pca compact trick

PCA (Principal Component Analysis) is a dimensionality reduction technique commonly used in machine learning and data analysis. The "compact trick" refers to a method for efficiently computing the eigenvectors and eigenvalues of a covariance matrix.

Here are the steps involved in the PCA compact trick:

  1. Compute the mean vector: Calculate the mean vector of the dataset by taking the average of each feature across all samples. This is done to center the data around the origin.

  2. Subtract the mean: Subtract the mean vector from each sample in the dataset. This step ensures that the data is centered around the origin, which is important for PCA.

  3. Compute the covariance matrix: Calculate the covariance matrix of the centered dataset. The covariance matrix quantifies the relationships between different features and is used to determine the principal components.

  4. Compute the eigenvectors and eigenvalues: Find the eigenvectors and eigenvalues of the covariance matrix. The eigenvectors represent the principal components, which are the directions of maximum variance in the data. The eigenvalues represent the amount of variance explained by each principal component.

  5. Sort eigenvectors by eigenvalues: Sort the eigenvectors in descending order based on their corresponding eigenvalues. This step is important as it helps identify the principal components that explain the most variance in the data.

  6. Select the desired number of principal components: Choose the number of principal components to retain based on the amount of variance you want to preserve. Typically, a threshold is set (e.g., retaining components that explain 95% of the variance) or a fixed number of components are chosen.

  7. Compute the projection matrix: Construct the projection matrix by selecting the eigenvectors corresponding to the desired number of principal components. The projection matrix maps the original dataset onto the lower-dimensional space spanned by the principal components.

  8. Transform the data: Finally, transform the original dataset by multiplying it with the projection matrix. This results in a lower-dimensional representation of the data with reduced dimensionality.

By following these steps, the PCA compact trick efficiently computes the eigenvectors and eigenvalues of the covariance matrix, enabling dimensionality reduction of high-dimensional datasets.