Principal Components Analysis
目录
- Principal Components Analysis
- Intuition
- Formalization
Intuition
PCA tries to identify the subspace in which the data approximately lies.
Intuitively, we choose a direction for projection and we reserve the most variance / difference.
Formalization
\[\frac{1}{m}\sum_{i=1}^m (x^{{(i)}^T} u)^2=u^T(\frac{1}{m}\sum_{i=1}^m x^{(i)}x^{(i)^T})u
\]
so the problem is transferred to choosing a eigenvector that maximize eigenvalue.
choose top k eigenvalue to reduce data dimension from \(\R^n\) down to \(\R^k\)
