrepancy between pairwise distances in the high-dimensional space ($�$) and the embedded space ($�$):

   $$\text{Minimize Stress}(�)=\sqrt{\frac{\sum _{�=1}^{�}\sum _{�=1}^{�}({�}_{��}-\mathrm{\parallel }{�}_{�}-{�}_{�}{\mathrm{\parallel }}_2{)}^2}{\sum _{�=1}^{�}\sum _{�=1}^{�}{�}_{��}^2}}$$

   Here, ${�}_{��}$ represents the pairwise distance between black holes $�$ and $�$ in the high-dimensional space, and $\mathrm{\parallel }{�}_{�}-{�}_{�}{\mathrm{\parallel }}_2$ represents the Euclidean distance between their embedded points in the lower-dimensional space.

2. Equation using T-distributed Stochastic Neighbor Embedding (t-SNE):

   Similar to MDS, t-SNE aims to embed high-dimensional data ($�$) into a lower-dimensional space ($�$ dimensions) while preserving pairwise similarities. Let ${�}_{��}$ be the conditional similarity between data points $�$ and $�$ in the high-dimensional space and ${�}_{��}$ be the similarity between their embedded points in the lower-dimensional space. The t-SNE objective function minimizes the Kullback-Leibler divergence between the conditional probabilities $�$ and $�$:

   $$\text{Minimize KL}(�\mathrm{\mid }\mathrm{\mid }�)=\sum _{�}\sum _{�}{�}_{��}\log \left(\frac{{�}_{��}}{{�}_{��}}\right)$$

   Where $�$ is the pairwise similarity matrix in the high-dimensional space, computed using Gaussian kernels, and $�$ is the pairwise similarity matrix in the lower-dimensional space, computed using the Student's t-distribution.

These equations represent the optimization objectives for embedding black hole data into lower-dimensional spaces using MDS and t-SNE, preserving pairwise distances or similarities. The specific computation of $�$ and $�$ matrices and the optimization process are involved and may require iterative numerical techniques for solving the objectives.