arXiv subjects

Eduardo Pavez

Publications and source records attributed to Eduardo Pavez.

48 records · Page 3Linked to original sources

arXiv2021

Learning Sparse Graph with Minimax Concave Penalty under Gaussian Markov Random Fields

This paper presents a convex-analytic framework to learn sparse graphs from data. While our problem formulation is inspired by an extension of the graphical lasso using the so-called combinatorial graph Laplacian framework, a key difference is the use of a nonconvex alternative to the $\ell_1$ norm to attain graphs with better interpretability. Specifically, we use the weakly-convex minimax concave penalty (the difference between the $\ell_1$ norm and the Huber function) which is known to yield sparse solutions with lower estimation bias than $\ell_1$ for regression problems. In our framework, the graph Laplacian is replaced in the optimization by a linear transform of the vector corresponding to its upper triangular part. Via a reformulation relying on Moreau's decomposition, we show that overall convexity is guaranteed by introducing a quadratic function to our cost function. The problem can be solved efficiently by the primal-dual splitting method, of which the admissible conditions for provable convergence are presented. Numerical examples show that the proposed method significantly outperforms the existing graph learning methods with reasonable CPU time.

eess.SP↗

arXiv2021

Cylindrical coordinates for LiDAR point cloud compression

We present an efficient voxelization method to encode the geometry and attributes of 3D point clouds obtained from autonomous vehicles. Due to the circular scanning trajectory of sensors, the geometry of LiDAR point clouds is inherently different from that of point clouds captured from RGBD cameras. Our method exploits these specific properties to representing points in cylindrical coordinates instead of conventional Cartesian coordinates. We demonstrate thatRegion Adaptive Hierarchical Transform (RAHT) can be extended to this setting, leading to attribute encoding based on a volumetric partition in cylindrical coordinates. Experimental results show that our proposed voxelization outperforms conventional approaches based on Cartesian coordinates for this type of data. We observe a significant improvement in attribute coding performance with 5-10%reduction in bitrate and octree representation with 35-45% reduction in bits.

eess.IV↗

arXiv2021

Multi-resolution intra-predictive coding of 3D point cloud attributes

We propose an intra frame predictive strategy for compression of 3D point cloud attributes. Our approach is integrated with the region adaptive graph Fourier transform (RAGFT), a multi-resolution transform formed by a composition of localized block transforms, which produces a set of low pass (approximation) and high pass (detail) coefficients at multiple resolutions. Since the transform operations are spatially localized, RAGFT coefficients at a given resolution may still be correlated. To exploit this phenomenon, we propose an intra-prediction strategy, in which decoded approximation coefficients are used to predict uncoded detail coefficients. The prediction residuals are then quantized and entropy coded. For the 8i dataset, we obtain gains up to 0.5db as compared to intra predicted point cloud compresion based on the region adaptive Haar transform (RAHT).

eess.IV↗

arXiv2021

Covariance Matrix Estimation with Non Uniform and Data Dependent Missing Observations

In this paper we study covariance estimation with missing data. We consider missing data mechanisms that can be independent of the data, or have a time varying dependency. Additionally, observed variables may have arbitrary (non uniform) and dependent observation probabilities. For each mechanism, we construct an unbiased estimator and obtain bounds for the expected value of their estimation error in operator norm. Our bounds are equivalent, up to constant and logarithmic factors, to state of the art bounds for complete and uniform missing observations. Furthermore, for the more general non uniform and dependent cases, the proposed bounds are new or improve upon previous results. Our error estimates depend on quantities we call scaled effective rank, which generalize the effective rank to account for missing observations. All the estimators studied in this work have the same asymptotic convergence rate (up to logarithmic factors).

math.ST↗

arXiv2020

Spectral folding and two-channel filter-banks on arbitrary graphs

In the past decade, several multi-resolution representation theories for graph signals have been proposed. Bipartite filter-banks stand out as the most natural extension of time domain filter-banks, in part because perfect reconstruction, orthogonality and bi-orthogonality conditions in the graph spectral domain resemble those for traditional filter-banks. Therefore, many of the well known orthogonal and bi-orthogonal designs can be easily adapted for graph signals. A major limitation is that this framework can only be applied to the normalized Laplacian of bipartite graphs. In this paper we extend this theory to arbitrary graphs and positive semi-definite variation operators. Our approach is based on a different definition of the graph Fourier transform (GFT), where orthogonality is defined with the respect to the Q inner product. We construct GFTs satisfying a spectral folding property, which allows us to easily construct orthogonal and bi-orthogonal perfect reconstruction filter-banks. We illustrate signal representation and computational efficiency of our filter-banks on 3D point clouds with hundreds of thousands of points.

eess.SP↗

arXiv2020

Region adaptive graph fourier transform for 3d point clouds

We introduce the Region Adaptive Graph Fourier Transform (RA-GFT) for compression of 3D point cloud attributes. The RA-GFT is a multiresolution transform, formed by combining spatially localized block transforms. We assume the points are organized by a family of nested partitions represented by a rooted tree. At each resolution level, attributes are processed in clusters using block transforms. Each block transform produces a single approximation (DC) coefficient, and various detail (AC) coefficients. The DC coefficients are promoted up the tree to the next (lower resolution) level, where the process can be repeated until reaching the root. Since clusters may have a different numbers of points, each block transform must incorporate the relative importance of each coefficient. For this, we introduce the $\mathbf{Q}$-normalized graph Laplacian, and propose using its eigenvectors as the block transform. The RA-GFT achieves better complexity-performance trade-offs than previous approaches. In particular, it outperforms the Region Adaptive Haar Transform (RAHT) by up to 2.5 dB, with a small complexity overhead.

