Search ResultsShowing 1-20 of 26
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arXiv:2301.11059 (Published 2023-01-26)
Global existence for perturbations of the 2D stochastic Navier-Stokes equations with space-time white noise
Comments: 38 PagesSubjects: 60H15We prove global in time well-posedness for perturbations of the 2D stochastic Navier-Stokes equations \begin{equation*} \partial_t u + u \cdot \nabla u = \Delta u - \nabla p + \zeta + \xi \;, \quad u (0, \cdot) = u_{0}(\cdot) \;, \quad \mathrm{div} (u) = 0 \;, \end{equation*} driven by additive space-time white noise $ \xi $, with perturbation $ \zeta $ in the H\"older-Besov space $\mathcal{C}^{-2 + 3\kappa} $, periodic boundary conditions and initial condition $ u_{0} \in \mathcal{C}^{-1 + \kappa} $ for any $ \kappa >0 $. The proof relies on an energy estimate which in turn builds on a dynamic high-low frequency decomposition and tools from paracontrolled calculus. Our argument uses that the solution to the linear equation is a $ \log$-correlated field, yielding a double exponential growth bound on the solution. Notably, our method does not rely on any explicit knowledge of the invariant measure to the SPDE, hence the perturbation $ \zeta $ is not restricted to the Cameron-Martin space of the noise, and the initial condition may be anticipative. Finally, we introduce a notion of weak solution that leads to well-posedness for all initial data $u_{0}$ in $ L^{2} $, the critical space of initial conditions.
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arXiv:2211.16167 (Published 2022-11-29)
Comparison theorem and stability under perturbation of transition rate matrices for regime-switching processes
Comments: 25 pagesCategories: math.PRA comparison theorem for state-dependent regime-switching diffusion processes is established, which enables us to control pathwisely the evolution of the state-dependent switching component simply by Markov chains. Moreover, a sharp estimate on the stability of Markovian regime-switching processes under the perturbation of transition rate matrices is provided. Our approach is based on the elaborate constructions of switching processes in the spirit of Skorokhod's representation theorem varying according to the problem being dealt with. In particular, this method can cope with the switching processes in an infinite state space and not necessarily being of birth-death type. As an application, some known results on ergodicity and stability of state-dependent regime-switching processes can be improved.
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arXiv:2209.11539 (Published 2022-09-23)
Quantile-constrained Wasserstein projections for robust interpretability of numerical and machine learning models
Robustness studies of black-box models is recognized as a necessary task for numerical models based on structural equations and predictive models learned from data. These studies must assess the model's robustness to possible misspecification of regarding its inputs (e.g., covariate shift). The study of black-box models, through the prism of uncertainty quantification (UQ), is often based on sensitivity analysis involving a probabilistic structure imposed on the inputs, while ML models are solely constructed from observed data. Our work aim at unifying the UQ and ML interpretability approaches, by providing relevant and easy-to-use tools for both paradigms. To provide a generic and understandable framework for robustness studies, we define perturbations of input information relying on quantile constraints and projections with respect to the Wasserstein distance between probability measures, while preserving their dependence structure. We show that this perturbation problem can be analytically solved. Ensuring regularity constraints by means of isotonic polynomial approximations leads to smoother perturbations, which can be more suitable in practice. Numerical experiments on real case studies, from the UQ and ML fields, highlight the computational feasibility of such studies and provide local and global insights on the robustness of black-box models to input perturbations.
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arXiv:2101.04573 (Published 2021-01-12)
Perturbations of copulas and Mixing properties
Comments: 22 pages, 3 figures, journal articleThis paper explores the impact of perturbations of copulas on the dependence properties of the Markov chains they generate. We consider Markov chains generated by perturbed copulas. Results are provided for the mixing coefficients $\beta_n$, $\psi_n$ and $\phi_n$. Several results are provided on mixing for the considered perturbations. New copula functions are provided in connection with perturbations of variables that induce other types of perturbation of copulas not considered in the literature.
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arXiv:1905.11329 (Published 2019-05-27)
Barabási-Albert random graph with multiple type edges with perturbation
Comments: 12 pagesCategories: math.PRIn this paper we introduce the perturbed version of the Barab\'asi-Albert random graph with multiple type edges and prove the existence of the (generalized) asymptotic degree distribution. Similarly to the non-perturbed case, the asymptotic degree distribution depends on the almost sure limit of the proportion of edges of different types. However, if there is perturbation then the resulting degree distribution will be deterministic, which is a major difference compared to the non-perturbed case.
