Tensetheshifted

Tensetheshifted is a term used in tensor algebra to describe a modified tensor obtained by applying a fixed index shift to one or more modes of a tensor. It refers to the action of a shift operator that reindexes the entries of a tensor along specified dimensions, producing a new tensor that preserves the overall shape but repositions the data within it.

Construction and formal definition: Let T be an order-m tensor with dimensions n1 x n2 x ... x nm. A tensetheshifted tensor S is defined by a shift vector d = (d1, ..., dm), where each di is an integer in the range 0 to ni−1. The components of S are given by S_{i1,...,im} = T_{i1+d1 mod n1, ..., im+dm mod nm}, using modular wrap-around. Variants may use non-wrapping shifts with appropriate boundary conditions. If all di are zero, tensetheshifted(T) equals T. If the shift is invertible, the transformation is a permutation of the tensor entries.

Properties: The tensetheshifted operation is linear with respect to tensor addition, so S(T + U) = S(T) + S(U). When wrap-around is used, the operation is a cyclic permutation of indices and commutes with other index-permutation operations. The shift can be applied to a single mode or multiple modes, yielding a family of shifted tensors that differ only by index positions.

Applications: Tensetheshifted is used in signal processing, data augmentation for machine learning, and studies of invariance where shifted variants of data tensors are needed. It also appears in theoretical discussions of index-alignment transformations and in the construction of circulant or quasi-circulant tensor structures.

See also: tensor, index permutation, circulant tensor, data augmentation, shift operator. Etymology: the name combines “tensor” with “shifted,” reflecting the core operation of reindexing tensor entries. Usage notes: the term is not universally standardized and may be described by alternative phrases such as index-shifted tensor or shifted-tensor operator in different sources.