Eduardo Pavez

Generalized Laplacian precision matrix estimation for graph signal processing

Graph signal processing models high dimensional data as functions on the vertices of a graph. This theory is constructed upon the interpretation of the eigenvectors of the Laplacian matrix as the Fourier transform for graph signals. We formulate the graph learning problem as a precision matrix estimation with generalized Laplacian constraints, and we propose a new optimization algorithm. Our formulation takes a covariance matrix as input and at each iteration updates one row/column of the precision matrix by solving a non-negative quadratic program. Experiments using synthetic data with generalized Laplacian precision matrix show that our method detects the nonzero entries and it estimates its values more precisely than the graphical Lasso. For texture images we obtain graphs whose edges follow the orientation. We show our graphs are more sparse than the ones obtained using other graph learning methods.

GTT: Graph template transforms with applications to image coding

The Karhunen-Loeve transform (KLT) is known to be optimal for decorrelating stationary Gaussian processes, and it provides effective transform coding of images. Although the KLT allows efficient representations for such signals, the transform itself is completely data-driven and computationally complex. This paper proposes a new class of transforms called graph template transforms (GTTs) that approximate the KLT by exploiting a priori information known about signals represented by a graph-template. In order to construct a GTT (i) a design matrix leading to a class of transforms is defined, then (ii) a constrained optimization framework is employed to learn graphs based on given graph templates structuring a priori known information. Our experimental results show that some instances of the proposed GTTs can closely achieve the rate-distortion performance of KLT with significantly less complexity.

Graph Learning From Data Under Laplacian and Structural Constraints

Graphs are fundamental mathematical structures used in various fields to represent data, signals, and processes. In this paper, we propose a novel framework for learning/estimating graphs from data. The proposed framework includes (i) formulation of various graph learning problems, (ii) their probabilistic interpretations, and (iii) associated algorithms. Specifically, graph learning problems are posed as the estimation of graph Laplacian matrices from some observed data under given structural constraints (e.g., graph connectivity and sparsity level). From a probabilistic perspective, the problems of interest correspond to maximum a posteriori parameter estimation of Gaussian–Markov random field models, whose precision (inverse covariance) is a graph Laplacian matrix. For the proposed graph learning problems, specialized algorithms are developed by incorporating the graph Laplacian and structural constraints. The experimental results demonstrate that the proposed algorithms outperform the current state-of-the-art methods in terms of accuracy and computational efficiency.

Dynamic polygon cloud compression

We introduce a compressible representation of 3D geometry (including its attributes, such as color texture) intermediate between polygonal meshes and point clouds called a polygon cloud. Polygon clouds, compared to polygonal meshes, are more robust to live capture noise and artifacts. Furthermore, dynamic polygon clouds, compared to dynamic point clouds, are easier to compress, if certain challenges are addressed. In this paper, we propose methods for compressing dynamic polygon clouds using transform coding of color and motion residuals. We find that, compared to static polygon clouds and a fortiori static point clouds, dynamic polygon clouds can improve color compression by up to 2–3 dB in fidelity, and can improve geometry compression up to a factor of 2–5 in bit rate.

Graph learning with Laplacian constraints: Modeling attractive Gaussian Markov random fields

Graphs are fundamental mathematical structures used in various fields to represent data, signals and processes. This paper proposes a novel framework for learning graphs from data. The proposed framework (i) poses the graph learning problem as estimation of generalized graph Laplacian matrices and (ii) develops an efficient algorithm. Under specific statistical assumptions, the proposed formulation leads to modeling attractive Gaussian Markov random fields. Our experimental results show that the proposed algorithm outperforms sparse inverse covariance estimation methods in terms of graph learning performance.

Markov chain sparsification with independent sets for approximate value iteration

The ever-increasing size of wireless networks poses a significant computational challenge for policy optimization schemes. In this paper, we propose a technique to reduce the dimensionality of the value iteration problem, and thereby reduce computational complexity, by exploiting certain structural properties of the logical state transition network. Specifically, our method involves approximating the original Markov chain by a simplified one whose state transition graph contains an independent set of a prespecified size, thus resulting in a sparsification of the transition probability matrix. As a result, value iteration needs to be performed only on the vertex cover of the network, from which the value function on the independent set can be obtained in a one-step process via interpolation. The Markov chain approximation process presented in this paper, for a given choice of independent set, involves minimizing matrix distance defined in terms of Frobenius norm or the Kullback-Leibler distance. This minimum distance then helps us to define a cost that can be minimized through an iterative greedy algorithm to obtain an approximately optimal independent set. Our method provides a tradeoff between accuracy and complexity that one can exploit by choosing the size of the independent set. Numerical results show that for a class of “collision” networks the value function approximation is accurate, even with a large independent set.