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arXiv:1808.07248 (Published 2018-08-22)
Stability of regime-switching processes under perturbation of transition rate matrices
Comments: 25 pagesCategories: math.PRThis work is concerned with the stability of regime-switching processes under the perturbation of the transition rate matrices. From the viewpoint of application, two kinds of perturbations are studied: the size of the transition rate matrix is fixed, and only the values of entries are perturbed; the values of entries and the size of the transition matrix are all perturbed. Moreover, both regular and irregular coefficients of the underlying system are investigated, which clarifies the impact of the regularity of the coefficients on the stability of the underlying system.
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arXiv:1806.07411 (Published 2018-06-19)
Desynchronization of Random Dynamical System under Perturbation by an Intrinsic Noise
Comments: 26 pages, 11 figuresCategories: math.PRIn the theory of random dynamical systems (RDS), individuals with different initial states follow a same law of motion that is stochastically changing with time | called extrinsic noise. In the present work, intrin- sic noises for each individual are considered as a perturbation to an RDS. This gives rise to random Markov systems (RMS) in which the law of mo- tion is still stochastically changing with time, but individuals also exhibit statistically independent variations, with each transition having a small probability not to follow the law. As a consequence, two individuals in an RMS system go through stochastically distributed periods of synchro- nization and desynchronization, driven by extrinsic and intrinsic noises respectively. We show that in-sync time, e.g., escaping from a random attractor, has a symptotic geometric distribution.
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arXiv:1709.05137 (Published 2017-09-15)
Random walk on a perturbation of the infinitely-fast mixing interchange process
Comments: 21 pagesCategories: math.PRWe consider a random walk in dimension $d\geq 1$ in a dynamic random environment evolving as an interchange process with rate $\gamma>0$. We only assume that the annealed drift is non-zero. We prove that the empirical velocity of the walker $X_t/t$ eventually lies in an arbitrary small ball around the annealed drift if we choose $\gamma$ large enough. This statement is thus a perturbation of the case $\gamma =+\infty$ where the environment is refreshed between each step of the walker. We extend three-way part of the results of HS15, where the environment was given by the $1$ dimensional exclusion process: $(i)$ We deal with any dimension $d\geq 1$; $(ii)$ Each particle of the interchange process carries a transition vector chosen according to an arbitrary law $\mu$; $(iii)$ We show that $X_t/t$ is not only in the same direction of the annealed drift, but that it is also close to it.
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arXiv:1707.01207 (Published 2017-07-05)
The $\ell_\infty$ Perturbation of HOSVD and Low Rank Tensor Denoising
The higher order singular value decomposition (HOSVD) of tensors is a generalization of matrix SVD. The perturbation analysis of HOSVD under random noise is more delicate than its matrix counterpart. Recent progress has been made in Richard and Montanari (2014), Zhang and Xia (2017) and Liu et al. (2017) demonstrating that minimax optimal singular spaces estimation and low rank tensor recovery in $\ell_2$-norm can be obtained through polynomial time algorithms. In this paper, we analyze the HOSVD perturbation under Gaussian noise based on a second order method, which leads to an estimator of singular vectors with sharp bound in $\ell_\infty$-norm. A low rank tensor denoising estimator is then proposed which achieves a fast convergence rate characterizing the entry-wise deviations. The advantages of these $\ell_\infty$-norm bounds are displayed in applications including high dimensional clustering and sub-tensor localizations.
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arXiv:1701.05978 (Published 2017-01-21)
Perturbations and Projections of Kalman-Bucy Semigroups Motivated by Methods in Data Assimilation
Categories: math.PRThe purpose of this work is to analyse the effect of various perturbations and projections of Kalman-Bucy semigroups and Riccati equations. The original motivation was to understand the behaviour of various regulation methods used in ensemble Kalman filtering (EnKF). For example, covariance inflation-type methods (perturbations) and covariance localisation methods (projections) are commonly used in the EnKF literature to ensure well-posedness of the sample covariance (e.g. sufficient rank) and to `move' the sample covariance closer (in some sense) to the Riccati flow of the true Kalman filter. In the limit, as the number of samples tends to infinity, these methods drive the sample covariance toward a solution of a perturbed, or projected, version of the standard (Kalman-Bucy) differential Riccati equation. The behaviour of this modified Riccati equation is investigated here. Results concerning continuity (in terms of the perturbations), boundedness, and convergence of the Riccati flow to a limit are given. In terms of the limiting filters, results characterising the error between the perturbed/projected and nominal conditional distributions are given. New projection-type models and ideas are also discussed within the EnKF framework; e.g. projections onto so-called Bose-Mesner algebras. This work is generally important in understanding the limiting bias in both the EnKF empirical mean and covariance when applying regularisation. Finally, we note the perturbation and projection models considered herein are also of interest on their own, and in other applications such as differential games, control of stochastic and jump processes, and robust control theory, etc.
