högredimensional

högredimensional, or high-dimensional, is a term used in mathematics, statistics, and data science to describe spaces or datasets that have a large number of dimensions or features. Unlike the familiar three spatial dimensions, high-dimensional contexts involve many coordinates per observation, often with more features than observations.

This high dimensionality introduces challenges collectively referred to as the curse of dimensionality. As dimensionality grows, data become sparse in feature space, distance measures lose discriminative power, and the volume of the space grows rapidly, making statistical estimation harder and increasing the risk of overfitting.

To cope, researchers use dimensionality reduction and feature selection. Techniques such as principal component analysis (PCA), independent component analysis (ICA), t-distributed stochastic neighbor embedding (t-SNE), and uniform manifold approximation and projection (UMAP) reduce dimensionality while preserving structure. Regularization methods like Lasso and ridge help control model complexity.

High-dimensional data are common in genomics, image and text analysis, finance, and sensor networks, where the number of features can far exceed samples. Proper handling often relies on domain knowledge, cross-validation, and theoretical results from high-dimensional statistics.

Important theoretical ideas include the Johnson-Lindenstrauss lemma, which asserts that a set of points can be mapped into a lower-dimensional space with controlled distortion, and the concentration of measure phenomenon, which explains distance and density behavior in high dimensions.

High-dimensional analysis is distinct from infinite-dimensional contexts, such as function spaces in analysis. The distinction matters for both theory and practice, particularly in the interpretation of results and the design of algorithms.

Example: a gene-expression dataset with thousands of genes and a few dozen samples typifies high-dimensional data and motivates regularization and feature selection.