1,258 research outputs found

    Towards Reliable and Accurate Global Structure-from-Motion

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    Reconstruction of objects or scenes from sparse point detections across multiple views is one of the most tackled problems in computer vision. Given the coordinates of 2D points tracked in multiple images, the problem consists of estimating the corresponding 3D points and cameras\u27 calibrations (intrinsic and pose), and can be solved by minimizing reprojection errors using bundle adjustment. However, given bundle adjustment\u27s nonlinear objective function and iterative nature, a good starting guess is required to converge to global minima. Global and Incremental Structure-from-Motion methods appear as ways to provide good initializations to bundle adjustment, each with different properties. While Global Structure-from-Motion has been shown to result in more accurate reconstructions compared to Incremental Structure-from-Motion, the latter has better scalability by starting with a small subset of images and sequentially adding new views, allowing reconstruction of sequences with millions of images. Additionally, both Global and Incremental Structure-from-Motion methods rely on accurate models of the scene or object, and under noisy conditions or high model uncertainty might result in poor initializations for bundle adjustment. Recently pOSE, a class of matrix factorization methods, has been proposed as an alternative to conventional Global SfM methods. These methods use VarPro - a second-order optimization method - to minimize a linear combination of an approximation of reprojection errors and a regularization term based on an affine camera model, and have been shown to converge to global minima with a high rate even when starting from random camera calibration estimations.This thesis aims at improving the reliability and accuracy of global SfM through different approaches. First, by studying conditions for global optimality of point set registration, a point cloud averaging method that can be used when (incomplete) 3D point clouds of the same scene in different coordinate systems are available. Second, by extending pOSE methods to different Structure-from-Motion problem instances, such as Non-Rigid SfM or radial distortion invariant SfM. Third and finally, by replacing the regularization term of pOSE methods with an exponential regularization on the projective depth of the 3D point estimations, resulting in a loss that achieves reconstructions with accuracy close to bundle adjustment

    Essays on noncausal and noninvertible time series

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    Over the last two decades, there has been growing interest among economists in nonfundamental univariate processes, generally represented by noncausal and non-invertible time series. These processes have become increasingly popular due to their ability to capture nonlinear dynamics such as volatility clustering, asymmetric cycles, and local explosiveness - all of which are commonly observed in Macroeconomics and Finance. In particular, the incorporation of both past and future components into noncausal and noninvertible processes makes them attractive options for modeling forward-looking behavior in economic activities. However, the classical techniques used for analyzing time series models are largely limited to causal and invertible counterparts. This dissertation seeks to contribute to the field by providing theoretical tools robust to noncausal and noninvertible time series in testing and estimation. In the first chapter, "Quantile Autoregression-Based Non-causality Testing", we investigate the statistical properties of empirical conditional quantiles of non-causal processes. Specifically, we show that the quantile autoregression (QAR) estimates for non-causal processes do not remain constant across different quantiles in contrast to their causal counterparts. Furthermore, we demonstrate that non-causal autoregressive processes admit nonlinear representations for conditional quantiles given past observations. Exploiting these properties, we propose three novel testing strategies of non-causality for non-Gaussian processes within the QAR framework. The tests are constructed either by verifying the constancy of the slope coefficients or by applying a misspecification test of the linear QAR model over different quantiles of the process. Some numerical experiments are included to examine the finite sample performance of the testing strategies, where we compare different specification tests for dynamic quantiles with the Kolmogorov-Smirnov constancy test. The new methodology is applied to some time series from financial markets to investigate the presence of speculative bubbles. The extension of the approach based on the specification tests to AR processes driven by innovations with heteroskedasticity is studied through simulations. The performance of QAR estimates of non-causal processes at extreme quantiles is also explored. In the second chapter, "Estimation of Time Series Models Using the Empirical Distribution of Residuals", we introduce a novel estimation technique for general linear time series models, potentially noninvertible and noncausal, by utilizing the empirical cumulative distribution function of residuals. The proposed method relies on the generalized spectral cumulative function to characterize the pairwise dependence of residuals at all lags. Model identification can be achieved by exploiting the information in the joint distribution of residuals under the iid assumption. This method yields consistent estimates of the model parameters without imposing stringent conditions on the higher-order moments or any distributional assumptions on the innovations beyond non-Gaussianity. We investigate the asymptotic distribution of the estimates by employing a smoothed cumulative distribution function to approximate the indicator function, considering the non-differentiability of the original loss function. Efficiency improvements can be achieved by properly choosing the scaling parameter for residuals. Finite sample properties are explored through Monte Carlo simulations. An empirical application to illustrate this methodology is provided by fitting the daily trading volume of Microsoft stock by autoregressive models with noncausal representation. The flexibility of the cumulative distribution function permits the proposed method to be extended to more general dependence structures where innovations are only conditional mean or quantile independent. In the third chapter, "Directional Predictability Tests", joint with Carlos Velasco, we propose new tests of predictability for non-Gaussian sequences that may display general nonlinear dependence in higher-order properties. We test the null of martingale difference against parametric alternatives which can introduce linear or nonlinear dependence as generated by ARMA and all-pass restricted ARMA models, respectively. We also develop tests to check for linear predictability under the white noise null hypothesis parameterized by an all-pass model driven by martingale difference innovations and tests of non-linear predictability on ARMA residuals. Our Lagrange Multiplier tests are developed from a loss function based on pairwise dependence measures that identify the predictability of levels. We provide asymptotic and finite sample analysis of the properties of the new tests and investigate the predictability of different series of financial returns.This thesis has been possible thanks to the financial support from the grant BES-2017-082695 from the Ministerio de Economía Industria y Competitividad.Programa de Doctorado en Economía por la Universidad Carlos III de MadridPresidente: Miguel ángel Delgado González.- Secretario: Manuel Domínguez Toribio.- Vocal: Majid M. Al Sadoo

