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The primate brain contains a hierarchy of visual areas, dubbed the ventral stream, which rapidly computes object representations that are both specific for object identity and robust against identity-preserving transformations, like depth rotations [1, 2]. Current computational models of object recognition, including recent deep-learning networks, generate these properties through a hierarchy of alternating selectivity-increasing filtering and tolerance-increasing pooling operations, similar to simple-complex cells operations [3–6]. Here, we prove that a class of hierarchical architectures and a broad set of biologically plausible learning rules generate approximate invariance to identity-preserving transformations at the top level of the processing hierarchy. However, all past models tested failed to reproduce the most salient property of an intermediate representation of a three-level face-processing hierarchy in the brain …