TruncatedSVD

Dimensionality reduction via truncated Singular Value Decomposition. Unlike PCA, TruncatedSVD does not center the data before decomposition, making it suitable for sparse matrices where centering would destroy sparsity.

Algorithm

Given an $n \times p$ data matrix $X$ (not centered), compute the rank-$k$ SVD approximation:

$$X \approx U_k \Sigma_k V_k^\top$$

The projection to $k$ components is:

$$Z = X V_k \in \mathbb{R}^{n \times k}$$

The explained variance for component $j$ is the variance of the $j$-th column of $Z$, and the explained variance ratio is its proportion of the total variance in $X$.

Difference from PCA

	PCA	TruncatedSVD
Centering	Centers data ($X - \bar{x}$)	No centering
Sparse data	Destroys sparsity	Preserves sparsity
Use case	Dense data, general purpose	TF-IDF, count matrices, LSA

TruncatedSVD applied to a TF-IDF matrix is known as Latent Semantic Analysis (LSA), a classic technique in information retrieval and topic modeling.

When to Use

• Text data: When working with sparse TF-IDF or bag-of-words representations.
• Non-negative or sparse data: When centering would be inappropriate or computationally expensive.
• For dense data where centering is desired, prefer PCA.

Mirrors sklearn.decomposition.TruncatedSVD.

Constructor

Skigen::TruncatedSVD<Scalar> svd(Eigen::Index n_components = 2);

Parameter	Default	Description
n_components	2	Number of components to keep

Methods

Method	Description
fit(X)	Compute the truncated SVD of $X$
transform(X)	Project $X$ onto the $k$ components
fit_transform(X)	Fit and project in one call
inverse_transform(Z)	Approximate reconstruction: $\hat{X} = Z V_k^\top$

Fitted Attributes

Accessor	Type	Description
components()	MatrixType	Component directions (rows = components)
explained_variance()	VectorType	Variance explained by each component
explained_variance_ratio()	VectorType	Fraction of total variance per component
singular_values()	VectorType	Singular values $\sigma_1 \ge \sigma_2 \ge \cdots$

Example

#include <Skigen/Decomposition>

Skigen::TruncatedSVD svd(5);
auto X_reduced = svd.fit_transform(X);  // Project to 5 dimensions
std::cout << "Explained variance ratio: "
          << svd.explained_variance_ratio().transpose() << "\n";