    Distributed AC Optimal Power Flow for Generic Integrated Transmission-Distribution Systems

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    Coordination of transmission and distribution power systems is increasingly critical in the context of the ongoing energy transition. However, traditional centralized energy management faces challenges related to privacy and/or sovereignty concerns, leading to growing research interests in distributed approaches. Nevertheless, solving distributed AC optimal power flow (OPF) problems encounters difficulties due to their nonlinearity and nonconvexity, making it challenging for state-of-the-art distributed approaches. To solve this issue, the present paper focuses on investigating the distributed AC OPF problem of generic integrated transmission-distribution (ITD) systems, considering complex grid topology, by employing a new variant of the Augmented Lagrangian based Alternating Direction Inexact Newton method (ALADIN). In contrast to the standard ALADIN, we introduce a second-order correction into ALADIN to enhance its numerical robustness and properly convexify distribution subproblems within the ALADIN framework for computing efficiency. Moreover, a rigorous proof shows that the locally quadratic convergence rate can be preserved for solving the resulting distributed nonconvex problems. Extensive numerical simulations with varying problem sizes and grid topologies demonstrate the effectiveness of the proposed algorithm, outperforming state-of-the-art approaches in terms of numerical robustness, convergence speed, and scalability

    Implicit Loss of Surjectivity and Facial Reduction: Theory and Applications

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    Facial reduction, pioneered by Borwein and Wolkowicz, is a preprocessing method that is commonly used to obtain strict feasibility in the reformulated, reduced constraint system. The importance of strict feasibility is often addressed in the context of the convergence results for interior point methods. Beyond the theoretical properties that the facial reduction conveys, we show that facial reduction, not only limited to interior point methods, leads to strong numerical performances in different classes of algorithms. In this thesis we study various consequences and the broad applicability of facial reduction. The thesis is organized in two parts. In the first part, we show the instabilities accompanied by the absence of strict feasibility through the lens of facially reduced systems. In particular, we exploit the implicit redundancies, revealed by each nontrivial facial reduction step, resulting in the implicit loss of surjectivity. This leads to the two-step facial reduction and two novel related notions of singularity. For the area of semidefinite programming, we use these singularities to strengthen a known bound on the solution rank, the Barvinok-Pataki bound. For the area of linear programming, we reveal degeneracies caused by the implicit redundancies. Furthermore, we propose a preprocessing tool that uses the simplex method. In the second part of this thesis, we continue with the semidefinite programs that do not have strictly feasible points. We focus on the doubly-nonnegative relaxation of the binary quadratic program and a semidefinite program with a nonlinear objective function. We closely work with two classes of algorithms, the splitting method and the Gauss-Newton interior point method. We elaborate on the advantages in building models from facial reduction. Moreover, we develop algorithms for real-world problems including the quadratic assignment problem, the protein side-chain positioning problem, and the key rate computation for quantum key distribution. Facial reduction continues to play an important role for providing robust reformulated models in both the theoretical and the practical aspects, resulting in successful numerical performances

    Gaussian Control Barrier Functions : A Gaussian Process based Approach to Safety for Robots

