Analysdimension

Analysdimension is a term used in some branches of mathematics and data science to describe a measure of the complexity or inherent dimensionality of an analytic object when investigated with analytic methods. The term is not yet standardized, and different researchers may adopt different formal definitions depending on the context.

One interpretation treats analysdimension as the effective dimension of a dataset or space as revealed by analytic techniques, such as the spectrum of the covariance or kernel matrix. In this view, the analysdimension is the number of principal directions with substantial variance, aligning with the concept of intrinsic dimension in dimensionality reduction.

A second interpretation arises from the study of analytic varieties: analysdimension could denote the minimal ambient dimension required to realize an object as an analytic set while preserving its essential features. Under this view, simple analytic objects (like a smooth curve) have low analysdimension, while more complex or chaotic structures have higher analysdimensions.

Properties commonly discussed include invariance under coordinate changes, stability under small perturbations, and relationships to other dimension notions such as topological, Hausdorff, or algebraic dimension. In practice, analysdimension is often estimated empirically in data science or discussed in theoretical contexts within analysis.

Examples: a straight line in Euclidean space has analysdimension 1; a plane has analysdimension 2; a smooth analytic surface in three-dimensional space typically has analysdimension 2 with respect to its intrinsic structure, while certain fractal-like analytic objects may exhibit non-integer analysdimension under appropriate definitions.