What Is the Key Difference Between PCA and SVD?

Principal Component Analysis (PCA) and Singular Value Decomposition (SVD) are two matrix factorization methods used in machine learning and data analysis for dimensionality reduction. Though they are used for similar purposes, there are some key differences between the two. The key difference between Principal Component Analysis (PCA) and Singular Value Decomposition (SVD) lies in their respective applications and the matrices they operate on.

Dealing with the data

• PCA primarily deals with the covariance structure of the data. It's a statistical procedure that transforms the coordinates of a dataset into a new coordinate system. In the new system, the first axis corresponds to the first principal component that accounts for the maximum variance in the data. The second axis, perpendicular to the first, aligns with the direction of the second largest variance, and so on. PCA effectively tries to find orthogonal axes (the principal components) along which the variance of the data is maximized.
• SVD, on the other hand, does not rely on a covariance matrix. It is a factorization of the original data matrix, and it decomposes the original data into three matrices. This can be done without computing covariance, and even allows to work with missing data.

Computations

• Both PCA and SVD involve eigen-decomposition. For PCA, the eigen-decomposition is on the covariance matrix of the data which is a square symmetric matrix of size d x d (where d is the number of features). This could be an issue if d is large, since calculating the covariance matrix and performing subsequent eigen-decomposition could be computationally expensive.

• In contrast, SVD performs the decomposition on the data matrix itself (of size n x d where n is the number of observations and d is the number of features), theoretically making the computation more efficient, especially when d is much larger than n.

In summary, while the two techniques are related (PCA can actually be solved using SVD), they approach the problem of dimensionality reduction differently. PCA focuses on the covariance structure and tries to maximize variance along orthogonal axes, while SVD focuses on matrix factorization and can handle cases where data is missing. However, from an application perspective, they are generally used interchangeably.

PCA is a specific application of SVD, primarily used for dimensionality reduction, while SVD is a more general matrix decomposition technique with broader applications in linear algebra and data analysis.