HighDimensional

Highdimensional refers to spaces or datasets characterized by a large number of dimensions or features. In mathematics and data science, it commonly implies dimensionality d that is large, often much larger than the number of observations n, or simply large enough to challenge ordinary intuition.

In high-dimensional spaces, many geometric and statistical properties behave differently from those in low dimensions. Distances can concentrate, making distinctions between near and far points less pronounced; the volume of a unit ball grows rapidly relative to the space, and data become sparse relative to the volume. This leads to the curse of dimensionality: learning from data becomes harder as the number of dimensions increases, due to overfitting, increased computational cost, and degraded generalization.

Approaches to cope include dimensionality reduction and feature selection. Dimensionality reduction projects data into a lower-dimensional representation, using linear methods such as principal component analysis or independent component analysis, or nonlinear methods like t-SNE and UMAP for visualization. Feature selection aims to identify a subset of informative variables. Regularization techniques such as L1 and L2 penalties, ridge regression, and sparse models help prevent overfitting in high dimensions. In some settings, compressed sensing exploits sparsity to recover signals from undersampled measurements.

Applications span image and text data, genomics, neuroscience, finance, and engineering. The theoretical side includes high-dimensional statistics and probability, random matrix theory, and concentration of measure, which study estimation, hypothesis testing, and asymptotics when dimensionality grows with or exceeds the sample size.