Locally Stationary Graph Processes

Stationary graph process models are commonly used in the analysis and inference of data sets collected on irregular network topologies. While most of the existing methods represent graph signals with a single stationary process model that is globally valid on the entire graph, in many practical problems, the characteristics of the process may be subject to local variations in different regions of the graph. In this work, we propose a locally stationary graph process (LSGP) model that aims to extend the classical concept of local stationarity to irregular graph domains. We characterize local stationarity by expressing the overall process as the combination of a set of component processes such that the extent to which the process adheres to each component varies smoothly over the graph. We propose an algorithm for computing LSGP models from realizations of the process, and also study the approximation of LSGPs locally with WSS processes. Experiments on signal interpolation problems show that the proposed process model provides accurate signal representations competitive with the state of the art.

Learning Graph ARMA Processes from Time-Vertex Spectra

The modeling of time-varying graph signals as stationary time-vertex stochastic processes permits the inference of missing signal values by efficiently employing the correlation patterns of the process across different graph nodes and time instants. In this study, we first propose an algorithm for computing graph autoregressive moving average (graph ARMA) processes based on learning the joint time-vertex power spectral density of the process from its incomplete realizations. Our solution relies on first roughly estimating the joint spectrum of the process from partially observed realizations and then refining this estimate by projecting it onto the spectrum manifold of the ARMA process. We then present a theoretical analysis of the sample complexity of learning graph ARMA processes. Experimental results show that the proposed approach achieves improvement in the time-vertex signal estimation performance in comparison with reference approaches in the literature.

Learning Multi-Modal Nonlinear Embeddings: Performance Bounds and an Algorithm

While many approaches exist in the literature to learn representations for data collections in multiple modalities, the generalizability of the learnt representations to previously unseen data is a largely overlooked subject. In this work, we first present a theoretical analysis of learning multi-modal nonlinear embeddings in a supervised setting. Our performance bounds indicate that for successful generalization in multi-modal classification and retrieval problems, the regularity of the interpolation functions extending the embedding to the whole data space is as important as the between-class separation and cross-modal alignment criteria. We then propose a multi-modal nonlinear representation learning algorithm that is motivated by these theoretical findings, where the embeddings of the training samples are optimized jointly with the Lipschitz regularity of the interpolators. Experimental comparison to recent multi-modal and single-modal learning algorithms suggests that the proposed method yields promising performance in multi-modal image classification and cross-modal image-text retrieval applications.

Graph Domain Adaptation with Localized Graph Signal Representations

In this paper we propose a domain adaptation algorithm designed for graph domains. Given a source graph with many labeled nodes and a target graph with few or no labeled nodes, we aim to estimate the target labels by making use of the similarity between the characteristics of the variation of the label functions on the two graphs. Our assumption about the source and the target domains is that the local behaviour of the label function, such as its spread and speed of variation on the graph, bears resemblance between the two graphs. We estimate the unknown target labels by solving an optimization problem where the label information is transferred from the source graph to the target graph based on the prior that the projections of the label functions onto localized graph bases be similar between the source and the target graphs. In order to efficiently capture the local variation of the label functions on the graphs, spectral graph wavelets are used as the graph bases. Experimentation on various data sets shows that the proposed method yields quite satisfactory classification accuracy compared to reference domain adaptation methods.

Mask Combination of Multi-layer Graphs for Global Structure Inference

Structure inference is an important task for network data processing and analysis in data science. In recent years, quite a few approaches have been developed to learn the graph structure underlying a set of observations captured in a data space. Although real world data is often acquired in settings where relationships are influenced by a priori known rules, this domain knowledge is still not well exploited in structure inference problems. In this paper, we identify the structure of signals defined in a data space whose inner relationships are encoded by multi-layer graphs. We aim at properly exploiting the information originating from each layer to infer the global structure underlying the signals. We thus present a novel method for combining the multiple graphs into a global graph using mask matrices, which are estimated through an optimization problem that accommodates the multi-layer graph information and a signal representation model. The proposed mask combination method also estimates the contribution of each graph layer in the structure of signals. The experiments conducted both on synthetic and real world data suggest that integrating the multi-layer graph representation of the data in the structure inference framework enhances the learning procedure considerably by adapting to the quality and the quantity of the input data

Domain Adaptation on Graphs by Learning Graph Topologies: Theoretical Analysis and an Algorithm

