`-Graph has been proven to be effective in data clustering, which partitions the data space by using the sparse representation of the data as the similarity measure. However, the sparse representation is performed for each datum separately without taking into account the geometric structure of the data. Motivated by `-Graph and manifold leaning, we propose Laplacian Regularized `-Graph (LR`-Gra...

We study details of geometry noncommutative de Sitter space: we determine the Riemann and Ricci curvature tensors, energy Laplacian. find, in particular, that fuzzy space is an Einstein space, $R_{ab}=-3\zeta\,\eta_{ab}$. The Laplacian, defined frame formalism, not hermitian gives nonunitary evolution. When symmetrically ordered, it has usual quadratic form $\Delta=\Pi_a\Pi^a$ (when acting on f...

In this study, a novel procedure is presented for control and analysis of a group of autonomous agents with point mass dynamics achieving flocking motion by using a fuzzy-logic-based attractive/repulsive function. Two cooperative control laws are proposed for a group of autonomous agents to achieve flocking formations related to two different centers (mass center and geometric center) of the fl...

We focus on the multi-class segmentation problem using the piecewise constant Mumford-Shah model in a graph setting. After formulating a graph version of the Mumford-Shah energy, we propose an efficient algorithm called the MBO scheme using threshold dynamics. Theoretical analysis is developed and a Lyapunov functional is proven to decrease as the algorithm proceeds. Furthermore, to reduce the ...

An oriented hypergraph is a hypergraph where each vertex-edge incidence is given a label of +1 or −1. The adjacency and Laplacian eigenvalues of an oriented hypergraph are studied. Eigenvalue bounds for both the adjacency and Laplacian matrices of an oriented hypergraph which depend on structural parameters of the oriented hypergraph are found. An oriented hypergraph and its incidence dual are ...

Discussions about different graph Laplacians, mainly normalized and unnormalized versions of the graph Laplacians, have been ardent with respect to various methods in clustering and graph based semi-supervised learning. Previous research on the graph Laplacians investigated their convergence properties to Laplacian operators on continuous manifolds. There is still no strong proof on convergence...

Let qmin(G) stand for the smallest eigenvalue of the signless Laplacian of a graph G of order n: This paper gives some results on the following extremal problem: How large can qmin (G) be if G is a graph of order n; with no complete subgraph of order r + 1? It is shown that this problem is related to the well-known topic of making graphs bipartite. Using known classical results, several bounds ...

In recent years manifold methods have attracted a considerable amount of attention in machine learning. However most algorithms in that class may be termed “manifold-motivated” as they lack any explicit theoretical guarantees. In this paper we take a step towards closing the gap between theory and practice for a class of Laplacian-based manifold methods. These methods utilize the graph Laplacia...

Commute time embedding involves computing eigenfunctions of the graph Laplacian matrix. Spectral decomposition requires computational burden proportional to 3 ( ) O n , which may not be suitable for large scale dataset. This paper proposes computationally efficient commute time embedding by applying Nyström method to the normalized graph Laplacian. The performance of the proposed algorithms is ...

In this partly expository paper we discuss and describe some of our old and recent results on partial orders on the set G n of graphs with n vertices and m edges and some operations on graphs within G n that are monotone with respect to these partial orders. The partial orders under consideration include those related with some Laplacian characteristics of graphs as well as with some probabilis...