W2AOOnormv

W2AOOnormv is a norm-based measure used in discussions of weighted vector normalization and energy distribution in feature spaces. The designation appears in mathematical notes and some algorithm descriptions to denote a specific normalized energy ratio derived from a vector under a weighting profile.

Definition: For a real-valued vector v ∈ R^n and a weight vector a ∈ R^n_+ with a_i > 0, define w2AOOnormv(v; a) = || sqrt(a) ∘ v ||_2 / || v ||_2, where sqrt(a) ∘ v denotes the elementwise (Hadamard) product. If v = 0, the value is defined as 0. Here, sqrt(a) means taking the square root of each weight component, and || · ||_2 denotes the Euclidean norm.

Properties: Invariant to scaling of v: w2AOOnormv(cv; a) = w2AOOnormv(v; a) for c ≠ 0. The value depends on the weight pattern: if a is uniform with all a_i = α, the measure equals sqrt(α); otherwise it reflects how the weight emphasizes certain components and alters the energy concentration captured by the ratio.

Applications: The measure is used as a regularization term that penalizes dispersion of energy away from high-weight components, or to compare energy concentration across different feature weighting schemes in machine learning pipelines. It also appears in signal processing contexts for assessing weighted energy distribution across channels.

History and notes: The term w2AOOnormv is not a widely standardized construct; it appears in a limited set of papers and tutorials to illustrate weighted normalization concepts.