Traditional machine learning algorithms assume that the training and test data have the same distribution, while this assumption can be easily violated in real applications. Learning by taking into account the changes in the data distribution is called domain adaptation. In this work, we treat the domain adaptation problem in a graph setting. We consider a source and a target data graph that are constructed with samples drawn from a source and a target data manifold. We study the problem of estimating the unknown labels on the target graph by employing the label information in the source graph and the similarity between the two graphs. We particularly focus on a setting where the target label function is learnt such that its spectrum (frequency content when regarded as a graph signal) is similar to that of the source label function. We first present an overview of the recent field of graph signal processing and introduce concepts such as the Fourier transform on graphs. We then propose a theoretical analysis of domain adaptation over graphs, and present performance bounds relating the target classification error to the properties of the graph topologies and the manifold geometries. Finally, we propose a graph domain adaptation algorithm inspired by our theoretical findings, which estimates the label functions while learning the source and target graph topologies at the same time. Experiments on synthetic and real data sets suggest that the proposed method outperforms baseline approaches.

Nonlinear Supervised Dimensionality Reduction via Smooth Regular Embeddings

The recovery of the intrinsic geometric structures of data collections is an important problem in data analysis. Supervised extensions of several manifold learning approaches have been proposed in the recent years. Meanwhile, existing methods primarily focus on the embedding of the training data, and the generalization of the embedding to initially unseen test data is rather ignored. In this work, we build on recent theoretical results on the generalization performance of supervised manifold learning algorithms. Motivated by these performance bounds, we propose a supervised manifold learning method that computes a nonlinear embedding while constructing a smooth and regular interpolation function that extends the embedding to the whole data space in order to achieve satisfactory generalization. The embedding and the interpolator are jointly learnt such that the Lipschitz regularity of the interpolator is imposed while ensuring the separation between different classes. Experimental results on several image data sets show that the proposed method outperforms traditional classifiers and the supervised dimensionality reduction algorithms in comparison in terms of classification accuracy in most settings.

Domain Adaptation on Graphs by Learning Aligned Graph Bases

We propose a method for domain adaptation on graphs. Given sufficiently many observations of the label function on a source graph, we study the problem of transferring the label information from the source graph to a target graph for estimating the target label function. Our assumption about the relation between the two domains is that the frequency content of the label function, regarded as a graph signal, has similar characteristics over the source and the target graphs. We propose a method to learn a pair of coherent bases on the two graphs, such that the corresponding source and target graph basis vectors have similar spectral content, while "aligning" the two graphs at the same time so that the reconstructed source and target label functions have similar coefficients over the bases. Experiments on several types of data sets suggest that the proposed method compares quite favorably to reference domain adaptation methods. To the best of our knowledge, our treatment is the first to study the domain adaptation problem in a purely graph-based setting with no need for embedding the data in an ambient space. This feature is particularly convenient for many problems of interest concerning learning on graphs or networks.

Learning Discriminative Multilevel Structured Dictionaries for Supervised Image Classification

Sparse representations using overcomplete dictionaries have proved to be a powerful tool in many signal processing applications such as denoising, super-resolution, inpainting, compression or classification. The sparsity of the representation very much depends on how well the dictionary is adapted to the data at hand. In this paper, we propose a method for learning structured multilevel dictionaries with discriminative constraints to make them well suited for the supervised pixelwise classification of images. A multilevel tree-structured discriminative dictionary is learnt for each class, with a learning objective concerning the reconstruction errors of the image patches around the pixels over each class-representative dictionary. After the initial assignment of the class labels to image pixels based on their sparse representations over the learnt dictionaries, the final classification is achieved by smoothing the label image with a graph cut method and an erosion method. Applied to a common set of texture images, our supervised classification method shows competitive results with the state of the art.

A study of the classification of low-dimensional data with supervised manifold learning

Supervised manifold learning methods learn data representations by preserving the geometric structure of data while enhancing the separation between data samples from different classes. In this work, we propose a theoretical study of supervised manifold learning for classification. We consider nonlinear dimensionality reduction algorithms that yield linearly separable embeddings of training data and present generalization bounds for this type of algorithms. A necessary condition for satisfactory generalization performance is that the embedding allow the construction of a sufficiently regular interpolation function in relation with the separation margin of the embedding. We show that for supervised embeddings satisfying this condition, the classification error decays at an exponential rate with the number of training samples. Finally, we examine the separability of supervised nonlinear embeddings that aim to preserve the low-dimensional geometric structure of data based on graph representations. The proposed analysis is supported by experiments on several real data sets.