cs.CV↗

arXiv2020

An efficient algorithm for graph Laplacian optimization based on effective resistances

In graph signal processing, data samples are associated to vertices on a graph, while edge weights represent similarities between those samples. We propose a convex optimization problem to learn sparse well connected graphs from data. We prove that each edge weight in our solution is upper bounded by the inverse of the distance between data features of the corresponding nodes. We also show that the effective resistance distance between nodes is upper bounded by the distance between nodal data features. Thus, our proposed method learns a sparse well connected graph that encodes geometric properties of the data. We also propose a coordinate minimization algorithm that, at each iteration, updates an edge weight using exact minimization. The algorithm has a simple and low complexity implementation based on closed form expressions.

eess.SP↗

arXiv2018

Active covariance estimation by random sub-sampling of variables

We study covariance matrix estimation for the case of partially observed random vectors, where different samples contain different subsets of vector coordinates. Each observation is the product of the variable of interest with a $0-1$ Bernoulli random variable. We analyze an unbiased covariance estimator under this model, and derive an error bound that reveals relations between the sub-sampling probabilities and the entries of the covariance matrix. We apply our analysis in an active learning framework, where the expected number of observed variables is small compared to the dimension of the vector of interest, and propose a design of optimal sub-sampling probabilities and an active covariance matrix estimation algorithm.

stat.ML↗

arXiv2018

Graph Learning from Filtered Signals: Graph System and Diffusion Kernel Identification

This paper introduces a novel graph signal processing framework for building graph-based models from classes of filtered signals. In our framework, graph-based modeling is formulated as a graph system identification problem, where the goal is to learn a weighted graph (a graph Laplacian matrix) and a graph-based filter (a function of graph Laplacian matrices). In order to solve the proposed problem, an algorithm is developed to jointly identify a graph and a graph-based filter (GBF) from multiple signal/data observations. Our algorithm is valid under the assumption that GBFs are one-to-one functions. The proposed approach can be applied to learn diffusion (heat) kernels, which are popular in various fields for modeling diffusion processes. In addition, for specific choices of graph-based filters, the proposed problem reduces to a graph Laplacian estimation problem. Our experimental results demonstrate that the proposed algorithm outperforms the current state-of-the-art methods. We also implement our framework on a real climate dataset for modeling of temperature signals.

cs.LG↗

arXiv2018

Learning Graphs with Monotone Topology Properties and Multiple Connected Components

Recent papers have formulated the problem of learning graphs from data as an inverse covariance estimation with graph Laplacian constraints. While such problems are convex, existing methods cannot guarantee that solutions will have specific graph topology properties (e.g., being $k$-partite), which are desirable for some applications. In fact, the problem of learning a graph with given topology properties, e.g., finding the $k$-partite graph that best matches the data, is in general non-convex. In this paper, we develop novel theoretical results that provide performance guarantees for an approach to solve these problems. Our solution decomposes this problem 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, i.e., one having the desired property. 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 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 (e.g., state of the art approaches for graph coloring) which are typically based on minimizing the total weight of removed edges. Since the GWE stage is formulated as an inverse covariance estimation problem with linear constraints, it can be solved using existing convex optimization methods. We demonstrate that our two step approach can achieve good results for both synthetic and texture image data.

stat.ML↗

arXiv2017

Graph Learning from Data under Structural and Laplacian 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 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 (MAP) parameter estimation of Gaussian-Markov random field (GMRF) 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.

cs.LG↗

arXiv2017

Dynamic Polygon Clouds: Representation and Compression for VR/AR

We introduce the {\em polygon cloud}, also known as a polygon set or {\em soup}, as a compressible representation of 3D geometry (including its attributes, such as color texture) intermediate between polygonal meshes and point clouds. Dynamic or time-varying polygon clouds, like dynamic polygonal meshes and dynamic point clouds, can take advantage of temporal redundancy for compression, if certain challenges are addressed. In this paper, we propose methods for compressing both static and dynamic polygon clouds, specifically triangle clouds. We compare triangle clouds to both triangle meshes and point clouds in terms of compression, for live captured dynamic colored geometry. We find that triangle clouds can be compressed nearly as well as triangle meshes, while being far more robust to noise and other structures typically found in live captures, which violate the assumption of a smooth surface manifold, such as lines, points, and ragged boundaries. We also find that triangle clouds can be used to compress point clouds with significantly better performance than previously demonstrated point cloud compression methods. In particular, for intra-frame coding of geometry, our method improves upon octree-based intra-frame coding by a factor of 5-10 in bit rate. Inter-frame coding improves this by another factor of 2-5. Overall, our dynamic triangle cloud compression improves over the previous state-of-the-art in dynamic point cloud compression by 33\% or more.

cs.GR↗