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arXiv:1701.02597 (Published 2017-01-10)
Perturbations by random matrices
Comments: 25 pages, 4 figuresCategories: math.PRWe provide a perturbative expansion for the empirical spectral distribution of a Hermitian matrix with large size perturbed by a random matrix with small operator norm whose entries in the eigenvector basis of the first one are independent with a variance profile. We prove that, depending on the order of magnitude of the perturbation, several regimes can appear, called perturbative and semi-perturbative regimes. Depending on the regime, the leading terms of the expansion are either related to free probability theory or to the one-dimensional Gaussian free field.
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arXiv:1603.07464 (Published 2016-03-24)
On Perturbations of Stein Operator
Categories: math.PRIn this paper, we obtain Stein operator for sum of $n$ independent random variables (rvs) which is shown as perturbation of negative binomial (NB) operator. Comparing the operator with NB operator, we derive the error bounds for total variation distance by matching parameters. Also, three parameters approximation for such a sum is considered and is shown to improve the existing bounds in the literature. Finally, an application of our results to a function of waiting time for $(k_1,k_2)$-events is given.
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arXiv:1602.05247 (Published 2016-02-17)
The Computation of Key Properties of Markov Chains via Perturbations
Computational procedures for the stationary probability distribution, the group inverse of the Markovian kernel and the mean first passage times of an irreducible Markov chain, are developed using perturbations. The derivation of these expressions involves the solution of systems of linear equations and, structurally, inevitably the inverses of matrices. By using a perturbation technique, starting from a simple base where no such derivations are formally required, we update a sequence of matrices, formed by linking the solution procedures via generalized matrix inverses and utilising matrix and vector multiplications. Six different algorithms are given, some modifications are discussed, and numerical comparisons made using a test example. The derivations are based upon the ideas outlined in Hunter, J.J., The computation of stationary distributions of Markov chains through perturbations, Journal of Applied Mathematics and Stochastic Analysis, 4, 29-46, (1991).
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arXiv:1508.05299 (Published 2015-08-21)
Stable states of perturbed Markov chains
Given an infinitesimal perturbation of a discrete-time finite Markov chain, we seek the states that are stable despite the perturbation, \textit{i.e.} the states whose weights in the stationary distributions can be bounded away from $0$ as the noise fades away. Chemists, economists, and computer scientists have been studying irreducible perturbations built with exponential maps. Under these assumptions, Young proved the existence of and computed the stable states in cubic time. We fully drop these assumptions, generalize Young's technique, and show that stability is decidable as long as $f\in O(g)$ is. Furthermore, if the perturbation maps (and their multiplications) satisfy $f\in O(g)$ or $g\in O(f)$, we prove the existence of and compute the stable states and the metastable dynamics at all time scales where some states vanish. Conversely, if the big-$O$ assumption does not hold, we build a perturbation with these maps and no stable state. Our algorithm also runs in cubic time despite the general assumptions and the additional work. Proving the correctness of the algorithm relies on new or rephrased results in Markov chain theory, and on algebraic abstractions thereof.
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arXiv:1506.02764 (Published 2015-06-09)
Perturbation of linear forms of singular vectors under Gaussian noise
Categories: math.PRLet $A\in\mathbb{R}^{m\times n}$ be a matrix of rank $r$ with singular value decomposition (SVD) $A=\sum_{k=1}^r\sigma_k (u_k\otimes v_k),$ where $\{\sigma_k, k=1,\ldots,r\}$ are singular values of $A$ (arranged in a non-increasing order) and $u_k\in {\mathbb R}^m, v_k\in {\mathbb R}^n, k=1,\ldots, r$ are the corresponding left and right orthonormal singular vectors. Let $\tilde{A}=A+X$ be a noisy observation of $A,$ where $X\in\mathbb{R}^{m\times n}$ is a random matrix with i.i.d. Gaussian entries, $X_{ij}\sim\mathcal{N}(0,\tau^2),$ and consider its SVD $\tilde{A}=\sum_{k=1}^{m\wedge n}\tilde{\sigma}_k(\tilde{u}_k\otimes\tilde{v}_k)$ with singular values $\tilde{\sigma}_1\geq\ldots\geq\tilde{\sigma}_{m\wedge n}$ and singular vectors $\tilde{u}_k,\tilde{v}_k,k=1,\ldots, m\wedge n.$ The goal of this paper is to develop sharp concentration bounds for linear forms $\langle \tilde u_k,x\rangle, x\in {\mathbb R}^m$ and $\langle \tilde v_k,y\rangle, y\in {\mathbb R}^n$ of the perturbed (empirical) singular vectors in the case when the singular values of $A$ are distinct and, more generally, concentration bounds for bilinear forms of projection operators associated with SVD. In particular, the results imply upper bounds of the order $O\biggl(\sqrt{\frac{\log(m+n)}{m\vee n}}\biggr)$ (holding with a high probability) on $$\max_{1\leq i\leq m}\big|\big<\tilde{u}_k-\sqrt{1+b_k}u_k,e_i^m\big>\big|\ \ {\rm and} \ \ \max_{1\leq j\leq n}\big|\big<\tilde{v}_k-\sqrt{1+b_k}v_k,e_j^n\big>\big|,$$ where $b_k$ are properly chosen constants characterizing the bias of empirical singular vectors $\tilde u_k, \tilde v_k$ and $\{e_i^m,i=1,\ldots,m\}, \{e_j^n,j=1,\ldots,n\}$ are the canonical bases of $\mathbb{R}^m, {\mathbb R}^n,$ respectively.
