While the universal approximation property holds both for hierarchical and shallow networks, deep networks can approximate the class of compositional functions as well as shallow networks but with exponentially lower number of training parameters and sample complexity. Compositional functions are obtained as a hierarchy of local constituent functions, where "local functions'' are functions with low dimensionality. This theorem proves an old conjecture by Bengio on the role of depth in networks, characterizing precisely the conditions under which it holds. It also suggests possible answers to the the puzzle of why high-dimensional deep networks trained on large training sets often do not seem to show overfit.

Image instance retrieval is the problem of retrieving images from a database which contain the same object. Convolutional Neural Network (CNN) based descriptors are becoming the dominant approach for generating global image descriptors for the instance retrieval problem. One major drawback of CNN-based global descriptors is that uncompressed deep neural network models require hundreds of megabytes of storage making them inconvenient to deploy in mobile applications or in custom hardware. In this work, we study the problem of neural network model compression focusing on the image instance retrieval task. We study quantization, coding, pruning and weight sharing techniques for reducing model size for the instance retrieval problem. We provide extensive experimental results on the trade-off between retrieval performance and model size for different types of networks on several data sets providing the most comprehensive study on this topic. We compress models to the order of a few MBs: two orders of magnitude smaller than the uncompressed models while achieving negligible loss in retrieval performance1.

Gradient backpropagation (BP) requires symmetric feedforward and feedback connections — the same weights must be used for forward and backward passes. This "weight transport problem'' (Grossberg 1987) is thought to be one of the main reasons to doubt BP's biologically plausibility. Using 15 different classification datasets, we systematically investigate to what extent BP really depends on weight symmetry. In a study that turned out to be surprisingly similar in spirit to Lillicrap et al.'s demonstration (Lillicrap et al. 2014) but orthogonal in its results, our experiments indicate that: (1) the magnitudes of feedback weights do not matter to performance (2) the signs of feedback weights do matter — the more concordant signs between feedforward and their corresponding feedback connections, the better (3) with feedback weights having random magnitudes and 100% concordant signs, we were able to achieve the same or even better performance than SGD. (4) some normalizations/stabilizations are indispensable for such asymmetric BP to work, namely Batch Normalization (BN) (Ioffe and Szegedy 2015) and/or a "Batch Manhattan'' (BM) update rule.

Is visual cortex made up of general-purpose information processing machinery, or does it consist of a collection of specialized modules? If prior knowledge, acquired from learning a set of objects is only transferable to new objects that share properties with the old, then the recognition system's optimal organization must be one containing specialized modules for different object classes. Our analysis starts from a premise we call the invariance hypothesis: that the computational goal of the ventral stream is to compute an invariant-to-transformations and discriminative signature for recognition. The key condition enabling approximate transfer of invariance without sacrificing discriminability turns out to be that the learned and novel objects transform similarly. This implies that the optimal recognition system must contain subsystems trained only with data from similarly-transforming objects and suggests a novel interpretation of domain-specific regions like the fusiform face area (FFA). Furthermore, we can define an index of transformation-compatibility, computable from videos, that can be combined with information about the statistics of natural vision to yield predictions for which object categories ought to have domain-specific regions in agreement with the available data. The result is a unifying account linking the large literature on view-based recognition with the wealth of experimental evidence concerning domain-specific regions.

One approach to computer object recognition and modeling the brain's ventral stream involves unsupervised learning of representations that are invariant to common transformations. However, applications of these ideas have usually been limited to 2D affine transformations, e.g., translation and scaling, since they are easiest to solve via convolution. In accord with a recent theory of transformation-invariance, we propose a model that, while capturing other common convolutional networks as special cases, can also be used with arbitrary identity-preserving transformations. The model's wiring can be learned from videos of transforming objects---or any other grouping of images into sets by their depicted object. Through a series of successively more complex empirical tests, we study the invariance/discriminability properties of this model with respect to different transformations. First, we empirically confirm theoretical predictions for the case of 2D affine transformations. Next, we apply the model to non-affine transformations: as expected, it performs well on face verification tasks requiring invariance to the relatively smooth transformations of 3D rotation-in-depth and changes in illumination direction. Surprisingly, it can also tolerate clutter transformations'' which map an image of a face on one background to an image of the same face on a different background. Motivated by these empirical findings, we tested the same model on face verification benchmark tasks from the computer vision literature: Labeled Faces in the Wild, PubFig and a new dataset we gathered---achieving strong performance in these highly unconstrained cases as well."