We study the problem of learning from data representations that are invariant to transformations, and at the same time selective, in the sense that two points have the same representation if one is the transformation of the other. The mathematical results here sharpen some of the key claims of *i-theory*—a recent theory of feedforward processing in sensory cortex (Anselmi *et al.*, 2013, *Theor. Comput. Sci.* and arXiv:1311.4158; Anselmi *et al.*, 2013, Magic materials: a theory of deep hierarchical architectures for learning sensory representations. *CBCL Paper*; Anselmi & Poggio, 2010, Representation learning in sensory cortex: a theory. *CBMM Memo No.* 26).