Low Complexity Regularization of Linear Inverse Problems

Inverse problems and regularization theory is a central theme in contemporary signal processing, where the goal is to reconstruct an unknown signal from partial indirect, and possibly noisy, measurements of it. A now standard method for recovering the unknown signal is to solve a convex optimization problem that enforces some prior knowledge about its structure. This has proved efficient in many problems routinely encountered in imaging sciences, statistics and machine learning. This chapter delivers a review of recent advances in the field where the regularization prior promotes solutions conforming to some notion of simplicity/low-complexity. These priors encompass as popular examples sparsity and group sparsity (to capture the compressibility of natural signals and images), total variation and analysis sparsity (to promote piecewise regularity), and low-rank (as natural extension of sparsity to matrix-valued data). Our aim is to provide a unified treatment of all these regularizations under a single umbrella, namely the theory of partial smoothness. This framework is very general and accommodates all low-complexity regularizers just mentioned, as well as many others. Partial smoothness turns out to be the canonical way to encode low-dimensional models that can be linear spaces or more general smooth manifolds. This review is intended to serve as a one stop shop toward the understanding of the theoretical properties of the so-regularized solutions. It covers a large spectrum including: (i) recovery guarantees and stability to noise, both in terms of $\ell^2$-stability and model (manifold) identification; (ii) sensitivity analysis to perturbations of the parameters involved (in particular the observations), with applications to unbiased risk estimation ; (iii) convergence properties of the forward-backward proximal splitting scheme, that is particularly well suited to solve the corresponding large-scale regularized optimization problem

Integrating OLAP and Ranking: The Ranking-Cube Methodology

Recent years have witnessed an enormous growth of data in business, industry, and Web applications. Database search often returns a large collection of results, which poses challenges to both efficient query processing and effective digest of the query results. To address this problem, ranked search has been introduced to database systems. We study the problem of On-Line Analytical Processing (OLAP) of ranked queries, where ranked queries are conducted in the arbitrary subset of data defined by multi-dimensional selections. While pre-computation and multi-dimensional aggregation is the standard solution for OLAP, materializing dynamic ranking results is unrealistic because the ranking criteria are not known until the query time. To overcome such difficulty, we develop a new ranking cube method that performs semi on-line materialization and semi online computation in this thesis. Its complete life cycle, including cube construction, incremental maintenance, and query processing, is also discussed. We further extend the ranking cube in three dimensions. First, how to answer queries in high-dimensional data. Second, how to answer queries which involves joins over multiple relations. Third, how to answer general preference queries (besides ranked queries, such as skyline queries). Our performance studies show that ranking-cube is orders of magnitude faster than previous approaches

Fine-grained complexity and algorithm engineering of geometric similarity measures

Point sets and sequences are fundamental geometric objects that arise in any application that considers movement data, geometric shapes, and many more. A crucial task on these objects is to measure their similarity. Therefore, this thesis presents results on algorithms, complexity lower bounds, and algorithm engineering of the most important point set and sequence similarity measures like the Fréchet distance, the Fréchet distance under translation, and the Hausdorff distance under translation. As an extension to the mere computation of similarity, also the approximate near neighbor problem for the continuous Fréchet distance on time series is considered and matching upper and lower bounds are shown.Punktmengen und Sequenzen sind fundamentale geometrische Objekte, welche in vielen Anwendungen auftauchen, insbesondere in solchen die Bewegungsdaten, geometrische Formen, und ähnliche Daten verarbeiten. Ein wichtiger Bestandteil dieser Anwendungen ist die Berechnung der Ähnlichkeit von Objekten. Diese Dissertation präsentiert Resultate, genauer gesagt Algorithmen, untere Komplexitätsschranken und Algorithm Engineering der wichtigsten Ähnlichkeitsmaße für Punktmengen und Sequenzen, wie zum Beispiel Fréchetdistanz, Fréchetdistanz unter Translation und Hausdorffdistanz unter Translation. Als eine Erweiterung der bloßen Berechnung von Ähnlichkeit betrachten wir auch das Near Neighbor Problem für die kontinuierliche Fréchetdistanz auf Zeitfolgen und zeigen obere und untere Schranken dafür

Algorithms for the Analysis of Spatio-Temporal Data from Team Sports

Modern object tracking systems are able to simultaneously record trajectories—sequences of time-stamped location points—for large numbers of objects with high frequency and accuracy. The availability of trajectory datasets has resulted in a consequent demand for algorithms and tools to extract information from these data. In this thesis, we present several contributions intended to do this, and in particular, to extract information from trajectories tracking football (soccer) players during matches. Football player trajectories have particular properties that both facilitate and present challenges for the algorithmic approaches to information extraction. The key property that we look to exploit is that the movement of the players reveals information about their objectives through cooperative and adversarial coordinated behaviour, and this, in turn, reveals the tactics and strategies employed to achieve the objectives. While the approaches presented here naturally deal with the application-specific properties of football player trajectories, they also apply to other domains where objects are tracked, for example behavioural ecology, traffic and urban planning

Spatiotemporal Big Data Analytics for Future Mobility

University of Minnesota Ph.D. dissertation. May 2019. Major: Computer Science. Advisor: Shashi Shekhar. 1 computer file (PDF); xii, 161 pages.Recent years have witnessed the explosion of spatiotemporal big data (e.g. GPS trajectories, vehicle engine measurements, remote sensing imagery, and geotagged tweets) which has a potential to transform our societies. Terabytes of earth observation data are collected every day from thousands of places across the world. Modern vehicles are increasingly equipped with rich sensors that measure hundreds of engine variables (e.g., emissions, fuel consumption, speed, etc) annotated with timestamps and location data for every second of the vehicle’s trip. According to reports by McKinsey and Cisco, leveraging such data is potentially worth hundreds of billions of dollars annually in fuel savings. Spatiotemporal big data are also enabling many modern technologies such as on-demand transportation (e.g. Uber, Lyft). Today, the on-demand economy attracts millions of consumers annually and over $50 billion in spending. Even more growth is expected with the emergence of self-driving cars. However, spatiotemporal big data are of volume, velocity, variety, and veracity that exceed the capability of common spatiotemporal data analytic techniques. My thesis investigates spatiotemporal big data analytics that address the volume and velocity challenges of spatiotemporal big data in the context of novel applications in transportation and engine science, future mobility, and the on-demand economy. The thesis proposes scalable algorithms for mining “Non-compliant Window Co-occurrence Patterns”, which allow the discovery of correlations in spatiotemporal big data with a large number of variables. Novel upper bounds were introduced for a statistical interest measure of association to efficiently prune uninteresting candidate patterns. Case studies with real world engine data demonstrated the ability of the proposed approaches to discover patterns which are of interest to engine scientists. To address the high velocity challenge, the thesis explored online optimization heuristics for matching supply and demand in an on-demand spatial service broker. The proposed algorithms maximize the matching size while also maintaining a balanced provider utilization to ensure robustness against variations in the supply-demand ratio and that providers do not drop out. Proposed algorithms were shown to outperform related work on multiple performance measures. In addition, the thesis proposed a scalable matching and scheduling algorithm for an on-demand pickup and delivery broker for moving consumers with multiple candidate delivery locations and time intervals. Extensive evaluation showed that the proposed approach yields significant computational savings without sacrificing the solution quality