gradability

Gradability is a term used informally to describe the extent to which a quantity, function, or object can be described by a gradient or by a graded scale. Because the notion of a gradient or gradient-like structure varies by discipline, gradability has different concrete meanings in mathematics, measurement theory, and applied sciences.

In mathematics, gradability relates to differentiability. A real-valued function defined on a region is gradientable there if it has a gradient vector at each point, i.e., it is differentiable with partial derivatives that form a well-defined gradient. The gradient points in the direction of steepest ascent, and the existence of a gradient implies a level of smoothness; functions with sharp corners or cusps fail to be gradientable at those points. In vector fields, a gradient field is a field that can be expressed as the gradient of a potential function.

In measurement and statistics, gradability concerns the possibility of ordering or assigning meaningful step sizes. Variables on an ordinal scale are gradable in terms of order but do not necessarily support arithmetic on intervals. Interval and ratio scales provide meaningful distances between values and hence support gradient-like interpretations in optimization or gradient-based modeling, whereas nominal scales do not.

In optimization and machine learning, gradient-based methods require loss functions that are differentiable almost everywhere or admit subgradients. The term gradability is often used to describe whether a function is amenable to gradient-based optimization.

Other uses include material science and geology, where gradation describes the distribution of particle sizes in soils or aggregates. Gradability in this sense relates to how finely graded or well-graded the material is, influencing mechanical behavior.

Because gradability is not a single formal term, its precise meaning depends on context and discipline.