Antonio Ortega

Bio: Antonio Ortega is an academic researcher from University of Southern California. The author has contributed to research in topics: Graph (abstract data type) & Laplacian matrix. The author has an hindex of 15, co-authored 113 publications receiving 665 citations.

Eigendecomposition-Free Sampling Set Selection for Graph Signals

Abstract: This paper addresses the problem of selecting an optimal sampling set for signals on graphs The proposed sampling set selection (SSS) is based on a localization operator that can consider both vertex domain and spectral domain localizations We clarify the relationships among the proposed method, sensor position selection methods in machine learning, and existing SSS methods based on graph frequency In contrast to the alternative graph signal processing-based approaches, the proposed method does not need to compute the eigendecomposition of a variation operator, while still considering (graph) frequency information We evaluate the performance of our approach through comparisons of prediction errors and execution time

Sampling Signals on Graphs: From Theory to Applications

Abstract: The study of sampling signals on graphs, with the goal of building an analog of sampling for standard signals in the time and spatial domains, has attracted considerable attention recently. Beyond adding to the growing theory on graph signal processing (GSP), sampling on graphs has various promising applications. In this article, we review the current progress on sampling over graphs, focusing on theory and potential applications.

Two-Channel Critically Sampled Graph Filter Banks With Spectral Domain Sampling

Abstract: We propose two-channel critically-sampled filter banks for signals on undirected graphs that utilize spectral domain sampling. Unlike conventional approaches based on vertex domain sampling, our transforms have the following desirable properties: first, perfect reconstruction regardless of the characteristics of the underlying graphs and graph variation operators; and second, a symmetric structure; i.e., both analysis and synthesis filter banks are built using similar building blocks. Along with the structure of the filter banks, this paper also proves the general criterion for perfect reconstruction and theoretically shows that the vertex and spectral domain sampling coincide for a special case. The effectiveness of our approach is evaluated by comparing its performance in nonlinear approximation and denoising with various conventional graph transforms.

Sampling Signals on Graphs: From Theory to Applications

Abstract: The study of sampling signals on graphs, with the goal of building an analog of sampling for standard signals in the time and spatial domains, has attracted considerable attention recently. Beyond adding to the growing theory on graph signal processing (GSP), sampling on graphs has various promising applications. In this article, we review current progress on sampling over graphs focusing on theory and potential applications. Although most methodologies used in graph signal sampling are designed to parallel those used in sampling for standard signals, sampling theory for graph signals significantly differs from the theory of Shannon--Nyquist and shift-invariant sampling. This is due in part to the fact that the definitions of several important properties, such as shift invariance and bandlimitedness, are different in GSP systems. Throughout this review, we discuss similarities and differences between standard and graph signal sampling and highlight open problems and challenges.

Learning Graphs With Monotone Topology Properties and Multiple Connected Components

Abstract: Recent papers have formulated the problem of learning graphs from data as an inverse covariance estimation problem with graph Laplacian constraints While such problems are convex, existing methods cannot guarantee that solutions will have specific graph topology properties (eg, being a tree), which are desirable for some applications The problem of learning a graph with topology properties is in general non-convex In this paper, we propose an approach to solve these problems by decomposing them into two sub-problems for which efficient solutions are known Specifically, a graph topology inference (GTI) step is employed to select a feasible graph topology Then, a graph weight estimation (GWE) step is performed by solving a generalized graph Laplacian estimation problem, where edges are constrained by the topology found in the GTI step Our main result is a bound on the error of the GWE step as a function of the error in the GTI step This error bound indicates that the GTI step should be solved using an algorithm that approximates the data similarity matrix by another matrix whose entries have been thresholded to zero to have the desired type of graph topology The GTI stage can leverage existing methods, which are typically based on minimizing the total weight of removed edges Since the GWE stage is an inverse covariance estimation problem with linear constraints, it can be solved using existing convex optimization methods We demonstrate that our approach can achieve good results for both synthetic and texture image data

I and i

Abstract: There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality. Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

Random graphs

Abstract: We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Graph Signal Processing: Overview, Challenges, and Applications

Abstract: Research in graph signal processing (GSP) aims to develop tools for processing data defined on irregular graph domains. In this paper, we first provide an overview of core ideas in GSP and their connection to conventional digital signal processing, along with a brief historical perspective to highlight how concepts recently developed in GSP build on top of prior research in other areas. We then summarize recent advances in developing basic GSP tools, including methods for sampling, filtering, or graph learning. Next, we review progress in several application areas using GSP, including processing and analysis of sensor network data, biological data, and applications to image processing and machine learning.

Principal Manifolds and Nonlinear Dimensionality Reduction via Tangent Space Alignment

Abstract: We present a new algorithm for manifold learning and nonlinear dimensionality reduction. Based on a set of unorganized da-ta points sampled with noise from a parameterized manifold, the local geometry of the manifold is learned by constructing an approxi-mation for the tangent space at each point, and those tangent spaces are then aligned to give the global coordinates of the data pointswith respect to the underlying manifold. We also present an error analysis of our algorithm showing that reconstruction errors can bequite small in some cases. We illustrate our algorithm using curves and surfaces both in 2D/3D Euclidean spaces and higher dimension-al Euclidean spaces. We also address several theoretical and algorithmic issues for further research and improvements.