亚洲男人的天堂2018av,欧美草比,久久久久久免费视频精选,国色天香在线看免费,久久久久亚洲av成人片仓井空

We provide a clear and concise introduction to the subjects of inverse problems and data assimilation, and their inter-relations. The first part of our notes covers inverse problems; this refers to the study of how to estimate unknown model parameters from data. The second part of our notes covers data assimilation; this refers to a particular class of inverse problems in which the unknown parameter is the initial condition (and/or state) of a dynamical system, and the data comprises partial and noisy observations of the state. The third and final part of our notes describes the use of data assimilation methods to solve generic inverse problems by introducing an artificial algorithmic time. Our notes cover, among other topics, maximum a posteriori estimation, (stochastic) gradient descent, variational Bayes, Monte Carlo, importance sampling and Markov chain Monte Carlo for inverse problems; and 3DVAR, 4DVAR, extended and ensemble Kalman filters, and particle filters for data assimilation. Each of parts one and two starts with a chapter on the Bayesian formulation, in which the problem solution is given by a posterior distribution on the unknown parameter. Then the following chapter specializes the Bayesian formulation to a linear-Gaussian setting where explicit characterization of the posterior is possible and insightful. The next two chapters explore methods to extract information from the posterior in nonlinear and non-Gaussian settings using optimization and Gaussian approximations. The final two chapters describe sampling methods that can reproduce the full posterior in the large sample limit. Each chapter closes with a bibliography containing citations to alternative pedagogical literature and to relevant research literature. We also include a set of exercises at the end of parts one and two. Our notes are thus useful for both classroom teaching and self-guided study.

Diffusion models achieve state-of-the-art performance in various generation tasks. However, their theoretical foundations fall far behind. This paper studies score approximation, estimation, and distribution recovery of diffusion models, when data are supported on an unknown low-dimensional linear subspace. Our result provides sample complexity bounds for distribution estimation using diffusion models. We show that with a properly chosen neural network architecture, the score function can be both accurately approximated and efficiently estimated. Furthermore, the generated distribution based on the estimated score function captures the data geometric structures and converges to a close vicinity of the data distribution. The convergence rate depends on the subspace dimension, indicating that diffusion models can circumvent the curse of data ambient dimensionality.

Modern time series analysis requires the ability to handle datasets that are inherently high-dimensional; examples include applications in climatology, where measurements from numerous sensors must be taken into account, or inventory tracking of large shops, where the dimension is defined by the number of tracked items. The standard way to mitigate computational issues arising from the high dimensionality of the data is by applying some dimension reduction technique that preserves the structural properties of the ambient space. The dissimilarity between two time series is often measured by ``discrete'' notions of distance, e.g. the dynamic time warping or the discrete Fr\'echet distance. Since all these distance functions are computed directly on the points of a time series, they are sensitive to different sampling rates or gaps. The continuous Fr\'echet distance offers a popular alternative which aims to alleviate this by taking into account all points on the polygonal curve obtained by linearly interpolating between any two consecutive points in a sequence. We study the ability of random projections \`a la Johnson and Lindenstrauss to preserve the continuous Fr\'echet distance of polygonal curves by effectively reducing the dimension. In particular, we show that one can reduce the dimension to $O(\epsilon^{-2} \log N)$, where $N$ is the total number of input points while preserving the continuous Fr\'echet distance between any two determined polygonal curves within a factor of $1\pm \epsilon$. We conclude with applications on clustering.

We tackle the problem of establishing the soundness of approximate bisimilarity with respect to PCTL and its relaxed semantics. To this purpose, we consider a notion of bisimilarity inspired by the one introduced by Desharnais, Laviolette, and Tracol, and parametric with respect to an approximation error $\delta$, and to the depth $n$ of the observation along traces. Essentially, our soundness theorem establishes that, when a state $q$ satisfies a given formula up-to error $\delta$ and steps $n$, and $q$ is bisimilar to $q'$ up-to error $\delta'$ and enough steps, we prove that $q'$ also satisfies the formula up-to a suitable error $\delta"$ and steps $n$. The new error $\delta"$ is computed from $\delta$, $\delta'$ and the formula, and only depends linearly on $n$. We provide a detailed overview of our soundness proof. We extend our bisimilarity notion to families of states, thus obtaining an asymptotic equivalence on such families. We then consider an asymptotic satisfaction relation for PCTL formulae, and prove that asymptotically equivalent families of states asymptotically satisfy the same formulae.

2-Opt is probably the most basic local search heuristic for the TSP. This heuristic achieves amazingly good results on real world Euclidean instances both with respect to running time and approximation ratio. There are numerous experimental studies on the performance of 2-Opt. However, the theoretical knowledge about this heuristic is still very limited. Not even its worst case running time on 2-dimensional Euclidean instances was known so far. We clarify this issue by presenting, for every $p\in\mathbb{N}$, a family of $L_p$ instances on which 2-Opt can take an exponential number of steps. Previous probabilistic analyses were restricted to instances in which $n$ points are placed uniformly at random in the unit square $[0,1]^2$. We consider a more advanced model in which the points can be placed independently according to general distributions on $[0,1]^d$, for an arbitrary $d\ge 2$. In particular, we allow different distributions for different points. We study the expected number of local improvements in terms of the number $n$ of points and the maximal density $\phi$ of the probability distributions. We show an upper bound on the expected length of any 2-Opt improvement path of $\tilde{O}(n^{4+1/3}\cdot\phi^{8/3})$. When starting with an initial tour computed by an insertion heuristic, the upper bound on the expected number of steps improves even to $\tilde{O}(n^{4+1/3-1/d}\cdot\phi^{8/3})$. If the distances are measured according to the Manhattan metric, then the expected number of steps is bounded by $\tilde{O}(n^{4-1/d}\cdot\phi)$. In addition, we prove an upper bound of $O(\sqrt[d]{\phi})$ on the expected approximation factor with respect to all $L_p$ metrics. Let us remark that our probabilistic analysis covers as special cases the uniform input model with $\phi=1$ and a smoothed analysis with Gaussian perturbations of standard deviation $\sigma$ with $\phi\sim1/\sigma^d$.

