dimRp

DimRp, short for dimensionality reduction by random projections, refers to techniques that map high-dimensional data into a lower-dimensional space by multiplying with a random projection matrix. The goal is to preserve approximately the distances between data points while enabling faster computation and lower memory usage, drawing on ideas from the Johnson-Lindenstrauss lemma.

The typical workflow starts with a dataset X in R^{n×d}. A random matrix R in R^{d×k} is generated, with k < d, and the projected data X' = X R lies in R^{n×k}. Elements of R are drawn from distributions such as Gaussian N(0,1) or sparse sign distributions to reduce multiplications. In many cases, centering is unnecessary, and the projection can be applied in streaming or in memory-limited environments.

Variants include dense random projections and sparse variants (for example, schemes using entries from {−1, 0, +1}) that reduce computation and storage. More recent implementations use structured or fast transforms to accelerate multiplication.

Key properties include probabilistic guarantees on distance preservation: with appropriate choice of k, pairwise distances are preserved within (1±ε) with high probability for all pairs. The method is data-independent and fast, but may not capture covariance structure as PCA does, and distortions can occur for small sample sizes or in sensitive tasks.

Applications span large-scale machine learning pipelines, text mining, clustering, approximate nearest-neighbor search, and information retrieval, where speed and scalability are important.

See also: Johnson-Lindenstrauss lemma; random projection; dimensionality reduction; PCA.

References: Johnson and Lindenstrauss (1984); Achlioptas (2003); Bingham and Mannila (2001).