We consider goodness-of-fit methods for multivariate symmetric and asymmetric stable Paretian random vectors in arbitrary dimension. The methods are based on the empirical characteristic function and are implemented both in the i.i.d. context as well as for innovations in GARCH models. Asymptotic properties of the proposed procedures are discussed, while the finite-sample properties are illustrated by means of an extensive Monte Carlo study. The procedures are also applied to real data from the financial markets.
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This work puts forth low-complexity Riemannian subspace descent algorithms for the minimization of functions over the symmetric positive definite (SPD) manifold. Different from the existing Riemannian gradient descent variants, the proposed approach utilizes carefully chosen subspaces that allow the update to be written as a product of the Cholesky factor of the iterate and a sparse matrix. The resulting updates avoid the costly matrix operations like matrix exponentiation and dense matrix multiplication, which are generally required in almost all other Riemannian optimization algorithms on SPD manifold. We further identify a broad class of functions, arising in diverse applications, such as kernel matrix learning, covariance estimation of Gaussian distributions, maximum likelihood parameter estimation of elliptically contoured distributions, and parameter estimation in Gaussian mixture model problems, over which the Riemannian gradients can be calculated efficiently. The proposed uni-directional and multi-directional Riemannian subspace descent variants incur per-iteration complexities of $\O(n)$ and $\O(n^2)$ respectively, as compared to the $\O(n^3)$ or higher complexity incurred by all existing Riemannian gradient descent variants. The superior runtime and low per-iteration complexity of the proposed algorithms is also demonstrated via numerical tests on large-scale covariance estimation and matrix square root problems.
We extend the use of piecewise orthogonal collocation to computing periodic solutions of renewal equations, which are particularly important in modeling population dynamics. We prove convergence through a rigorous error analysis. Finally, we show some numerical experiments confirming the theoretical results, and a couple of applications in view of bifurcation analysis.
Current approaches to generic segmentation start by creating a hierarchy of nested image partitions and then specifying a segmentation from it. Our first contribution is to describe several ways, most of them new, for specifying segmentations using the hierarchy elements. Then, we consider the best hierarchy-induced segmentation specified by a limited number of hierarchy elements. We focus on a common quality measure for binary segmentations, the Jaccard index (also known as IoU). Optimizing the Jaccard index is highly non-trivial, and yet we propose an efficient approach for doing exactly that. This way we get algorithm-independent upper bounds on the quality of any segmentation created from the hierarchy. We found that the obtainable segmentation quality varies significantly depending on the way that the segments are specified by the hierarchy elements, and that representing a segmentation with only a few hierarchy elements is often possible. (Code is available).
We present a rigorous and precise analysis of the maximum degree and the average degree in a dynamic duplication-divergence graph model introduced by Sol\'e, Pastor-Satorras et al. in which the graph grows according to a duplication-divergence mechanism, i.e. by iteratively creating a copy of some node and then randomly alternating the neighborhood of a new node with probability $p$. This model captures the growth of some real-world processes e.g. biological or social networks. In this paper, we prove that for some $0 < p < 1$ the maximum degree and the average degree of a duplication-divergence graph on $t$ vertices are asymptotically concentrated with high probability around $t^p$ and $\max\{t^{2 p - 1}, 1\}$, respectively, i.e. they are within at most a polylogarithmic factor from these values with probability at least $1 - t^{-A}$ for any constant $A > 0$.
A semi-implicit in time, entropy stable finite volume scheme for the compressible barotropic Euler system is designed and analyzed and its weak convergence to a dissipative measure-valued (DMV) solution [E. Feireisl et al., Dissipative measure-valued solutions to the compressible Navier-Stokes system, Calc. Var. Partial Differential Equations, 2016] of the Euler system is shown. The entropy stability is achieved by introducing a shifted velocity in the convective fluxes of the mass and momentum balances, provided some CFL-like condition is satisfied to ensure stability. A consistency analysis is performed in the spirit of the Lax's equivalence theorem under some physically reasonable boundedness assumptions. The concept of K-convergence [E. Feireisl et al., K-convergence as a new tool in numerical analysis, IMA J. Numer. Anal., 2020] is used in order to obtain some strong convergence results, which are then illustrated via rigorous numerical case studies. The convergence of the scheme to a DMV solution, a weak solution and a strong solution of the Euler system using the weak-strong uniqueness principle and relative entropy are presented.
