Union Of Manifolds Research Articles

Tensor Manifold with Tucker Rank Constraints

Low-rank tensor approximation plays a crucial role in various tensor analysis tasks ranging from science to engineering applications. There are several important problems facing low-rank tensor approximation. First, the rank of an approximating tensor is given without checking feasibility. Second, even such approximating tensors exist, however, current proposed algorithms cannot provide global optimality guarantees. In this work, we define the low-rank tensor set (LRTS) for Tucker rank which is a union of manifolds of tensors with specific Tucker rank. We propose a procedure to describe LRTS semi-algebraically and characterize the properties of this LRTS, e.g., feasibility of tensors manifold, the equations/inequations size of LRTS, algebraic dimensions, etc. Furthermore, if the cost function for tensor approximation is polynomial type, e.g., Frobenius norm, we propose an algorithm to approximate a given tensor with Tucker rank constraints and prove the global optimality of the proposed algorithm through critical sets determined by the semi-algebraic characterization of LRTS.

Axiomatic S1 Morse–Bott theory

In various situations in Floer theory, one extracts homological invariants from "Morse-Bott" data in which the "critical set" is a union of manifolds, and the moduli spaces of "flow lines" have evaluation maps taking values in the critical set. This requires a mix of analytic arguments (establishing properties of the moduli spaces and evaluation maps) and formal arguments (defining or computing invariants from the analytic data). The goal of this paper is to isolate the formal arguments, in the case when the critical set is a union of circles. Namely, we state axioms for moduli spaces and evaluation maps (encoding a minimal amount of analytical information that one needs to verify in any given Floer-theoretic situation), and using these axioms we define homological invariants. More precisely, we define a (almost) category of "Morse-Bott systems". We construct a "cascade homology" functor on this category, based on ideas of Bourgeois and Frauenfelder, which is "homotopy invariant". This machinery is used in our work on cylindrical contact homology.

On Clustering and Embedding Mixture Manifolds Using a Low Rank Neighborhood Approach

Spectra from a single intimate (nonlinear) mixture can be modeled as data points drawn from a smooth manifold. Spectral data sets containing hyperspectral observations of multiple intimate mixtures with some constituent materials in common can, therefore, be modeled as data clouds, in which each point is drawn from a union of manifolds that share a boundary. Two important steps in the processing of such data are to: 1) identify the different mixture manifolds present in the data and 2) invert the nonlinear mixing function by mapping each mixture manifold into some low-dimensional Euclidean space (manifold embedding). The present state-of-the-art algorithms for joint manifold clustering and embedding perform poorly for hyperspectral data, particularly in the embedding task. We propose a novel reconstruction-based algorithm for the improved clustering and the embedding of mixture manifolds. The algorithm attempts to reconstruct each target point as an affine combination of its nearest neighbors with an additional rank penalty on the neighborhood to ensure that only the neighbors on the same manifold as the target point are used in the reconstruction. The reconstruction matrix generated by this technique is both block diagonal and neighborhood-based, leading to improved clustering and embedding. The improved performance of the algorithm against its competitors is exhibited on a variety of simulated and real mixture data sets.

Minimal heteroclinics for a class of fourth order O.D.E. systems

We prove the existence of minimal heteroclinic orbits for a class of fourth order O.D.E. systems with variational structure. In our general set-up, the set of equilibria of these systems is a union of manifolds, and the heteroclinic orbits connect two disjoint components of this set.

Kernel Low-Rank and Sparse Graph for Unsupervised and Semi-Supervised Classification of Hyperspectral Images

In this paper, we present a graph representation that is based on the assumption that data live on a union of manifolds. Such a representation is based on sample proximities in reproducing kernel Hilbert spaces and is thus linear in the feature space and nonlinear in the original space. Moreover, it also expresses sample relationships under sparse and low-rank constraints, meaning that the resulting graph will have limited connectivity (sparseness) and that samples belonging to the same group will be likely to be connected together and not with those from other groups (low rankness). We present this graph representation as a general representation that can be then applied to any graph-based method. In the experiments, we consider the clustering of hyperspectral images and semi-supervised classification (one class and multiclass).

Bayesian Low-Rank and Sparse Nonlinear Representation for Manifold Clustering

Linear representation usually used by the optimization model about low-rankness and sparsity limits their applications to some extent. In this paper, we propose Bayesian low-rank and sparse nonlinear representation (BLSN) model exploiting nonlinear representation. Different from the optimization model, BLSN can be solved by traditional algorithm in Bayesian statistics easily without knowing the explicit mapping by kernel trick. Moreover, it can learn the parameters adaptively to choose the low-rank and sparse properties and also provides a way to enforce more properties on one quantity in a Bayesian model. Based on the observation that the data points drawn from a union of manifolds may gain more meaningful linear structure after a nonlinear mapping, we apply BLSN for manifold clustering. It can handle different problems by constructing various kernels. With respect to the case of linear manifold, known as subspace segmentation, we propose a kernel by the Veronese mapping. In addition, we also design the kernel matrices for the case of nonlinear manifold. Experimental results confirm the effectiveness and the potential of our model for manifold clustering.

Bifurcation diagrams and critical subsystems of the Kowalevski gyrostat in two constant fields

The Kowalevski gyrostat in two constant fields is known as the unique example of an integrable rigid body problem described by the Hamiltonian system with three degrees of freedom not reducible to a family of systems in fewer dimensions. The practical explicit integration of this system can hardly be obtained by the existing techniques. Then the challenging problem becomes to fulfill the qualitative investigation based on the study of the Liouville foliation of the phase space. As the first approach to topological analysis of this system we find the stratified critical set of the momentum map; this set is represented as the union of manifolds with induced almost Hamiltonian systems having less than three degrees of freedom. We obtain the equations of the bifurcation diagram in three-dimensional space. These equations have the form convenient for the classification of the bifurcation sets arising on 5-dimensional iso-energetic levels.

Sobolev problem in the complete scale of Banach Spaces

In a bounded domainG ⊂ ℝ n , whose boundary is the union of manifolds of different dimensions, we study the Sobolev problem for a properly elliptic expression of order 2m. The boundary conditions are given by linear differential expressions on manifolds of different dimensions. We study the Sobolev problem in the complete scale of Banach spaces. For this problem, we prove the theorem on a complete set of isomorphisms and indicate its applications.