Decomposition

FastICA: Independent Component Analysis using the fast fixed-point algorithm.

Separates a multivariate signal into additive, independent non-Gaussian components. Uses negentropy maximization with the logcosh or exp contrast functions (or the kurtosis-based cube function), symmetric decorrelation (parallel algorithm) and, by default, whitening to unit variance.

After fitting, components holds the unmixing matrix W·K of shape (n_components, n_features) so that transform(X) = (X - mean) @ componentsᵀ, and mixingMatrix is its pseudo-inverse of shape (n_features, n_components). Independent components are only defined up to sign, scale and order.

Latent Dirichlet Allocation (LDA) for topic modeling.

Decomposes a document-term matrix into document-topic and topic-term distributions using batch variational Bayes inference. This is the topic model, not Linear Discriminant Analysis (see LinearDiscriminantAnalysis).

Input X should be a non-negative matrix (e.g., from CountVectorizer).

Non-negative Matrix Factorization (NMF).

Factorizes a non-negative matrix X ≈ W * H where: - W is the transformed data (n_samples × n_components) - H is the components matrix (n_components × n_features) - Both W and H are non-negative

Uses multiplicative update rules (Lee & Seung, 2001) that minimize the squared Frobenius norm. With init: "random" the factors start from non-negative random values scaled to the data (sqrt(mean(X) / nComponents)); "nndsvd" and "nndsvda" use a deterministic SVD-based start (Boutsidis & Gallopoulos, 2008). Plain "nndsvd" produces exact zeros, which multiplicative updates never change, so "nndsvda" is usually the better choice.

Useful for topic modeling, recommendation systems, and signal separation.

Principal Component Analysis (PCA).

Linear dimensionality reduction using Singular Value Decomposition (SVD) to project data to a lower dimensional space.

Algorithm: 1. Center the data by subtracting the column means 2. Compute SVD: X = U * Σ * V^T (exact, or randomized with power iterations) 3. Principal components are the rows of V^T, signed so that the entry of largest magnitude in each component is positive (same convention as scikit-learn) 4. Transform data by projecting onto the principal components

Time Complexity: O(min(n*d^2, d*n^2)) for the exact solver, where n=samples, d=features

Truncated SVD (aka LSA: Latent Semantic Analysis).

Unlike PCA, TruncatedSVD does not center the data before computing SVD. This makes it suitable for sparse data (e.g., TF-IDF matrices from text), where centering would destroy sparsity.

Algorithm: 1. Compute SVD of X directly: X ≈ U * Σ * V^T (truncated to nComponents) 2. Components are rows of V^T (signed so the largest-magnitude entry is positive) 3. Transform: X_new = X @ V = U * Σ

explainedVariance is the variance of each transformed column (population variance, ddof = 0) and explainedVarianceRatio divides it by the total variance of the columns of X, as in scikit-learn.