Positron Emission Tomography (PET) enables functional imaging of deep brain structures, but the bulk and weight of current systems preclude their use during many natural human activities, such as locomotion. The proposed long-term solution is to construct a robotic system that can support an imaging system surrounding the subject's head, and then move the system to accommodate natural motion. This requires a system to measure the motion of the head with respect to the imaging ring, for use by both the robotic system and the image reconstruction software. We report here the design and experimental evaluation of a parallel string encoder mechanism for sensing this motion. Our preliminary results indicate that the measurement system may achieve accuracy within 0.5 mm, especially for small motions, with improved accuracy possible through kinematic calibration.
Quadratization of polynomial and nonpolynomial systems of ordinary differential equations is advantageous in a variety of disciplines, such as systems theory, fluid mechanics, chemical reaction modeling and mathematical analysis. A quadratization reveals new variables and structures of a model, which may be easier to analyze, simulate, control, and provides a convenient parametrization for learning. This paper presents novel theory, algorithms and software capabilities for quadratization of non-autonomous ODEs. We provide existence results, depending on the regularity of the input function, for cases when a quadratic-bilinear system can be obtained through quadratization. We further develop existence results and an algorithm that generalizes the process of quadratization for systems with arbitrary dimension that retain the nonlinear structure when the dimension grows. For such systems, we provide dimension-agnostic quadratization. An example is semi-discretized PDEs, where the nonlinear terms remain symbolically identical when the discretization size increases. As an important aspect for practical adoption of this research, we extended the capabilities of the QBee software towards both non-autonomous systems of ODEs and ODEs with arbitrary dimension. We present several examples of ODEs that were previously reported in the literature, and where our new algorithms find quadratized ODE systems with lower dimension than the previously reported lifting transformations. We further highlight an important area of quadratization: reduced-order model learning. This area can benefit significantly from working in the optimal lifting variables, where quadratic models provide a direct parametrization of the model that also avoids additional hyperreduction for the nonlinear terms. A solar wind example highlights these advantages.
While many phenomena in physics and engineering are formally high-dimensional, their long-time dynamics often live on a lower-dimensional manifold. The present work introduces an autoencoder framework that combines implicit regularization with internal linear layers and $L_2$ regularization (weight decay) to automatically estimate the underlying dimensionality of a data set, produce an orthogonal manifold coordinate system, and provide the mapping functions between the ambient space and manifold space, allowing for out-of-sample projections. We validate our framework's ability to estimate the manifold dimension for a series of datasets from dynamical systems of varying complexities and compare to other state-of-the-art estimators. We analyze the training dynamics of the network to glean insight into the mechanism of low-rank learning and find that collectively each of the implicit regularizing layers compound the low-rank representation and even self-correct during training. Analysis of gradient descent dynamics for this architecture in the linear case reveals the role of the internal linear layers in leading to faster decay of a "collective weight variable" incorporating all layers, and the role of weight decay in breaking degeneracies and thus driving convergence along directions in which no decay would occur in its absence. We show that this framework can be naturally extended for applications of state-space modeling and forecasting by generating a data-driven dynamic model of a spatiotemporally chaotic partial differential equation using only the manifold coordinates. Finally, we demonstrate that our framework is robust to hyperparameter choices.
We consider the low-rank alternating directions implicit (ADI) iteration for approximately solving large-scale algebraic Sylvester equations. Inside every iteration step of this iterative process a pair of linear systems of equations has to be solved. We investigate the situation when those inner linear systems are solved inexactly by an iterative methods such as, for example, preconditioned Krylov subspace methods. The main contribution of this work are thresholds for the required accuracies regarding the inner linear systems which dictate when the employed inner Krylov subspace methods can be safely terminated. The goal is to save computational effort by solving the inner linear system as inaccurate as possible without endangering the functionality of the low-rank Sylvester-ADI method. Ideally, the inexact ADI method mimics the convergence behaviour of the more expensive exact ADI method, where the linear systems are solved directly. Alongside the theoretical results, also strategies for an actual practical implementation of the stopping criteria are developed. Numerical experiments confirm the effectiveness of the proposed strategies.
Probabilistic variants of Model Order Reduction (MOR) methods have recently emerged for improving stability and computational performance of classical approaches. In this paper, we propose a probabilistic Reduced Basis Method (RBM) for the approximation of a family of parameter-dependent functions. It relies on a probabilistic greedy algorithm with an error indicator that can be written as an expectation of some parameter-dependent random variable. Practical algorithms relying on Monte Carlo estimates of this error indicator are discussed. In particular, when using Probably Approximately Correct (PAC) bandit algorithm, the resulting procedure is proven to be a weak greedy algorithm with high probability. Intended applications concern the approximation of a parameter-dependent family of functions for which we only have access to (noisy) pointwise evaluations. As a particular application, we consider the approximation of solution manifolds of linear parameter-dependent partial differential equations with a probabilistic interpretation through the Feynman-Kac formula.
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