# Dimension Reduction¶

Sanjiv R. Das

## A Matrix Reduction¶

Suppose we reduce the $k=11$ dimensional feature space $X$ to reduced factor space $R$ with $k=3$. We translate with a matrix $L$.

$$R = X \cdot L$$

where $F$ is $(64 \times 3)$, $X$ is $(64 \times 11)$, and $L$ is $(11 \times 3)$.

## Where does matrix L come from?¶

• From Principal Components Analysis.
• Based on an Eigenvalue Decomposition of the covariance matrix of the features, i.e., $C = Cov(X)$, which is size $(11 \times 11)$.
• Decomposition is based on solving the following equation:
$$\lambda l = C \cdot l$$
• There are 11 solutions $l$ and the first 3 will form the matrix $L$.