Observing and tracking bandlimited graph processes from sampled measurements

A critical challenge in graph signal processing is the sampling of bandlimited graph signals; signals that are sparse in a well-defined graph Fourier domain. Current works focused on sampling time-invariant graph signals and ignored their temporal evolution. However, time can bring new insights on sampling since sensor, biological, and financial network signals are correlated in both domains. Hence, in this work, we develop a sampling theory for time varying graph signals, named graph processes, to observe and track a process described by a linear state-space model. We provide a mathematical analysis to highlight the role of the graph, process bandwidth, and sample locations. We also propose sampling strategies that exploit the coupling between the topology and the corresponding process. Numerical experiments corroborate our theory and show the proposed methods trade well the number of samples with accuracy

Advanced data analysis for traction force microscopy and data-driven discovery of physical equations

The plummeting cost of collecting and storing data and the increasingly available computational power in the last decade have led to the emergence of new data analysis approaches in various scientific fields. Frequently, the new statistical methodology is employed for analyzing data involving incomplete or unknown information. In this thesis, new statistical approaches are developed for improving the accuracy of traction force microscopy (TFM) and data-driven discovery of physical equations. TFM is a versatile method for the reconstruction of a spatial image of the traction forces exerted by cells on elastic gel substrates. The traction force field is calculated from a linear mechanical model connecting the measured substrate displacements with the sought-for cell-generated stresses in real or Fourier space, which is an inverse and ill-posed problem. This inverse problem is commonly solved making use of regularization methods. Here, we systematically test the performance of new regularization methods and Bayesian inference for quantifying the parameter uncertainty in TFM. We compare two classical schemes, L1- and L2-regularization with three previously untested schemes, namely Elastic Net regularization, Proximal Gradient Lasso, and Proximal Gradient Elastic Net. We find that Elastic Net regularization, which combines L1 and L2 regularization, outperforms all other methods with regard to accuracy of traction reconstruction. Next, we develop two methods, Bayesian L2 regularization and Advanced Bayesian L2 regularization, for automatic, optimal L2 regularization. We further combine the Bayesian L2 regularization with the computational speed of Fast Fourier Transform algorithms to develop a fully automated method for noise reduction and robust, standardized traction-force reconstruction that we call Bayesian Fourier transform traction cytometry (BFTTC). This method is made freely available as a software package with graphical user-interface for intuitive usage. Using synthetic data and experimental data, we show that these Bayesian methods enable robust reconstruction of traction without requiring a difficult selection of regularization parameters specifically for each data set. Next, we employ our methodology developed for the solution of inverse problems for automated, data-driven discovery of ordinary differential equations (ODEs), partial differential equations (PDEs), and stochastic differential equations (SDEs). To find the equations governing a measured time-dependent process, we construct dictionaries of non-linear candidate equations. These candidate equations are evaluated using the measured data. With this approach, one can construct a likelihood function for the candidate equations. Optimization yields a linear, inverse problem which is to be solved under a sparsity constraint. We combine Bayesian compressive sensing using Laplace priors with automated thresholding to develop a new approach, namely automatic threshold sparse Bayesian learning (ATSBL). ATSBL is a robust method to identify ODEs, PDEs, and SDEs involving Gaussian noise, which is also referred to as type I noise. We extensively test the method with synthetic datasets describing physical processes. For SDEs, we combine data-driven inference using ATSBL with a novel entropy-based heuristic for discarding data points with high uncertainty. Finally, we develop an automatic iterative sampling optimization technique akin to Umbrella sampling. Therewith, we demonstrate that data-driven inference of SDEs can be substantially improved through feedback during the inference process if the stochastic process under investigation can be manipulated either experimentally or in simulations

Scalable Low-rank Matrix and Tensor Decomposition on Graphs

In many signal processing, machine learning and computer vision applications, one often has to deal with high dimensional and big datasets such as images, videos, web content, etc. The data can come in various forms, such as univariate or multivariate time series, matrices or high dimensional tensors. The goal of the data mining community is to reveal the hidden linear or non-linear structures in the datasets. Over the past couple of decades matrix factorization, owing to its intrinsic association with dimensionality reduction has been adopted as one of the key methods in this context. One can either use a single linear subspace to approximate the data (the standard Principal Component Analysis (PCA) approach) or a union of low dimensional subspaces where each data class belongs to a different subspace. In many cases, however, the low dimensional data follows some additional structure. Knowledge of such structure is beneficial, as we can use it to enhance the representativity of our models by adding structured priors. A nowadays standard way to represent pairwise affinity between objects is by using graphs. The introduction of graph-based priors to enhance matrix factorization models has recently brought them back to the highest attention of the data mining community. Representation of a signal on a graph is well motivated by the emerging field of signal processing on graphs, based on notions of spectral graph theory. The underlying assumption is that high-dimensional data samples lie on or close to a smooth low-dimensional manifold. Interestingly, the underlying manifold can be represented by its discrete proxy, i.e. a graph. A primary limitation of the state-of-the-art low-rank approximation methods is that they do not generalize for the case of non-linear low-rank structures. Furthermore, the standard low-rank extraction methods for many applications, such as low-rank and sparse decomposition, are computationally cumbersome. We argue, that for many machine learning and signal processing applications involving big data, an approximate low-rank recovery suffices. Thus, in this thesis, we present solutions to the above two limitations by presenting a new framework for scalable but approximate low-rank extraction which exploits the hidden structure in the data using the notion of graphs. First, we present a novel signal model, called `Multilinear low-rank tensors on graphs (MLRTG)' which states that a tensor can be encoded as a multilinear combination of the low-frequency graph eigenvectors, where the graphs are constructed along the various modes of the tensor. Since the graph eigenvectors have the interpretation of \textit{non-linear} embedding of a dataset on the low-dimensional manifold, we propose a method called `Graph Multilinear SVD (GMLSVD)' to recover PCA based linear subspaces from these eigenvectors. Finally, we propose a plethora of highly scalable matrix and tensor based problems for low-rank extraction which implicitly or explicitly make use of the GMLSVD framework. The core idea is to replace the expensive iterative SVD operations by updating the linear subspaces from the fixed non-linear ones via low-cost operations. We present applications in low-rank and sparse decomposition and clustering of the low-rank features to evaluate all the proposed methods. Our theoretical analysis shows that the approximation error of the proposed framework depends on the spectral properties of the graph Laplacian

Modelling and Classification of Motor Imagery EEG for BCI

Ph.DDOCTOR OF PHILOSOPH