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arXiv:1403.5125 (Published 2014-03-20)
Perturbation of the loop measure
Categories: math.PRThe loop measure is associated with a Markov generator. We compute the variation of the loop measure induced by an in nitesimal variation of the generator a ecting the killing rates or the jumping rates.
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arXiv:1311.3066 (Published 2013-11-13)
On the effect of perturbation of conditional probabilities in total variation
Journal: Statistics & Probability Letters 88 (2014), pp. 1-8Categories: math.PRKeywords: perturbation, total variation metric affect, regular conditional probabilities, resulting product probability measure, product spaceTags: journal articleA celebrated result by A. Ionescu Tulcea provides a construction of a probability measure on a product space given a sequence of regular conditional probabilities. We study how the perturbations of the latter in the total variation metric affect the resulting product probability measure.
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arXiv:1206.4983 (Published 2012-06-21)
Coupling from the past times with ambiguities and perturbations of interacting particle systems
Comments: This paper is an extended and revised version of an earlier manuscript available as arXiv:0712.0072, where the results were limited to perturbations of RN+YpR nucleotide substitution modelsCategories: math.PRWe discuss coupling from the past techniques (CFTP) for perturbations of interacting particle systems on the d-dimensional integer lattice, with a finite set of states, within the framework of the graphical construction of the dynamics based on Poisson processes. We first develop general results for what we call CFTP times with ambiguities. These are analogous to classical coupling (from the past) times, except that the coupling property holds only provided that some ambiguities concerning the stochastic evolution of the system are resolved. If these ambiguities are rare enough on average, CFTP times with ambiguities can be used to build actual CFTP times, whose properties can be controlled in terms of those of the original CFTP time with ambiguities. We then prove a general perturbation result, which can be stated informally as follows. Start with an interacting particle system possessing a CFTP time whose definition involves the exploration of an exponentially integrable number of points in the graphical construction, and which satisfies the positive rates property. Then consider a perturbation obtained by adding new transitions to the original dynamics. Our result states that, provided that the perturbation is small enough (in the sense of small enough rates), the perturbed interacting particle system too possesses a CFTP time (with nice properties such as an exponentially decaying tail). The proof consists in defining a CFTP time with ambiguities for the perturbed dynamics, from the CFTP time for the unperturbed dynamics. Finally, we discuss examples of particle systems to which this result can be applied. Concrete examples include a class of neighbor-dependent nucleotide substitution model, and variations of the classical voter model, illustrating the ability of our approach to go beyond the case of weakly interacting particle systems.
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arXiv:1102.3905 (Published 2011-02-18)
Perturbation of Burkholder's martingale transform and Monge--Ampère equation
Comments: 45 pages, 13 figuresLet $\{d_k\}_{k \geq 0}$ be a complex martingale difference in $L^p[0,1],$ where $1<p<\infty,$ and $\{\e_k\}_{k \geq 0}$ a sequence in $\{\pm 1\}.$ We obtain the following generalization of Burkholder's famous result. If $\tau \in [-\frac 12, \frac 12]$ and $n \in \Z_+$ then $$|\sum_{k=0}^n{(\{c} \e_k \tau) d_k}|_{L^p([0,1], \C^2)} \leq ((p^*-1)^2 + \tau^2)^{\frac 12}|\sum_{k=0}^n{d_k}|_{L^p([0,1], \C)},$$ where $((p^*-1)^2 + \tau^2)^{\frac 12}$ is sharp and $p^*-1 = \max\{p-1, \frac 1{p-1}\}.$ For $2\leq p<\infty$ the result is also true with sharp constant for $\tau \in \R.$
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Perturbation of matrices and non-negative rank with a view toward statistical models
Comments: 13 pages, 3 figures. A theorem has been rewritten, and some improvements in the presentations have been implementedIn this paper we study how perturbing a matrix changes its non-negative rank. We prove that the non-negative rank is upper-semicontinuos and we describe some special families of perturbations. We show how our results relate to Statistics in terms of the study of Maximum Likelihood Estimation for mixture models.