arXiv subjects

Antonio Ortega

Publications and source records attributed to Antonio Ortega.

At least 109 records · Page 6Linked to original sources

arXiv2019

Deep geometric knowledge distillation with graphs

In most cases deep learning architectures are trained disregarding the amount of operations and energy consumption. However, some applications, like embedded systems, can be resource-constrained during inference. A popular approach to reduce the size of a deep learning architecture consists in distilling knowledge from a bigger network (teacher) to a smaller one (student). Directly training the student to mimic the teacher representation can be effective, but it requires that both share the same latent space dimensions. In this work, we focus instead on relative knowledge distillation (RKD), which considers the geometry of the respective latent spaces, allowing for dimension-agnostic transfer of knowledge. Specifically we introduce a graph-based RKD method, in which graphs are used to capture the geometry of latent spaces. Using classical computer vision benchmarks, we demonstrate the ability of the proposed method to efficiently distillate knowledge from the teacher to the student, leading to better accuracy for the same budget as compared to existing RKD alternatives.

cs.LG↗

arXiv2019

Structural Robustness for Deep Learning Architectures

Deep Networks have been shown to provide state-of-the-art performance in many machine learning challenges. Unfortunately, they are susceptible to various types of noise, including adversarial attacks and corrupted inputs. In this work we introduce a formal definition of robustness which can be viewed as a localized Lipschitz constant of the network function, quantified in the domain of the data to be classified. We compare this notion of robustness to existing ones, and study its connections with methods in the literature. We evaluate this metric by performing experiments on various competitive vision datasets.

cs.LG↗

arXiv2019

Fast Graph Fourier Transforms Based on Graph Symmetry and Bipartition

The graph Fourier transform (GFT) is an important tool for graph signal processing, with applications ranging from graph-based image processing to spectral clustering. However, unlike the discrete Fourier transform, the GFT typically does not have a fast algorithm. In this work, we develop new approaches to accelerate the GFT computation. In particular, we show that Haar units (Givens rotations with angle $π/4$) can be used to reduce GFT computation cost when the graph is bipartite or satisfies certain symmetry properties based on node pairing. We also propose a graph decomposition method based on graph topological symmetry, which allows us to identify and exploit butterfly structures in stages. This method is particularly useful for graphs that are nearly regular or have some specific structures, e.g., line graphs, cycle graphs, grid graphs, and human skeletal graphs. Though butterfly stages based on graph topological symmetry cannot be used for general graphs, they are useful in applications, including video compression and human action analysis, where symmetric graphs, such as symmetric line graphs and human skeletal graphs, are used. Our proposed fast GFT implementations are shown to reduce computation costs significantly, in terms of both number of operations and empirical runtimes.

eess.SP↗

arXiv2019

Introducing Graph Smoothness Loss for Training Deep Learning Architectures

We introduce a novel loss function for training deep learning architectures to perform classification. It consists in minimizing the smoothness of label signals on similarity graphs built at the output of the architecture. Equivalently, it can be seen as maximizing the distances between the network function images of training inputs from distinct classes. As such, only distances between pairs of examples in distinct classes are taken into account in the process, and the training does not prevent inputs from the same class to be mapped to distant locations in the output domain. We show that this loss leads to similar performance in classification as architectures trained using the classical cross-entropy, while offering interesting degrees of freedom and properties. We also demonstrate the interest of the proposed loss to increase robustness of trained architectures to deviations of the inputs.

cs.LG↗

arXiv2019

A Sampling Theory Perspective of Graph-based Semi-supervised Learning

Graph-based methods have been quite successful in solving unsupervised and semi-supervised learning problems, as they provide a means to capture the underlying geometry of the dataset. It is often desirable for the constructed graph to satisfy two properties: first, data points that are similar in the feature space should be strongly connected on the graph, and second, the class label information should vary smoothly with respect to the graph, where smoothness is measured using the spectral properties of the graph Laplacian matrix. Recent works have justified some of these smoothness conditions by showing that they are strongly linked to the semi-supervised smoothness assumption and its variants. In this work, we reinforce this connection by viewing the problem from a graph sampling theoretic perspective, where class indicator functions are treated as bandlimited graph signals (in the eigenvector basis of the graph Laplacian) and label prediction as a bandlimited reconstruction problem. Our approach involves analyzing the bandwidth of class indicator signals generated from statistical data models with separable and nonseparable classes. These models are quite general and mimic the nature of most real-world datasets. Our results show that in the asymptotic limit, the bandwidth of any class indicator is also closely related to the geometry of the dataset. This allows one to theoretically justify the assumption of bandlimitedness of class indicator signals, thereby providing a sampling theoretic interpretation of graph-based semi-supervised classification.

cs.LG↗

arXiv2019

What's in a frequency: new tools for graph Fourier Transform visualization

Recent progress in graph signal processing (GSP) has addressed a number of problems, including sampling and filtering. Proposed methods have focused on generic graphs and defined signals with certain characteristics, e.g., bandlimited signals, based on t he graph Fourier transform (GFT). However, the effect of GFT properties (e.g., vertex localization) on the behavior of such methods is not as well understood. In this paper, we propose novel GFT visualization tools and provide some examples to illustrate certain GFT properties and their impact on sampling or wavelet transforms.

eess.SP↗

arXiv2019

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

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: 1) perfect reconstruction regardless of the characteristics of the underlying graphs and graph variation operators and 2) 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.

