Convolutional network are the de-facto standard for analysing spatio-temporal data such as images, videos, 3D shapes, etc. Whilst some of this data is naturally dense (for instance, photos), many other data sources are inherently sparse. Examples include penstrokes forming on a piece of paper, or (colored) 3D point clouds that were obtained using a LiDAR scanner or RGB-D camera. Standard “dense...

A calibration ¢ is a differential form on a Riemannian manifold with two additional properties. First, the form should be closed under exterior differentiation. Second, it should be less than or equal to the volume form on each oriented submanifold (of the same dimension as the degree of the form ¢). Each calibration ¢ determines a geometry of submanifolds, namely those oriented submanifolds fo...

Cieliebak, Mundet i Riera and Salamon recently formulated a definition of branched submanifold of Euclidean space in connection with their discussion of multivalued sections and the Euler class. This note proposes an intrinsic definition of a weighted branched manifold Z that is obtained from the usual definition of oriented orbifold groupoid by relaxing the properness condition and adding a we...

A calibration ¢ is a differential form on a Riemannian manifold with two additional properties. First, the form should be closed under exterior differentiation. Second, it should be less than or equal to the volume form on each oriented submanifold (of the same dimension as the degree of the form ¢). Each calibration ¢ determines a geometry of submanifolds, namely those oriented submanifolds fo...

In [Ch1], Chern gives a generalization of projective geometry by considering foliations on the Grassman bundle of p-planes Gr(p, R) → R by p-dimensional submanifolds that are integrals of the canonical contact differential system. The equivalence method yields an sl(n + 1, R)valued Cartan connection whose curvature captures the geometry of such foliation. In the flat case, the space of leaves o...

Twenty years ago it was not obvious that subspace approximations would be such a successful representation for faces and other phenomena whose measurement-space manifolds exhibit clear nonlinearities. Now PCA is ubiquitous in computer vision and, although its globally linear view of the data manifold can be a liability as systems scale up, it is not obvious that one can reliably construct a bet...

We consider a sequence of quantum states, $\rho_N$, associated with submanifold $\Lambda$ product two integral compact Kahler manifolds. show that, when is submanifold, then, in the semiclassical limit, these states are not entangled. discuss whether geometry (specifically being submanifold) has any relationship entanglement properties $\rho_N$.

We characterize the boundaries of positive holomorphic chains in an arbitrary complex manifold. We then consider a compact oriented real submanifold of dimension 2p − 1 in a compact Kähler manifold X and address the question of which relative homology classes inH2p(X,M ; Z) are represented by positive holomorphic chains. Specifically, we define what it means for a class τ ∈ H2p(X,M ; Z) to be o...