# Generalized Matrix Decomposition

##### Posted on (Update: )

This post is based on the talk given by Dr. Yue Wang at the Department of Statistics and Data Science, Southern University of Science and Technology on Jan. 04, 2020.

## Motivation

## Microbiome Data

Here is a great tutorial with python code to calculate $\beta$ diversity, including the UniFrac, and Bray-Curtis.

## Exploratory Analysis: Sample Clustering

Actually, it should be treated as **classical scaling**, an approach of **multidimensional scaling (MDS)**, but the classical scaling is equivalent to the principal analysis if the similarity is defined as the centered inner-products.

But here is still some differences. In the slide, it is the distance, the square root of the inner-product. And note that if $X=USV^T$, then $X^TX = VD^2V^T$ and $XX^T=UD^2U^T$.

The wikipedia page of MDS says that MDS is also known as Principal Coordinate Analysis.

## Exploratory Analysis: Important Variables

If we consider them in separate coordinate system respectively, then

but if we put them into a single coordinate system, then

where the biplot is

## The GMD-biplot

## Smokeless Tobacco Data

The author also writes a tutorial for the GMD-biplot.

## Supervised Learning with the GMD: GMDR

where the **variable importance** is calculated using the response.

why propose such a space of $\beta$? just want to exhibit the randomness of weights?

## Inference

what is $D$ here?