Robust PCA as Bilinear Decomposition with Outlier-Sparsity Regularization

Principal component analysis (PCA) is widely used for dimensionality reduction, with well-documented merits in various applications involving high-dimensional data, including computer vision, preference measurement, and bioinformatics. In this context, the fresh look advocated here permeates benefits from variable selection and compressive sampling, to robustify PCA against outliers. A least-trimmed squares estimator of a low-rank bilinear factor analysis model is shown closely related to that obtained from an $\ell_0$-(pseudo)norm-regularized criterion encouraging sparsity in a matrix explicitly modeling the outliers. This connection suggests robust PCA schemes based on convex relaxation, which lead naturally to a family of robust estimators encompassing Huber's optimal M-class as a special case. Outliers are identified by tuning a regularization parameter, which amounts to controlling sparsity of the outlier matrix along the whole robustification path of (group) least-absolute shrinkage and selection operator (Lasso) solutions. Beyond its neat ties to robust statistics, the developed outlier-aware PCA framework is versatile to accommodate novel and scalable algorithms to: i) track the low-rank signal subspace robustly, as new data are acquired in real time; and ii) determine principal components robustly in (possibly) infinite-dimensional feature spaces. Synthetic and real data tests corroborate the effectiveness of the proposed robust PCA schemes, when used to identify aberrant responses in personality assessment surveys, as well as unveil communities in social networks, and intruders from video surveillance data.Comment: 30 pages, submitted to IEEE Transactions on Signal Processin

Takens-Bogdanov bifurcation of travelling wave solutions in pipe flow

The appearance of travelling-wave-type solutions in pipe Poiseuille flow that are disconnected from the basic parabolic profile is numerically studied in detail. We focus on solutions in the 2-fold azimuthally-periodic subspace because of their special stability properties, but relate our findings to other solutions as well. Using time-stepping, an adapted Krylov-Newton method and Arnoldi iteration for the computation and stability analysis of relative equilibria, and a robust pseudo-arclength continuation scheme we unfold a double-zero (Takens-Bogdanov) bifurcating scenario as a function of Reynolds number (Re) and wavenumber (k). This scenario is extended, by the inclusion of higher order terms in the normal form, to account for the appearance of supercritical modulated waves emanating from the upper branch of solutions at a degenerate Hopf bifurcation. These waves are expected to disappear in saddle-loop bifurcations upon collision with lower-branch solutions, thereby leaving stable upper-branch solutions whose subsequent secondary bifurcations could contribute to the formation of the phase space structures that are required for turbulent dynamics at higher Re.Comment: 26 pages, 15 figures (pdf and png). Submitted to J. Fluid Mec

Turbulence Transition in Shear Flows and Dynamical Systems Theory

Turbulence is allegedly “the most important unsolved problem of classical physics” (attributed to Richard Feynman). While the equations of motion are known since almost 150 years and despite the work of many physicists, in particular the transition to turbulence in linearly stable shear flows evades a satisfying description. In recent decades, the availability of more powerful computers and developments in chaos theory have provided the basis for considerable progress in our understanding of this issue. The successful work of many scientists proved dynamical systems theory to be a useful and important tool to analyze transitional turbulence in fluid mechanics, allowing to explain observed phenomena such as transition thresholds and transient lifetimes through bifurcation analyses and the identification of underlying state space structures. In this thesis we continue on that path with direct numerical simulations in plane Couette flow, the asymptotic suction boundary layer and Blasius boundary layers. We explore the state space structures and bifurcations in plane Couette flow, study the threshold dynamics in the ASBL and develop a model for the spatio-temporal dynamics in the boundary layers. The results show how the insights obtained for parallel, bounded shear flows can be transferred to spatially developing external flows

CLIPPER+: A Fast Maximal Clique Algorithm for Robust Global Registration

We present CLIPPER+, an algorithm for finding maximal cliques in unweighted graphs for outlier-robust global registration. The registration problem can be formulated as a graph and solved by finding its maximum clique. This formulation leads to extreme robustness to outliers; however, finding the maximum clique is an NP-hard problem, and therefore approximation is required in practice for large-size problems. The performance of an approximation algorithm is evaluated by its computational complexity (the lower the runtime, the better) and solution accuracy (how close the solution is to the maximum clique). Accordingly, the main contribution of CLIPPER+ is outperforming the state-of-the-art in accuracy while maintaining a relatively low runtime. CLIPPER+ builds on prior work (CLIPPER [1] and PMC [2]) and prunes the graph by removing vertices that have a small core number and cannot be a part of the maximum clique. This will result in a smaller graph, on which the maximum clique can be estimated considerably faster. We evaluate the performance of CLIPPER+ on standard graph benchmarks, as well as synthetic and real-world point cloud registration problems. These evaluations demonstrate that CLIPPER+ has the highest accuracy and can register point clouds in scenarios where over $99\%$ of associations are outliers. Our code and evaluation benchmarks are released at https://github.com/ariarobotics/clipperp

