1. A computer-implemented method to retarget facial expressions by providing a parameter-parallel retargeting space between an input facial expression and an output facial expression and via a plurality of facial layers, the computer-implemented method comprising: receiving input representing a facial expression of a first character; generating a plurality of facial layers for the first character, based on the received input and a composition function and by operation of one or more computer processors, wherein the plurality of facial layers includes a plurality of parameters extracted from the received input, the plurality of parameters including a simplex coefficient common to the plurality of facial layers and weighted by a respective measure of influence for each of the plurality of facial layers; wherein the simplex coefficient is extracted by an optimization operation that includes determining a set of inputs for which a given function attains a minimum value, wherein the optimization operation is subject to a set of constraints and is based on a sequential quadratic programming algorithm; wherein each facial layer encodes, in a simplicial basis, one or more semantically significant aspects of the facial expression of the first character; and wherein a simplex is formed from components of each simplicial basis; and generating, for a second character different from the first character in appearance, a facial expression corresponding to the facial expression of the first character, based on the plurality of facial layers.