Agoal of central importance in the study of hierarchical models for object recogni- tion – and indeed the mammalian visual cortex – is that of understanding quantita- tively the trade-off between invariance and selectivity, and howinvariance and dis- crimination properties contribute towards providing an improved representation useful for learning from data. In this work we provide a general group-theoretic framework for characterizing and understanding invariance in a family of hierar- chical models. We show that by taking an algebraic perspective, one can provide a concise set of conditions which must be met to establish invariance, as well as a constructive prescription for meeting those conditions. Analyses in specific cases of particular relevance to computer vision and text processing are given, yielding insight into how and when invariance can be achieved. We find that the minimal intrinsic properties of a hierarchical model needed to support a particular invariance can be clearly described, thereby encouraging efficient computational implementations.
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