Exhaustive and Efficient Constraint Propagation: A Semi-Supervised Learning Perspective and Its Applications

This paper presents a novel pairwise constraint propagation approach by decomposing the challenging constraint propagation problem into a set of independent semi-supervised learning subproblems which can be solved in quadratic time using label propagation based on k-nearest neighbor graphs. Considering that this time cost is proportional to the number of all possible pairwise constraints, our approach actually provides an efficient solution for exhaustively propagating pairwise constraints throughout the entire dataset. The resulting exhaustive set of propagated pairwise constraints are further used to adjust the similarity matrix for constrained spectral clustering. Other than the traditional constraint propagation on single-source data, our approach is also extended to more challenging constraint propagation on multi-source data where each pairwise constraint is defined over a pair of data points from different sources. This multi-source constraint propagation has an important application to cross-modal multimedia retrieval. Extensive results have shown the superior performance of our approach.Comment: The short version of this paper appears as oral paper in ECCV 201

Diagonalizability of Constraint Propagation Matrices

In order to obtain stable and accurate general relativistic simulations, re-formulations of the Einstein equations are necessary. In a series of our works, we have proposed using eigenvalue analysis of constraint propagation equations for evaluating violation behavior of constraints. In this article, we classify asymptotical behaviors of constraint-violation into three types (asymptotically constrained, asymptotically bounded, and diverge), and give their necessary and sufficient conditions. We find that degeneracy of eigenvalues sometimes leads constraint evolution to diverge (even if its real-part is not positive), and conclude that it is quite useful to check the diagonalizability of constraint propagation matrices. The discussion is general and can be applied to any numerical treatments of constrained dynamics.Comment: 4 pages, RevTeX, one figure, added one paragraph in concluding remarks. The version to appear in Class. Quant. Grav. (Lett

Algebraic stability analysis of constraint propagation

The divergence of the constraint quantities is a major problem in computational gravity today. Apparently, there are two sources for constraint violations. The use of boundary conditions which are not compatible with the constraint equations inadvertently leads to 'constraint violating modes' propagating into the computational domain from the boundary. The other source for constraint violation is intrinsic. It is already present in the initial value problem, i.e. even when no boundary conditions have to be specified. Its origin is due to the instability of the constraint surface in the phase space of initial conditions for the time evolution equations. In this paper, we present a technique to study in detail how this instability depends on gauge parameters. We demonstrate this for the influence of the choice of the time foliation in context of the Weyl system. This system is the essential hyperbolic part in various formulations of the Einstein equations.Comment: 25 pages, 5 figures; v2: small additions, new reference, publication number, classification and keywords added, address fixed; v3: update to match journal versio