The effective dimension of a set of measurement data (Q1571147)

The authors develop the concept of the effective dimension of the set \({\mathbf M}\) of measurements (experimental data) interpreted as elements of the Euclidean space \(\mathbb{R}\), \({\mathbf M}\subset\mathbb{R}\), and analyze its relation to the effective rank of the measurement model. The effective dimension of \({\mathbf M}\) is defined as a function \(\zeta(\cdot):[0,\infty)\rightarrow\{1,2,\dots\}\) whose value \(\zeta(\varepsilon)\) is equal to the minimum dimension of the linear subspace \(\mathbb{R}^{(\varepsilon)}\subset \mathbb{R}\) of the orthogonal components of measurements.