The A–polynomial of a manifold whose boundary consists of a single torus is generalised to an eigenvalue variety of a manifold whose boundary consists of a finite number of tori, and the set of strongly detected boundary curves is determined by Bergman’s logarithmic limit set, which describes the exponential behaviour of the eigenvalue variety at infinity. This enables one to read off the detec...

We present a convergence proof for higher order implementations of the projective integration method (PI) for a class of deterministic multi-scale systems in which fast variables quickly settle on a slow manifold. The error is shown to contain contributions associated with the length of the microsolver, the numerical accuracy of the macrosolver and the distance from the slow manifold caused by ...

We consider the multi-view data completion problem, i.e., to complete a matrix U = [U1|U2] where the ranks of U,U1, and U2 are given. In particular, we investigate the fundamental conditions on the sampling pattern, i.e., locations of the sampled entries for finite completability of such a multi-view data given the corresponding rank constraints. In contrast with the existing analysis on Grassm...

This paper presents a multi-material dual “contouring” method used to convert digital 3D voxel-based atlas of basal ganglia deformable discrete multi-surface model that supports surgical navigation for an intraoperative MRI-compatible robot, featuring fast deformation computation. It is vital the final surface maintain shared boundaries where appropriate so even as deep-brain deforms reflect ch...

Abstract Although low-rank approximation of multi-dimensional arrays has been widely discussed in linear algebra, its statistical properties remain unclear. In this paper, we use information geometry to uncover a picture non-negative approximations. First, treat each input array as probability distribution using log-linear model on poset, where structure an is realized partial order. We then de...

Deep belief networks (DBNs) have been widely applied in hyperspectral imagery (HSI) processing. However, the original DBN model fails to explore prior knowledge of training samples which limits discriminant capability extracted features for classification. In this paper, we proposed a new deep learning method, termed manifold-based multi-DBN (MMDBN), obtain manifold HSI. MMDBN designed hierarch...

If a given behavior of a multi-agent system restricts the phase variable to an invariant manifold, then we define a phase transition as a change of physical characteristics such as speed, coordination, and structure. We define such a phase transition as splitting an underlying manifold into two sub-manifolds with distinct dimensionalities around the singularity where the phase transition physic...

In this paper, we introduce a constructive rigorous numerical method to compute smooth manifolds implicitly defined by infinite dimensional nonlinear operators. We compute a simplicial triangulation of the manifold using a multi-parameter continuation method on a finite dimensional projection. The triangulation is then used to construct local charts and an atlas of the manifold in the infinite ...