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    In recent years, the need for safety of autonomous and intelligent robots has increased. Today, as robots are being increasingly deployed in closer proximity to humans, there is an exigency for safety since human lives may be at risk, e.g., self-driving vehicles or surgical robots. The objective of this thesis is to present a safety framework for dynamical systems that leverages tools from control theory and machine learning. More formally, the thesis presents a data-driven framework for designing safety function candidates which ensure properties of forward invariance. The potential benefits of the results presented in this thesis are expected to help applications such as safe exploration, collision avoidance problems, manipulation tasks, and planning, to name some. We utilize Gaussian processes (GP) to place a prior on the desired safety function candidate, which is to be utilized as a control barrier function (CBF). The resultant formulation is called Gaussian CBFs and they reside in a reproducing kernel Hilbert space. A key concept behind Gaussian CBFs is the incorporation of both safety belief as well as safety uncertainty, which former barrier function formulations did not consider. This is achieved by using robust posterior estimates from a GP where the posterior mean and variance serve as surrogates for the safety belief and uncertainty respectively. We synthesize safe controllers by framing a convex optimization problem where the kernel-based representation of GPs allows computing the derivatives in closed-form analytically. Finally, in addition to the theoretical and algorithmic frameworks in this thesis, we rigorously test our methods in hardware on a quadrotor platform. The platform used is a Crazyflie 2.1 which is a versatile palm-sized quadrotor. We provide our insights and detailed discussions on the hardware implementations which will be useful for large-scale deployment of the techniques presented in this dissertation.Ph.D

    Novel Representations of Semialgebraic Sets Arising in Planning and Control

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    The mathematical notion of a set arises frequently in planning and control of autonomous systems. A common challenge is how to best represent a given set in a manner that is efficient, accurate, and amenable to computational tools of interest. For example, ensuring a vehicle does not collide with an obstacle can be generically posed in multiple ways using techniques from optimization or computational geometry. However these representations generally rely on executing algorithms instead of evaluating closed-form expressions. This presents an issue when we wish to represent an obstacle avoidance condition within a larger motion planning problem which is solved using nonlinear optimization. These tools generally can only accept smooth, closed-form expressions. As such our available representations of obstacle avoidance conditions, while accurate, are not amenable to the relevant tools. A related problem is how to represent a set in a compact form without sacrificing accuracy. For example, we may be presented with point-cloud data representing the boundary of an object that our vehicle must avoid. Using the obstacle avoidance conditions directly on the point-cloud data would require performing these calculations with respect to each point individually. A more efficient approach is to first approximate the data with simple geometric shapes and perform later analysis with the approximation. Common shapes include bounding boxes, ellipsoids, and superquadrics. These shapes are convenient in that they have a compact representation and we have good heuristic objectives for fitting the data. However, their primitive nature means accuracy of representation may suffer. Most notably, their inherent symmetry makes them ill-suited for representing asymmetric shapes. In theory we could consider more complicated shapes given by an implicit function. However we lack reliable methods for ensuring a good fit. This thesis proposes novel approaches to these problems based on tools from convex optimization and convex analysis. Throughout, the sets of interest are described by polynomial inequalities, making them semialgebraic

    Cooperative Filtering and Parameter Identification for Advection-Diffusion Processes Using a Mobile Sensor Network

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    This article presents an online parameter identification scheme for advection-diffusion processes using data collected by a mobile sensor network. The advection-diffusion equation is incorporated into the information dynamics associated with the trajectories of the mobile sensors. A constrained cooperative Kalman filter is developed to provide estimates of the field values and gradients along the trajectories of the mobile sensors so that the temporal variations in the field values can be estimated. This leads to a co-design scheme for state estimation and parameter identification for advection-diffusion processes that is different from comparable schemes using sensors installed at fixed spatial locations. Using state estimates from the constrained cooperative Kalman filter, a recursive least-square (RLS) algorithm is designed to estimate unknown model parameters of the advection-diffusion processes. Theoretical justifications are provided for the convergence of the proposed cooperative Kalman filter by deriving a set of sufficient conditions regarding the formation shape and the motion of the mobile sensor network. Simulation and experimental results show satisfactory performance and demonstrate the robustness of the algorithm under realistic uncertainties and disturbances

    Learning Stable Koopman Models for Identification and Control of Dynamical Systems

