Motivated by information geometry, a distance function on the space of stochastic matrices is advocated. Starting with sequences of Markov chains the Bhattacharyya angle is advocated as the natural tool for comparing both short and long term Markov chain runs. Bounds on the convergence of the distance and mixing times are derived. Guided by the desire to compare different Markov chain models, especially in the setting of healthcare processes, a new distance function on the space of stochastic matrices is presented. It is a true distance measure which has a closed form and is efficient to implement for numerical evaluation. In the case of ergodic Markov chains, it is shown that considering either the Bhattacharyya angle on Markov sequences or the new stochastic matrix distance leads to the same distance between models.
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This work considers the Galerkin approximation and analysis for a hyperbolic integrodifferential equation, where the non-positive variable-sign kernel and nonlinear-nonlocal damping with both the weak and viscous damping effects are involved. We derive the long-time stability of the solution and its finite-time uniqueness. For the semi-discrete-in-space Galerkin scheme, we derive the long-time stability of the semi-discrete numerical solution and its finite-time error estimate by technical splitting of intricate terms. Then we further apply the centering difference method and the interpolating quadrature to construct a fully discrete Galerkin scheme and prove the long-time stability of the numerical solution and its finite-time error estimate by designing a new semi-norm. Numerical experiments are performed to verify the theoretical findings.
In this contribution, we address the numerical solutions of high-order asymptotic equivalent partial differential equations with the results of a lattice Boltzmann scheme for an inhomogeneous advection problem in one spatial dimension. We first derive a family of equivalent partial differential equations at various orders, and we compare the lattice Boltzmann experimental results with a spectral approximation of the differential equations. For an unsteady situation, we show that the initialization scheme at a sufficiently high order of the microscopic moments plays a crucial role to observe an asymptotic error consistent with the order of approximation. For a stationary long-time limit, we observe that the measured asymptotic error converges with a reduced order of precision compared to the one suggested by asymptotic analysis.
Physically motivated stochastic dynamics are often used to sample from high-dimensional distributions. However such dynamics often get stuck in specific regions of their state space and mix very slowly to the desired stationary state. This causes such systems to approximately sample from a metastable distribution which is usually quite different from the desired, stationary distribution of the dynamic. We rigorously show that, in the case of multi-variable discrete distributions, the true model describing the stationary distribution can be recovered from samples produced from a metastable distribution under minimal assumptions about the system. This follows from a fundamental observation that the single-variable conditionals of metastable distributions that satisfy a strong metastability condition are on average close to those of the stationary distribution. This holds even when the metastable distribution differs considerably from the true model in terms of global metrics like Kullback-Leibler divergence or total variation distance. This property allows us to learn the true model using a conditional likelihood based estimator, even when the samples come from a metastable distribution concentrated in a small region of the state space. Explicit examples of such metastable states can be constructed from regions that effectively bottleneck the probability flow and cause poor mixing of the Markov chain. For specific cases of binary pairwise undirected graphical models (i.e. Ising models), we extend our results to further rigorously show that data coming from metastable states can be used to learn the parameters of the energy function and recover the structure of the model.
A finite element method is introduced to track interface evolution governed by the level set equation. The method solves for the level set indicator function in a narrow band around the interface. An extension procedure, which is essential for a narrow band level set method, is introduced based on a finite element $L^2$- or $H^1$-projection combined with the ghost-penalty method. This procedure is formulated as a linear variational problem in a narrow band around the surface, making it computationally efficient and suitable for rigorous error analysis. The extension method is combined with a discontinuous Galerkin space discretization and a BDF time-stepping scheme. The paper analyzes the stability and accuracy of the extension procedure and evaluates the performance of the resulting narrow band finite element method for the level set equation through numerical experiments.
We systematically investigate the preservation of differential privacy in functional data analysis, beginning with functional mean estimation and extending to varying coefficient model estimation. Our work introduces a distributed learning framework involving multiple servers, each responsible for collecting several sparsely observed functions. This hierarchical setup introduces a mixed notion of privacy. Within each function, user-level differential privacy is applied to $m$ discrete observations. At the server level, central differential privacy is deployed to account for the centralised nature of data collection. Across servers, only private information is exchanged, adhering to federated differential privacy constraints. To address this complex hierarchy, we employ minimax theory to reveal several fundamental phenomena: from sparse to dense functional data analysis, from user-level to central and federated differential privacy costs, and the intricate interplay between different regimes of functional data analysis and privacy preservation. To the best of our knowledge, this is the first study to rigorously examine functional data estimation under multiple privacy constraints. Our theoretical findings are complemented by efficient private algorithms and extensive numerical evidence, providing a comprehensive exploration of this challenging problem.
