Heteroassociative memory ($x^{\mu }\mapsto t^{\mu }$)

A learned map retrieves an associated output pattern that need not equal its input cue. With normalized cue vectors $x^{\mu }$ and targets $t^{\mu }$, Hebbian learning gives $W={\sum }_{\mu }{t}^{\mu }(x^{\mu }{)}^{\mathsf{T}}$. For orthonormal cues, $W{x}^{\nu }=t^{\nu }$; taking the matrix product proves exact retrieval. Nonorthogonal cues add interference ${\sum }_{\mu \ne \nu }{t}^{\mu }((x^{\mu }{)}^{\mathsf{T}}{x}^{\nu })$. Additional context is required if one identical cue should select different outputs on different occasions.