Perdimension

Perdimension is a term used in some theoretical and applied contexts to refer to a per-dimension quantity that describes how a property scales with the number of dimensions. It is not a standardized concept with a single definition, and its meaning can vary across disciplines.

One common informal approach is to treat perdimension as a growth-rate measure with respect to the dimension count n. For a quantity Q that depends on n dimensions, the perdimension growth rate can be defined as PD(n) = d/dn log Q(n). This captures how rapidly Q grows or shrinks as dimensions are added, abstracting away the exact form of Q and focusing on its scaling behavior. In practice, PD(n) is used to compare how different systems or algorithms respond to increasing dimensionality.

Another use treats perdimension as the vector of dimension-specific contributions to a total quantity. For a differentiable function f(x1, ..., xn), the perdimension sensitivities can be described by the partial derivatives ∂f/∂xi. Normalizing these components to reflect their shares of total variation yields a perdimension profile that highlights which dimensions drive most of the change in f.

Applications of the concept appear in high-dimensional data analysis, algorithmic benchmarking, and theoretical physics, where researchers examine how costs, errors, or densities scale with dimensionality. Because perdimension is not formally standardized, practitioners should explicitly state their definition and context when employing it.

See also: dimensional analysis, scaling, curse of dimensionality, per-coordinate analysis.