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Complexity theory of circuits strongly suggests that deep architectures can be much more efficient (sometimes exponentially) than shallow architectures, in terms of computational elements required to represent some functions. Deep multi-layer neural networks have many levels of non-linearities allowing them to compactly represent highly non-linear and highly-varying functions. However, until recently it was not clear how to train such deep networks, since gradient-based optimization starting from random initialization appears to often get stuck in poor solutions. Hinton et al. recently introduced a greedy layer-wise unsupervised learning algorithm for Deep Belief Networks (DBN), a generative model with many layers of hidden causal variables. In the context of the above optimization problem, we study this algorithm empirically and explore variants to better understand its success and extend it to cases where the inputs are continuous or where the structure of the input distribution is not revealing enough about the variable to be predicted in a supervised task. Our experiments also confirm the hypothesis that the greedy layer-wise unsupervised training strategy mostly helps the optimization, by initializing weights in a region near a good local minimum, giving rise to internal distributed representations that are high-level abstractions of the input, bringing better generalization.

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Abstract. We propose the study of a Monge-Ampère-type equation in bidegree (n−1, n−1) rather than (1, 1) on a compact complex manifold X of dimension n for which we prove ellipticity and uniqueness of the solution subject to positivity and normalisation restrictions. Existence will hopefully be dealt with in future work. The aim is to construct a special Gauduchon metric uniquely associated with any Aeppli cohomology class of bidegree (n−1, n−1) lying in the Gauduchon cone of X that we hereby introduce as a subset of the real Aeppli cohomology group of type (n−1, n−1) and whose first properties we study. Two directions for applications of this new equation are envisaged : to moduli spaces of Calabi-Yau ∂∂-manifolds and to a further study of the deformation properties of the Gauduchon cone beyond those given in this paper.