We present weak approximations schemes of any order for the Heston model that are obtained by using the method developed by Alfonsi and Bally (2021). This method consists in combining approximation schemes calculated on different random grids to increase the order of convergence. We apply this method with either the Ninomiya-Victoir scheme (2008) or a second-order scheme that samples exactly the volatility component, and we show rigorously that we can achieve then any order of convergence. We give numerical illustrations on financial examples that validate the theoretical order of convergence. We also present promising numerical results for the multifactor/rough Heston model and hint at applications to other models, including the Bates model and the double Heston model.
We consider the problem of estimating the error when solving a system of differential algebraic equations. Richardson extrapolation is a classical technique that can be used to judge when computational errors are irrelevant and estimate the discretization error. We have simulated molecular dynamics with constraints using the GROMACS library and found that the output is not always amenable to Richardson extrapolation. We derive and illustrate Richardson extrapolation using a variety of numerical experiments. We identify two necessary conditions that are not always satisfied by the GROMACS library.
Clustering and outlier detection are two important tasks in data mining. Outliers frequently interfere with clustering algorithms to determine the similarity between objects, resulting in unreliable clustering results. Currently, only a few clustering algorithms (e.g., DBSCAN) have the ability to detect outliers to eliminate interference. For other clustering algorithms, it is tedious to introduce another outlier detection task to eliminate outliers before each clustering process. Obviously, how to equip more clustering algorithms with outlier detection ability is very meaningful. Although a common strategy allows clustering algorithms to detect outliers based on the distance between objects and clusters, it is contradictory to improving the performance of clustering algorithms on the datasets with outliers. In this paper, we propose a novel outlier detection approach, called ODAR, for clustering. ODAR maps outliers and normal objects into two separated clusters by feature transformation. As a result, any clustering algorithm can detect outliers by identifying clusters. Experiments show that ODAR is robust to diverse datasets. Compared with baseline methods, the clustering algorithms achieve the best on 7 out of 10 datasets with the help of ODAR, with at least 5% improvement in accuracy.
Many mathematical models of synaptic plasticity have been proposed to explain the diversity of plasticity phenomena observed in biological organisms. These models range from simple interpretations of Hebb's postulate, which suggests that correlated neural activity leads to increases in synaptic strength, to more complex rules that allow bidirectional synaptic updates, ensure stability, or incorporate additional signals like reward or error. At the same time, a range of learning paradigms can be observed behaviorally, from Pavlovian conditioning to motor learning and memory recall. Although it is difficult to directly link synaptic updates to learning outcomes experimentally, computational models provide a valuable tool for building evidence of this connection. In this chapter, we discuss several fundamental learning paradigms, along with the synaptic plasticity rules that might be used to implement them.
Multidimensional quaternion arrays (often referred to as "quaternion tensors") and their decompositions have recently gained increasing attention in various fields such as color and polarimetric imaging or video processing. Despite this growing interest, the theoretical development of quaternion tensors remains limited. This paper introduces a novel multilinear framework for quaternion arrays, which extends the classical tensor analysis to multidimensional quaternion data in a rigorous manner. Specifically, we propose a new definition of quaternion tensors as $\mathbb{H}\mathbb{R}$-multilinear forms, addressing the challenges posed by the non-commutativity of quaternion multiplication. Within this framework, we establish the Tucker decomposition for quaternion tensors and develop a quaternion Canonical Polyadic Decomposition (Q-CPD). We thoroughly investigate the properties of the Q-CPD, including trivial ambiguities, complex equivalent models, and sufficient conditions for uniqueness. Additionally, we present two algorithms for computing the Q-CPD and demonstrate their effectiveness through numerical experiments. Our results provide a solid theoretical foundation for further research on quaternion tensor decompositions and offer new computational tools for practitioners working with quaternion multiway data.
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