Differential machine learning (DML) is a recently proposed technique that uses samplewise state derivatives to regularize least square fits to learn conditional expectations of functionals of stochastic processes as functions of state variables. Exploiting the derivative information leads to fewer samples than a vanilla ML approach for the same level of precision. This paper extends the methodology to parametric problems where the processes and functionals also depend on model and contract parameters, respectively. In addition, we propose adaptive parameter sampling to improve relative accuracy when the functionals have different magnitudes for different parameter sets. For calibration, we construct pricing surrogates for calibration instruments and optimize over them globally. We discuss strategies for robust calibration. We demonstrate the usefulness of our methodology on one-factor Cheyette models with benchmark rate volatility specification with an extra stochastic volatility factor on (two-curve) caplet prices at different strikes and maturities, first for parametric pricing, and then by calibrating to a given caplet volatility surface. To allow convenient and efficient simulation of processes and functionals and in particular the corresponding computation of samplewise derivatives, we propose to specify the processes and functionals in a low-code way close to mathematical notation which is then used to generate efficient computation of the functionals and derivatives in TensorFlow.

The Laplacian-constrained Gaussian Markov Random Field (LGMRF) is a common multivariate statistical model for learning a weighted sparse dependency graph from given data. This graph learning problem is formulated as a maximum likelihood estimation (MLE) of the precision matrix, subject to Laplacian structural constraints, with a sparsity-inducing penalty term. This paper aims to solve this learning problem accurately and efficiently. First, since the commonly-used $\ell_1$-norm penalty is less appropriate in this setting, we employ the nonconvex minimax concave penalty (MCP), which promotes sparse solutions with lower estimation bias. Second, as opposed to most existing first-order methods for this problem, we base our method on the second-order proximal Newton approach to obtain an efficient solver for large-scale networks. This approach is considered the most efficient for the related graphical LASSO problem and allows for several algorithmic features we exploit, such as using Conjugate Gradients, preconditioning, and splitting to active/free sets. Numerical experiments demonstrate the advantages of the proposed method in terms of \emph{both} computational complexity and graph learning accuracy compared to existing methods.

This paper characterizes the impact of covariate serial dependence on the non-asymptotic estimation error bound of penalized regressions (PRs). Focusing on the direct relationship between the degree of cross-correlation between covariates and the estimation error bound of PRs, we show that orthogonal or weakly cross-correlated stationary AR processes can exhibit high spurious correlations caused by serial dependence. We provide analytical results on the distribution of the sample cross-correlation in the case of two orthogonal Gaussian AR(1) processes, and extend and validate them through an extensive simulation study. Furthermore, we introduce a new procedure to mitigate spurious correlations in a time series setting, applying PRs to pre-whitened (ARMA filtered) time series. We show that under mild assumptions our procedure allows both to reduce the estimation error and to develop an effective forecasting strategy. The estimation accuracy of our proposal is validated through additional simulations, as well as an empirical application to a large set of monthly macroeconomic time series relative to the Euro Area.

We study the interplay between the data distribution and Q-learning-based algorithms with function approximation. We provide a unified theoretical and empirical analysis as to how different properties of the data distribution influence the performance of Q-learning-based algorithms. We connect different lines of research, as well as validate and extend previous results. We start by reviewing theoretical bounds on the performance of approximate dynamic programming algorithms. We then introduce a novel four-state MDP specifically tailored to highlight the impact of the data distribution in the performance of Q-learning-based algorithms with function approximation, both online and offline. Finally, we experimentally assess the impact of the data distribution properties on the performance of two offline Q-learning-based algorithms under different environments. According to our results: (i) high entropy data distributions are well-suited for learning in an offline manner; and (ii) a certain degree of data diversity (data coverage) and data quality (closeness to optimal policy) are jointly desirable for offline learning.

The paper addresses the problem of estimation of the model parameters of the logistic exponential distribution based on progressive type-I hybrid censored sample. The maximum likelihood estimates are obtained and computed numerically using Newton-Raphson method. Further, the Bayes estimates are derived under squared error, LINEX and generalized entropy loss functions. Two types (independent and bivariate) of prior distributions are considered for the purpose of Bayesian estimation. It is seen that the Bayes estimates are not of explicit forms.Thus, Lindley's approximation technique is employed to get approximate Bayes estimates. Interval estimates of the parameters based on normal approximate of the maximum likelihood estimates and normal approximation of the log-transformed maximum likelihood estimates are constructed. The highest posterior density credible intervals are obtained by using the importance sampling method. Furthermore, numerical computations are reported to review some of the results obtained in the paper. A real life dataset is considered for the purpose of illustrations.