1  Introduction to PCA and SVD

1.1 Overview of PCA

Principal Component Analysis (PCA) is a widely-used technique for dimensionality reduction and data analysis. It transforms a dataset into a new coordinate system, where the axes (known as principal components) are ordered by the amount of variance in the data that they capture.

PCA allows us to reduce the dimensionality of the dataset while preserving as much of the variance as possible. This makes it especially useful for:

  • Data visualization: Reducing high-dimensional data to 2D or 3D for easy visualization.
  • Feature extraction: Identifying the most important directions in the data, which often correspond to the most significant features.
  • Noise reduction: By ignoring the principal components with small variances, we can filter out noise.

1.2 Why SVD for PCA?

While PCA can be computed using eigen decomposition on the covariance matrix, Singular Value Decomposition (SVD) is often preferred for both computational and theoretical reasons:

  • Numerical stability: SVD is numerically more stable than eigendecomposition.
  • Handling large datasets: SVD can handle large or sparse matrices efficiently without the need to compute the covariance matrix explicitly.
  • Flexibility: SVD can be applied to non-square matrices, making it more flexible in real-world applications.

1.2.1 Connection Between PCA and SVD

The relationship between PCA and SVD can be understood as follows. Given a centered data matrix \(X \in \mathbb{R}^{n \times p}\), where \(n\) is the number of samples and \(p\) is the number of features, we can compute its covariance matrix:

\[ S^2 = \dfrac{1}{n} X^T X \]

PCA is typically computed by finding the eigenvectors of the covariance matrix \(S^2\), but we can achieve the same result by directly applying SVD to the data matrix \(X\):

\[ X = U \Sigma V^T \]

Where:

  • \(U \in \mathbb{R}^{n \times n}\) contains the left singular vectors (associated with the principal components).
  • \(\Sigma \in \mathbb{R}^{n \times p}\) is the diagonal matrix of singular values (related to the eigen values of the covariance matrix).
  • \(V^T \in \mathbb{R}^{p \times p}\) contains the right singular vectors (directions of maximum variance).

By using SVD, we can directly obtain the principal components from the matrix \(V\) without explicitly computing the covariance matrix.

1.3 Use Cases and Benefits of PCA

PCA is widely applied in various domains, including:

  • Finance: PCA is used to analyze financial data, such as stock returns, by reducing the number of variables while retaining the most important trends.
  • Image Processing: PCA helps in compressing images by reducing the dimensionality of image data while preserving essential visual features.
  • Genomics: In genetics, PCA is used to analyze large datasets of gene expression or genetic variants.

The benefits of PCA include:

  • Dimensionality reduction: Simplifies models, reduces computational costs, and helps prevent overfitting.
  • Noise filtering: PCA helps remove irrelevant or noisy features.
  • Improved visualization: High-dimensional datasets can be visualized by projecting them onto the first two or three principal components.

In the next section, we will dive into the mathematical foundations of PCA and explain the role of SVD in efficiently computing the principal components.