Unsupervised learning of invariant representations with low sample complexity: the magic of sensory cortex or a new framework for machine learning?

The present phase of Machine Learning is characterized by supervised learning algorithms relying on large sets of labeled examples (n → ∞). The next phase is likely to focus on algorithms capable of learning from very few labeled examples (n → ∞), like humans seem able to do. We propose an approach to this problem and describe the underlying theory, based on the unsupervised, automatic learning of a "good" representation for supervised learning, characterized by small sample complexity (n). We consider the case of visual object recognition though the theory applies to other domains. The starting point is the conjecture, proved in specific cases, that image representations which are invariant to translations, scaling and other transformations can considerably reduce the sample complexity of learning. We prove that an invariant and unique (discriminative) signature can be computed for each image patch, I, in terms of empirical distributions of the dot-products between I and a set of templates stored during unsupervised learning. A module performing filtering and pooling, like the simple and complex cells described by Hubel and Wiesel, can compute such estimates. Hierarchical architectures consisting of this basic Hubel-Wiesel moduli inherit its properties of invariance, stability, and discriminability while capturing the compositional organization of the visual world in terms of wholes and parts. The theory extends existing deep learning convolutional architectures for image and speech recognition. It also suggests that the main computational goal of the ventral stream of visual cortex is to provide a hierarchical representation of new objects/images which is invariant to transformations, stable, and discriminative for recognition|and that this representation may be continuously learned in an unsupervised way during development and visual experience.This work was supported by the Center for Brains, Minds and Machines (CBMM), funded by NSF STC award CCF - 1231216

Mini-Workshop: Deep Learning and Inverse Problems

Machine learning and in particular deep learning offer several data-driven methods to amend the typical shortcomings of purely analytical approaches. The mathematical research on these combined models is presently exploding on the experimental side but still lacking on the theoretical point of view. This workshop addresses the challenge of developing a solid mathematical theory for analyzing deep neural networks for inverse problems

Unveiling the Dynamics of the Universe

We explore the dynamics and evolution of the Universe at early and late times, focusing on both dark energy and extended gravity models and their astrophysical and cosmological consequences. Modified theories of gravity not only provide an alternative explanation for the recent expansion history of the universe, but they also offer a paradigm fundamentally distinct from the simplest dark energy models of cosmic acceleration. In this review, we perform a detailed theoretical and phenomenological analysis of different modified gravity models and investigate their consistency. We also consider the cosmological implications of well motivated physical models of the early universe with a particular emphasis on inflation and topological defects. Astrophysical and cosmological tests over a wide range of scales, from the solar system to the observable horizon, severely restrict the allowed models of the Universe. Here, we review several observational probes -- including gravitational lensing, galaxy clusters, cosmic microwave background temperature and polarization, supernova and baryon acoustic oscillations measurements -- and their relevance in constraining our cosmological description of the Universe.Comment: 94 pages, 14 figures. Review paper accepted for publication in a Special Issue of Symmetry. "Symmetry: Feature Papers 2016". V2: Matches published version, now 79 pages (new format

Topological Photonics

Topological photonics is a rapidly emerging field of research in which geometrical and topological ideas are exploited to design and control the behavior of light. Drawing inspiration from the discovery of the quantum Hall effects and topological insulators in condensed matter, recent advances have shown how to engineer analogous effects also for photons, leading to remarkable phenomena such as the robust unidirectional propagation of light, which hold great promise for applications. Thanks to the flexibility and diversity of photonics systems, this field is also opening up new opportunities to realize exotic topological models and to probe and exploit topological effects in new ways. This article reviews experimental and theoretical developments in topological photonics across a wide range of experimental platforms, including photonic crystals, waveguides, metamaterials, cavities, optomechanics, silicon photonics, and circuit QED. A discussion of how changing the dimensionality and symmetries of photonics systems has allowed for the realization of different topological phases is offered, and progress in understanding the interplay of topology with non-Hermitian effects, such as dissipation, is reviewed. As an exciting perspective, topological photonics can be combined with optical nonlinearities, leading toward new collective phenomena and novel strongly correlated states of light, such as an analog of the fractional quantum Hall effect.Comment: 87 pages, 30 figures, published versio