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    Learning models of dynamical systems from data is a widely-studied problem in control theory and machine learning. One recent approach for modelling nonlinear systems considers the class of Koopman models, which embeds the nonlinear dynamics in a higher-dimensional linear subspace. Learning a Koopman embedding would allow for the analysis and control of nonlinear systems using tools from linear systems theory. Many recent methods have been proposed for data-driven learning of such Koopman embeddings, but most of these methods do not consider the stability of the Koopman model. Stability is an important and desirable property for models of dynamical systems. Unstable models tend to be non-robust to input perturbations and can produce unbounded outputs, which are both undesirable when the model is used for prediction and control. In addition, recent work has shown that stability guarantees may act as a regularizer for model fitting. As such, a natural direction would be to construct Koopman models with inherent stability guarantees. Two new classes of Koopman models are proposed that bridge the gap between Koopman-based methods and learning stable nonlinear models. The first model class is guaranteed to be stable, while the second is guaranteed to be stabilizable with an explicit stabilizing controller that renders the model stable in closed-loop. Furthermore, these models are unconstrained in their parameter sets, thereby enabling efficient optimization via gradient-based methods. Theoretical connections between the stability of Koopman models and forms of nonlinear stability such as contraction are established. To demonstrate the effect of the stability guarantees, the stable Koopman model is applied to a system identification problem, while the stabilizable model is applied to an imitation learning problem. Experimental results show empirically that the proposed models achieve better performance over prior methods without stability guarantees

    Geometric Learning on Graph Structured Data

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    Graphs provide a ubiquitous and universal data structure that can be applied in many domains such as social networks, biology, chemistry, physics, and computer science. In this thesis we focus on two fundamental paradigms in graph learning: representation learning and similarity learning over graph-structured data. Graph representation learning aims to learn embeddings for nodes by integrating topological and feature information of a graph. Graph similarity learning brings into play with similarity functions that allow to compute similarity between pairs of graphs in a vector space. We address several challenging issues in these two paradigms, designing powerful, yet efficient and theoretical guaranteed machine learning models that can leverage rich topological structural properties of real-world graphs. This thesis is structured into two parts. In the first part of the thesis, we will present how to develop powerful Graph Neural Networks (GNNs) for graph representation learning from three different perspectives: (1) spatial GNNs, (2) spectral GNNs, and (3) diffusion GNNs. We will discuss the model architecture, representational power, and convergence properties of these GNN models. Specifically, we first study how to develop expressive, yet efficient and simple message-passing aggregation schemes that can go beyond the Weisfeiler-Leman test (1-WL). We propose a generalized message-passing framework by incorporating graph structural properties into an aggregation scheme. Then, we introduce a new local isomorphism hierarchy on neighborhood subgraphs. We further develop a novel neural model, namely GraphSNN, and theoretically prove that this model is more expressive than the 1-WL test. After that, we study how to build an effective and efficient graph convolution model with spectral graph filters. In this study, we propose a spectral GNN model, called DFNets, which incorporates a novel spectral graph filter, namely feedback-looped filters. As a result, this model can provide better localization on neighborhood while achieving fast convergence and linear memory requirements. Finally, we study how to capture the rich topological information of a graph using graph diffusion. We propose a novel GNN architecture with dynamic PageRank, based on a learnable transition matrix. We explore two variants of this GNN architecture: forward-euler solution and invariable feature solution, and theoretically prove that our forward-euler GNN architecture is guaranteed with the convergence to a stationary distribution. In the second part of this thesis, we will introduce a new optimal transport distance metric on graphs in a regularized learning framework for graph kernels. This optimal transport distance metric can preserve both local and global structures between graphs during the transport, in addition to preserving features and their local variations. Furthermore, we propose two strongly convex regularization terms to theoretically guarantee the convergence and numerical stability in finding an optimal assignment between graphs. One regularization term is used to regularize a Wasserstein distance between graphs in the same ground space. This helps to preserve the local clustering structure on graphs by relaxing the optimal transport problem to be a cluster-to-cluster assignment between locally connected vertices. The other regularization term is used to regularize a Gromov-Wasserstein distance between graphs across different ground spaces based on degree-entropy KL divergence. This helps to improve the matching robustness of an optimal alignment to preserve the global connectivity structure of graphs. We have evaluated our optimal transport-based graph kernel using different benchmark tasks. The experimental results show that our models considerably outperform all the state-of-the-art methods in all benchmark tasks

    Microstructure modeling and crystal plasticity parameter identification for predicting the cyclic mechanical behavior of polycrystalline metals

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    Computational homogenization permits to capture the influence of the microstructure on the cyclic mechanical behavior of polycrystalline metals. In this work we investigate methods to compute Laguerre tessellations as computational cells of polycrystalline microstructures, propose a new method to assign crystallographic orientations to the Laguerre cells and use Bayesian optimization to find suitable parameters for the underlying micromechanical model from macroscopic experiments
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