Computational and Theoretical Issues of Multiparameter Persistent Homology for Data Analysis

The basic goal of topological data analysis is to apply topology-based descriptors to understand and describe the shape of data. In this context, homology is one of the most relevant topological descriptors, well-appreciated for its discrete nature, computability and dimension independence. A further development is provided by persistent homology, which allows to track homological features along a oneparameter increasing sequence of spaces. Multiparameter persistent homology, also called multipersistent homology, is an extension of the theory of persistent homology motivated by the need of analyzing data naturally described by several parameters, such as vector-valued functions. Multipersistent homology presents several issues in terms of feasibility of computations over real-sized data and theoretical challenges in the evaluation of possible descriptors. The focus of this thesis is in the interplay between persistent homology theory and discrete Morse Theory. Discrete Morse theory provides methods for reducing the computational cost of homology and persistent homology by considering the discrete Morse complex generated by the discrete Morse gradient in place of the original complex. The work of this thesis addresses the problem of computing multipersistent homology, to make such tool usable in real application domains. This requires both computational optimizations towards the applications to real-world data, and theoretical insights for finding and interpreting suitable descriptors. Our computational contribution consists in proposing a new Morse-inspired and fully discrete preprocessing algorithm. We show the feasibility of our preprocessing over real datasets, and evaluate the impact of the proposed algorithm as a preprocessing for computing multipersistent homology. A theoretical contribution of this thesis consists in proposing a new notion of optimality for such a preprocessing in the multiparameter context. We show that the proposed notion generalizes an already known optimality notion from the one-parameter case. Under this definition, we show that the algorithm we propose as a preprocessing is optimal in low dimensional domains. In the last part of the thesis, we consider preliminary applications of the proposed algorithm in the context of topology-based multivariate visualization by tracking critical features generated by a discrete gradient field compatible with the multiple scalar fields under study. We discuss (dis)similarities of such critical features with the state-of-the-art techniques in topology-based multivariate data visualization

Assessment and control of transition to turbulence in plane Couette flow

Transition to turbulence in shear flows is a puzzling problem regarding the motion of fluids flowing, for example, through the pipe (pipe flow), as in oil pipelines or blood vessels, or confined between two counter-moving walls (plane Couette flow). In this kind of flows, the initially laminar (ordered and layered) state of fluid motion is linearly stable, but turbulent (disordered and swirling) flows can also be observed if a suitable perturbation is imposed. This thesis concerns the assessment of transitional properties of such flows in the uncontrolled and controlled environments allowing for the quantitative comparisons of control strategies aimed at suppressing or trigerring transition to turbulence. Efficient finite-amplitude perturbations typically take the form of small patches of turbulence embedded in the laminar flow and called turbulent spots. Using direct numerical simulations, the nonlinear dynamics of turbulent spots, modelled as exact solutions, is investigated in the transitional regime of plane Couette flow and a detailed map of dynamics encompassing the main features found in transitional shear flows (self-sustained cycles, front propagation and spot splitting) is built. The map represents a quantitative assessment of transient dynamics of turbulent spots as a dependence of the relaminarisation time, i.e. the time it takes for a finite-amplitude perturbation, added to the laminar flow, to decay, on the Reynolds number and the width of a localised perturbation. By applying a simple passive control strategy, sinusoidal wall oscillations, the change in the spot dynamics with respect to the amplitude and frequency of the wall oscillations is assessed by the re-evaluation of the relaminarisation time for few selected localised initial conditions. Finally, a probabilistic protocol for the assessment of transition to turbulence and its control is suggested. The protocol is based on the calculation of the laminarisation probability, i.e. the probability that a random perturbation decays as a function of its energy. It is used to assess the robustness of the laminar flow to finite-amplitude perturbations in transitional plane Couette flow in a small computational domain in the absence of control and under the action of sinusoidal wall oscillations. The protocol is expected to be useful for a wide range of nonlinear systems exhibiting finite-amplitude instability