Foldins

Foldins are a hypothetical class of reversible folding transforms used to represent high-dimensional data in a compressed, invertible form. The concept treats folding as a sequence of reflections across chosen hyperplanes, producing a folded representation that retains sufficient information for reconstruction under suitable conditions. The term foldin combines folding and embedding, reflecting its role in mapping a space into a folded, lower-entropy form.

Construction: Let X be a dataset in R^d. A foldin of depth k is defined by a sequence of k folding operations along selected coordinate axes or directions. Each operation maps a subregion of the current space onto an overlapping region via a reflection and a possibly orientation-preserving transform. The resulting folded coordinates serve as the representation. Inversion is possible by applying the reverse sequence of folds.

Properties: For certain choices of folds, foldins are invertible and can preserve distances up to a distortion bound. Overlaps between folded regions introduce redundancy which can be controlled by the folding plan. Variants vary in whether folds are linear, nonlinear, deterministic, or probabilistic.

Variants: Linear foldins use piecewise linear folding functions; nonlinear foldins employ smooth or piecewise smooth transformations; probabilistic foldins select folds by a stochastic process, enabling randomized representations that may improve robustness to noise.

Applications: Foldins appear in discussions of dimensionality reduction, data compression, and shape representation. In theory, they offer a simple mechanism to encode spatial information; in practice, they are studied as a conceptual tool or as a component of broader folding-based schemes.

See also: folding, isometry, dimensionality reduction, space-partitioning, reflections.