View-tolerant face recognition and Hebbian learning imply mirror-symmetric neural tuning to head orientation

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 relatively robust against identity-preserving transformations like depth-rotations. 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. While simulations of these models recapitulate the ventral stream's progression from early view-specific to late view-tolerant representations, they fail to generate the most salient property of the intermediate representation for faces found in the brain: mirror-symmetric tuning of the neural population to head orientation. Here we prove that a class of hierarchical architectures and a broad set of biologically plausible learning rules can provide approximate invariance at the top level of the network. While most of the learning rules do not yield mirror-symmetry in the mid-level representations, we characterize a specific biologically-plausible Hebb-type learning rule that is guaranteed to generate mirror-symmetric tuning to faces tuning at intermediate levels of the architecture

The computational magic of the ventral stream

I argue that the sample complexity of (biological, feedforward) object recognition is mostly due to geometric image transformations and conjecture that a main goal of the ventral stream – V1, V2, V4 and IT – is to learn-and-discount image transformations.

In the first part of the paper I describe a class of simple and biologically plausible memory-based modules that learn transformations from unsupervised visual experience. The main theorems show that these modules provide (for every object) a signature which is invariant to local affine transformations and approximately invariant for other transformations. I also prove that,
in a broad class of hierarchical architectures, signatures remain invariant from layer to layer. The identification of these memory-based modules with complex (and simple) cells in visual areas leads to a theory of invariant recognition for the ventral stream.

In the second part, I outline a theory about hierarchical architectures that can learn invariance to transformations. I show that the memory complexity of learning affine transformations is drastically reduced in a hierarchical architecture that factorizes transformations in terms of the subgroup of translations and the subgroups of rotations and scalings. I then show how translations are automatically selected as the only learnable transformations during development by enforcing small apertures – eg small receptive fields – in the first layer.

In a third part I show that the transformations represented in each area can be optimized in terms of storage and robustness, as a consequence determining the tuning of the neurons in the area, rather independently (under normal conditions) of the statistics of natural images. I describe a model of learning that can be proved to have this property, linking in an elegant way the spectral properties of the signatures with the tuning of receptive fields in different areas. A surprising implication of these theoretical results is that the computational goals and some of the tuning properties of cells in the ventral stream may follow from symmetry properties (in the sense of physics) of the visual world through a process of unsupervised correlational learning, based on Hebbian synapses. In particular, simple and complex cells do not directly care about oriented bars: their tuning is a side effect of their role in translation invariance. Across the whole ventral stream the preferred features reported for neurons in different areas are only a symptom of the invariances computed and represented.

The results of each of the three parts stand on their own independently of each other. Together this theory-in-fieri makes several broad predictions, some of which are:

-invariance to small transformations in early areas (eg translations in V1) may underly stability of visual perception (suggested by Stu Geman);

-each cell’s tuning properties are shaped by visual experience of image transformations during developmental and adult plasticity;

-simple cells are likely to be the same population as complex cells, arising from different convergence of the Hebbian learning rule. The input to complex “complex” cells are dendritic branches with simple cell properties;

-class-specific transformations are learned and represented at the top of the ventral stream hierarchy; thus class-specific modules such as faces, places and possibly body areas should exist in IT;

-the type of transformations that are learned from visual experience depend on the size of the receptive fields and thus on the area (layer in the models) – assuming that the size increases with layers;

-the mix of transformations learned in each area influences the tuning properties of the cells oriented bars in V1+V2, radial and spiral patterns in V4 up to class specific tuning in AIT (eg face tuned cells);

-features must be discriminative and invariant: invariance to transformations is the primary determinant of the tuning of cortical neurons rather than statistics of natural images.

The theory is broadly consistent with the current version of HMAX. It explains it and extend it in terms of unsupervised learning, a broader class of transformation invariance and higher level modules. The goal of this paper is to sketch a comprehensive theory with little regard for mathematical niceties. If the theory turns out to be useful there will be scope for deep mathematics, ranging from group representation tools to wavelet theory to dynamics of learning

The Computational Magic of the Ventral Stream: Towards a Theory

I conjecture that the sample complexity of object recognition is mostly due to geometric image transformations and that a main goal of the ventral stream – V1, V2, V4 and IT – is to learn-and-discount image transformations. The most surprising implication of the theory emerging from these assumptions is that the computational goals and detailed properties of cells in the ventral stream follow from symmetry properties of the visual world through a process of unsupervised correlational learning.

From the assumption of a hierarchy of areas with receptive fields of increasing size the theory predicts that the size of the receptive fields determines which transformations are learned during development and then factored out during normal processing; that the transformation represented in each area determines the tuning of the neurons in the aerea, independently of the statistics of natural images; and that class-specific transformations are learned and represented at the top of the ventral stream hierarchy.

Some of the main predictions of this theory-in-fieri are:
1. the type of transformation that are learned from visual experience depend on the size (measured in terms of wavelength) and thus on the area (layer in the models) – assuming that the aperture size increases with layers;
2. the mix of transformations learned determine the properties of the receptive fields – oriented bars in V1+V2, radial and spiral patterns in V4 up to class specific tuning in AIT (eg face tuned cells);
3. invariance to small translations in V1 may underly stability of visual perception
4. class-specific modules – such as faces, places and possibly body areas – should exist in IT to process images of object classes

How Does Our Visual System Achieve Shift and Size Invariance?

The question of shift and size invariance in the primate visual system is discussed. After a short review of the relevant neurobiology and psychophysics, a more detailed analysis of computational models is given. The two main types of networks considered are the dynamic routing circuit model and invariant feature networks, such as the neocognitron. Some specific open questions in context of these models are raised and possible solutions discussed