Let G be a strongly connected, weighted directed graph. We define a product distance η(i, j) for pairs i, j of vertices and form the corresponding product distance matrix. We obtain a formula for the determinant and the inverse of the product distance matrix. The edge orientation matrix of a directed tree is defined and a formula for its determinant and its inverse, when it exists, is obtained....

The fused lasso, also known as (anisotropic) total variation denoising, is widely used for piecewise constant signal estimation with respect to a given undirected graph. The fused lasso estimate is highly nontrivial to compute when the underlying graph is large and has an arbitrary structure. But for a special graph structure, namely, the chain graph, the fused lasso—or simply, 1d fused lasso—c...

We define and study two new kinds of “effective resistances” based on hubs-biased – hubs-repelling hubs-attracting models navigating a graph/network. prove that these effective resistances are squared Euclidean distances between the vertices graph. They can be expressed in terms Moore–Penrose pseudoinverse Laplacian matrices analogous Kirchhoff indices graph resistance distances. several result...

Locality preserving projection (LPP) is an effective dimensionality reduction method based on manifold learning, which is defined over the graph weighted squared 2-norm distances in the projected subspace. Since squared 2-norm distance is prone to outliers, it is desirable to develop a robust LPP method. In this paper, motivated by existing studies that improve the robustness of statistical lea...

In this article we study the simultaneous estimation of the means in Poisson decomposable graphical models. We derive some classes of estimators which improve on the maximum likelihood estimator under the normalized squared losses. Our estimators are based on the argument in Chou[3] and shrink the maximum likelihood estimator depending on the marginal frequencies of variables forming a complete...