eess.SP↗

arXiv2018

Irregularity-Aware Graph Fourier Transforms

In this paper, we present a novel generalization of the graph Fourier transform (GFT). Our approach is based on separately considering the definitions of signal energy and signal variation, leading to several possible orthonormal GFTs. Our approach includes traditional definitions of the GFT as special cases, while also leading to new GFT designs that are better at taking into account the irregular nature of the graph. As an illustration, in the context of sensor networks we use the Voronoi cell area of vertices in our GFT definition, showing that it leads to a more sensible definition of graph signal energy even when sampling is highly irregular.

eess.SP↗

arXiv2018

Laplacian Networks: Bounding Indicator Function Smoothness for Neural Network Robustness

For the past few years, Deep Neural Network (DNN) robustness has become a question of paramount importance. As a matter of fact, in sensitive settings misclassification can lead to dramatic consequences. Such misclassifications are likely to occur when facing adversarial attacks, hardware failures or limitations, and imperfect signal acquisition. To address this question, authors have proposed different approaches aiming at increasing the robustness of DNNs, such as adding regularizers or training using noisy examples. In this paper we propose a new regularizer built upon the Laplacian of similarity graphs obtained from the representation of training data at each layer of the DNN architecture. This regularizer penalizes large changes (across consecutive layers in the architecture) in the distance between examples of different classes, and as such enforces smooth variations of the class boundaries. Since it is agnostic to the type of deformations that are expected when predicting with the DNN, the proposed regularizer can be combined with existing ad-hoc methods. We provide theoretical justification for this regularizer and demonstrate its effectiveness to improve robustness of DNNs on classical supervised learning vision datasets.

cs.LG↗

arXiv2018

Graph Variogram: A novel tool to measure spatial stationarity

Irregularly sampling a spatially stationary random field does not yield a graph stationary signal in general. Based on this observation, we build a definition of graph stationarity based on intrinsic stationarity, a less restrictive definition of classical stationarity. We introduce the concept of graph variogram, a novel tool for measuring spatial intrinsic stationarity at local and global scales for irregularly sampled signals by selecting subgraphs of local neighborhoods. Graph variograms are extensions of variograms used for signals defined on continuous Euclidean space. Our experiments with intrinsically stationary signals sampled on a graph, demonstrate that graph variograms yield estimates with small bias of true theoretical models, while being robust to sampling variation of the space.

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 Signal Processing: Overview, Challenges and Applications

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. We then summarize recent developments 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. We finish by providing a brief historical perspective to highlight how concepts recently developed in GSP build on top of prior research in other areas.

eess.SP↗

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↗

arXiv2018

Mosaicked multispectral image compression based on inter- and intra-band correlation

Multispectral imaging has been utilized in many fields, but the cost of capturing and storing image data is still high. Single-sensor cameras with multispectral filter arrays can reduce the cost of capturing images at the expense of slightly lower image quality. When multispectral filter arrays are used, conventional multispectral image compression methods can be applied after interpolation, but the compressed image data after interpolation has some redundancy because the interpolated data are computed from the captured raw data. In this paper, we propose an efficient image compression method for single-sensor multispectral cameras. The proposed method encodes the captured multispectral data before interpolation. We also propose a new spectral transform method for the compression of mosaicked multispectral images. This transform is designed by considering the filter arrangement and the spectral sensitivities of a multispectral filter array. The experimental results show that the proposed method achieves a higher peak signal-to-noise ratio at higher bit rates than a conventional compression method that encodes a multispectral image after interpolation, e.g., 3-dB gain over conventional compression when coding at rates of over 0.1 bit/pixel/bands.

eess.IV↗

arXiv2017

Closed Form Solutions of Combinatorial Graph Laplacian Estimation under Acyclic Topology Constraints

How to obtain a graph from data samples is an important problem in graph signal processing. One way to formulate this graph learning problem is based on Gaussian maximum likelihood estimation, possibly under particular topology constraints. To solve this problem, we typically require iterative convex optimization solvers. In this paper, we show that when the target graph topology does not contain any cycle, then the solution has a closed form in terms of the empirical covariance matrix. This enables us to efficiently construct a tree graph from data, even if there is only a single data sample available. We also provide an error bound of the objective function when we use the same solution to approximate a cyclic graph. As an example, we consider an image denoising problem, in which for each input image we construct a graph based on the theoretical result. We then apply low-pass graph filters based on this graph. Experimental results show that the weights given by the graph learning solution lead to better denoising results than the bilateral weights under some conditions.

eess.SP↗

arXiv2017

Object-based high contrast travel time tomography

We consider travel time tomography problems involving detection of high contrast, discrete high velocity structures. This results in a discrete nonlinear inverse problem, for which traditional grid-based models and iterative linearized least-squares reconstruction algorithms are not suitable. This is because travel paths change significantly near the high contrast velocity structure, making it more difficult to inversely calculate the travel path and infer the velocity along the path. We propose a model-based approach to describe the high velocity structure using pre-defined elementary objects. Compared to a grid-based model, our approach has complexity that increases as a function of the number of objects, rather than increasing with the number of cells (usually very large). A new reconstruction algorithm is developed that provides estimates of the probability that a high velocity structure appears at any point in the region of interest. Simulation results show that our method can efficiently sample the model parameter space, and we map the model parameters into the high velocity structures in spatial domain to generate a "probability map", which represent the appearance of the high velocity structure in different regions. We show the probability map not only gives the highest probability to the optimal solution, but also includes other possible models as well.

physics.data-an↗

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↗