sqrtw

Sqrtw is a mathematical construct used in some texts to denote the square root of a weighted quantity. In its scalar form, if t is a nonnegative real number and w is a nonnegative weight, sqrtw(t; w) is defined as sqrt(w t). This simple form is easily extended to vectors by applying a weighted sum inside the square root.

For a vector x in the nonnegative orthant (all components x_i ≥ 0) and a weight vector w with the same dimension and nonnegative components, sqrtw(x; w) is defined as the square root of the weighted sum, sqrt( ∑ i w_i x_i ). This generalizes the scalar case and provides a compact notation for taking the square root of a weighted inner product.

Key properties include nonnegativity (sqrtw ≥ 0 when inputs are nonnegative), monotonicity in both x and w (larger inputs or weights do not decrease the value), and homogeneity of degree 1/2: sqrtw(a x; b w) = sqrt(a b) sqrtw(x; w) for a, b ≥ 0. It is also subadditive in x: sqrtw(x + y; w) ≤ sqrtw(x; w) + sqrtw(y; w) when x, y ≥ 0.

Relation to other concepts: sqrtw is the square root of a weighted inner product and can be viewed as a blend between the L1 and L2 scales for nonnegative data. When all weights are equal to 1, sqrtw(x) reduces to the square root of the L1-norm, sqrt(∑ x_i). It is not a norm in the strict mathematical sense, as it is homogeneous of degree 1/2 rather than 1.

Applications include feature scaling, scoring schemes in machine learning, and optimization penalties where a square-root scale with weights can model diminishing returns. The term sqrtw is not universally standardized and may be defined differently in various contexts.