Geometric Wavelet Scattering Networks on Compact Riemannian Manifolds

The Euclidean scattering transform was introduced nearly a decade ago to improve the mathematical understanding of convolutional neural networks. Inspired by recent interest in geometric deep learning, which aims to generalize convolutional neural networks to manifold and graph-structured domains, we define a geometric scattering transform on manifolds. Similar to the Euclidean scattering transform, the geometric scattering transform is based on a cascade of wavelet filters and pointwise nonlinearities. It is invariant to local isometries and stable to certain types of diffeomorphisms. Empirical results demonstrate its utility on several geometric learning tasks. Our results generalize the deformation stability and local translation invariance of Euclidean scattering, and demonstrate the importance of linking the used filter structures to the underlying geometry of the data.Comment: 35 pages; 3 figures; 2 tables; v3: Revisions based on reviewer comment

On-Shell Structures of MHV Amplitudes Beyond the Planar Limit

We initiate an exploration of on-shell functions in $\mathcal{N}=4$ SYM beyond the planar limit by providing compact, combinatorial expressions for all leading singularities of MHV amplitudes and showing that they can always be expressed as a positive sum of differently ordered Parke-Taylor tree amplitudes. This is understood in terms of an extended notion of positivity in $G(2,n)$, the Grassmannian of 2-planes in $n$ dimensions: a single on-shell diagram can be associated with many different "positive" regions, of which the familiar positive region associated with planar diagrams is just one example. The decomposition into Parke-Taylor factors is simply a "triangulation" of these extended positive regions. The $U(1)$ decoupling and Kleiss-Kuijf (KK) relations satisfied by the Parke-Taylor amplitudes also follow naturally from this geometric picture. These results suggest that non-planar MHV amplitudes in $\mathcal{N}=4$ SYM at all loop orders can be expressed as a sum of polylogarithms weighted by color factors and (unordered) Parke-Taylor amplitudes.Comment: 16 pages, 19 figure

Geometric deep learning: going beyond Euclidean data

Many scientific fields study data with an underlying structure that is a non-Euclidean space. Some examples include social networks in computational social sciences, sensor networks in communications, functional networks in brain imaging, regulatory networks in genetics, and meshed surfaces in computer graphics. In many applications, such geometric data are large and complex (in the case of social networks, on the scale of billions), and are natural targets for machine learning techniques. In particular, we would like to use deep neural networks, which have recently proven to be powerful tools for a broad range of problems from computer vision, natural language processing, and audio analysis. However, these tools have been most successful on data with an underlying Euclidean or grid-like structure, and in cases where the invariances of these structures are built into networks used to model them. Geometric deep learning is an umbrella term for emerging techniques attempting to generalize (structured) deep neural models to non-Euclidean domains such as graphs and manifolds. The purpose of this paper is to overview different examples of geometric deep learning problems and present available solutions, key difficulties, applications, and future research directions in this nascent field

Stratifying On-Shell Cluster Varieties: the Geometry of Non-Planar On-Shell Diagrams

The correspondence between on-shell diagrams in maximally supersymmetric Yang-Mills theory and cluster varieties in the Grassmannian remains largely unexplored beyond the planar limit. In this article, we describe a systematic program to survey such 'on-shell varieties', and use this to provide a complete classification in the case of $G(3,6)$. In particular, we find exactly 24 top-dimensional varieties and 10 co-dimension one varieties in $G(3,6)$---up to parity and relabeling of the external legs. We use this case to illustrate some of the novelties found for non-planar varieties relative to the case of positroids, and describe some of the features that we expect to hold more generally.Comment: 35 pages, 70 figures, and 1 table; also included is a file with explicit details for our classification. Signs corrected in two residue theorems, and a new interpretation (and formula) given for the las