Error bounds for Lanczos-based matrix function approximation

We analyze the Lanczos method for matrix function approximation (Lanczos-FA), an iterative algorithm for computing $f(\mathbf{A}) \mathbf{b}$ when $\mathbf{A}$ is a Hermitian matrix and $\mathbf{b}$ is a given mathbftor. Assuming that $f : \mathbb{C} \rightarrow \mathbb{C}$ is piecewise analytic, we give a framework, based on the Cauchy integral formula, which can be used to derive {\em a priori} and \emph{a posteriori} error bounds for Lanczos-FA in terms of the error of Lanczos used to solve linear systems. Unlike many error bounds for Lanczos-FA, these bounds account for fine-grained properties of the spectrum of $\mathbf{A}$, such as clustered or isolated eigenvalues. Our results are derived assuming exact arithmetic, but we show that they are easily extended to finite precision computations using existing theory about the Lanczos algorithm in finite precision. We also provide generalized bounds for the Lanczos method used to approximate quadratic forms $\mathbf{b}^\textsf{H} f(\mathbf{A}) \mathbf{b}$, and demonstrate the effectiveness of our bounds with numerical experiments

Robust and Efficient Data Clustering with Signal Processing on Graphs

Data is pervasive in today's world and has actually been for quite some time. With the increasing volume of data to process, there is a need for faster and at least as accurate techniques than what we already have. In particular, the last decade recorded the effervescence of social networks and ubiquitous sensing (through smartphones and the Internet of Things). These phenomena, including also the progresses in bioinformatics and traffic monitoring, pushed forward the research on graph analysis and called for more efficient techniques. Clustering is an important field of machine learning because it belongs to the unsupervised techniques (i.e., one does not need to possess a ground truth about the data to start learning). With it, one can extract meaningful patterns from large data sources without requiring an expert to annotate a portion of the data, which can be very costly. However, the techniques of clustering designed so far all tend to be computationally demanding and have trouble scaling with the size of today's problems. The emergence of Graph Signal Processing, attempting to apply traditional signal processing techniques on graphs instead of time, provided additional tools for efficient graph analysis. By considering the clustering assignment as a signal lying on the nodes of the graph, one may now apply the tools of GSP to the improvement of graph clustering and more generally data clustering at large. In this thesis, we present several techniques using some of the latest developments of GSP in order to improve the scalability of clustering, while aiming for an accuracy resembling that of Spectral Clustering, a famous graph clustering technique that possess a solid mathematical intuition. On the one hand, we explore the benefits of random signal filtering on a practical and theoretical aspect for the determination of the eigenvectors of the graph Laplacian. In practice, this attempt requires the design of polynomial approximations of the step function for which we provided an accelerated heuristic. We used this series of work in order to reduce the complexity of dynamic graphs clustering, the problem of defining a partition to a graph which is evolving in time at each snapshot. We also used them to propose a fast method for the determination of the subspace generated by the first eigenvectors of any symmetrical matrix. This element is useful for clustering as it serves in Spectral Clustering but it goes beyond that since it also serves in graph visualization (with Laplacian Eigenmaps) and data mining (with Principal Components Projection). On the other hand, we were inspired by the latest works on graph filter localization in order to propose an extremely fast clustering technique. We tried to perform clustering by only using graph filtering and combining the results in order to obtain a partition of the nodes. These different contributions are completed by experiments using both synthetic datasets and real-world problems. Since we think that research should be shared in order to progress, all the experiments made in this thesis are publicly available on my personal Github account