patternsbetween

Patternsbetween is a term used in pattern analysis to denote the collection of intermediate patterns that lie between two given patterns in a defined space. It is a conceptual tool rather than a fixed algorithm, applicable in domains such as data mining, machine learning, and design optimization. The idea is to formalize what it means for one pattern to exist between another two in terms of similarity, progression, or transformation.

Definition: Given patterns P and Q, patternsbetween(P, Q) represents the set of patterns R that are considered between P and Q according to a chosen criterion. This criterion may be a distance metric, a path on a manifold, or a sequence of incremental transformations. The precise definition depends on the representation of patterns and the goals of the analysis.

Formalization: In vector spaces, a common instantiation is to take R = (1−t)P + tQ for t in [0,1], yielding a linear interpolation of the two patterns. In non-Euclidean spaces, one uses geodesics or other interpolation paths. In discrete domains, patternsbetween may be the set of stepwise transitions that connect P to Q through valid intermediate states.

Computation: Computationally, constructing patternsbetween requires a representation for patterns, a metric or topology, and an interpolation method. Practitioners often sample a finite sequence of parameters t to generate a tractable set of intermediate patterns for analysis or visualization.

Applications: Applications include visualizing gradual changes in time-series or images, exploring design morphings, and studying evolution of patterns in evolutionary computation. It also informs clustering or curriculum learning by revealing gradual transitions between cluster prototypes or skill states.

See also: interpolation, morphing, latent space interpolation, pattern mining, distance metrics.