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This paper proposes a new approach to identifying the effective cointegration rank in high-dimensional unit-root (HDUR) time series from a prediction perspective using reduced-rank regression. For a HDUR process $\mathbf{x}_t\in \mathbb{R}^N$ and a stationary series $\mathbf{y}_t\in \mathbb{R}^p$ of interest, our goal is to predict future values of $\mathbf{y}_t$ using $\mathbf{x}_t$ and lagged values of $\mathbf{y}_t$. The proposed framework consists of a two-step estimation procedure. First, the Principal Component Analysis is used to identify all cointegrating vectors of $\mathbf{x}_t$. Second, the co-integrated stationary series are used as regressors, together with some lagged variables of $\mathbf{y}_t$, to predict $\mathbf{y}_t$. The estimated reduced rank is then defined as the effective cointegration rank of $\mathbf{x}_t$. Under the scenario that the autoregressive coefficient matrices are sparse (or of low-rank), we apply the Least Absolute Shrinkage and Selection Operator (or the reduced-rank techniques) to estimate the autoregressive coefficients when the dimension involved is high. Theoretical properties of the estimators are established under the assumptions that the dimensions $p$ and $N$ and the sample size $T \to \infty$. Both simulated and real examples are used to illustrate the proposed framework, and the empirical application suggests that the proposed procedure fares well in predicting stock returns.

We present a comprehensive computational study of a class of linear system solvers, called {\it Triangle Algorithm} (TA) and {\it Centering Triangle Algorithm} (CTA), developed by Kalantari \cite{kalantari23}. The algorithms compute an approximate solution or minimum-norm solution to $Ax=b$ or $A^TAx=A^Tb$, where $A$ is an $m \times n$ real matrix of arbitrary rank. The algorithms specialize when $A$ is symmetric positive semi-definite. Based on the description and theoretical properties of TA and CTA from \cite{kalantari23}, we give an implementation of the algorithms that is easy-to-use for practitioners, versatile for a wide range of problems, and robust in that our implementation does not necessitate any constraints on $A$. Next, we make computational comparisons of our implementation with the Matlab implementations of two state-of-the-art algorithms, GMRES and ``lsqminnorm". We consider square and rectangular matrices, for $m$ up to $10000$ and $n$ up to $1000000$, encompassing a variety of applications. These results indicate that our implementation outperforms GMRES and ``lsqminnorm" both in runtime and quality of residuals. Moreover, the relative residuals of CTA decrease considerably faster and more consistently than GMRES, and our implementation provides high precision approximation, faster than GMRES reports lack of convergence. With respect to ``lsqminnorm", our implementation runs faster, producing better solutions. Additionally, we present a theoretical study in the dynamics of iterations of residuals in CTA and complement it with revealing visualizations. Lastly, we extend TA for LP feasibility problems, handling non-negativity constraints. Computational results show that our implementation for this extension is on par with those of TA and CTA, suggesting applicability in linear programming and related problems.

A common goal throughout science and engineering is to solve optimization problems constrained by computational models. However, in many cases a high-fidelity numerical emulation of systems cannot be optimized due to code complexity and computational costs which prohibit the use of intrusive and many query algorithms. Rather, lower-fidelity models are constructed to enable intrusive algorithms for large-scale optimization. As a result of the discrepancy between high and low-fidelity models, optimal solutions determined using low-fidelity models are frequently far from true optimality. In this article we introduce a novel approach that uses post-optimality sensitivities with respect to model discrepancy to update the optimization solution. Limited high-fidelity data is used to calibrate the model discrepancy in a Bayesian framework which in turn is propagated through post-optimality sensitivities of the low-fidelity optimization problem. Our formulation exploits structure in the post-optimality sensitivity operator to achieve computational scalability. Numerical results demonstrate how an optimal solution computed using a low-fidelity model may be significantly improved with limited evaluations of a high-fidelity model.

Noiseless compressive sensing is a protocol that enables undersampling and later recovery of a signal without loss of information. This compression is possible because the signal is usually sufficiently sparse in a given basis. Currently, the algorithm offering the best tradeoff between compression rate, robustness, and speed for compressive sensing is the LASSO (l1-norm bias) algorithm. However, many studies have pointed out the possibility that the implementation of lp-norms biases, with p smaller than one, could give better performance while sacrificing convexity. In this work, we focus specifically on the extreme case of the l0-based reconstruction, a task that is complicated by the discontinuity of the loss. In the first part of the paper, we describe via statistical physics methods, and in particular the replica method, how the solutions to this optimization problem are arranged in a clustered structure. We observe two distinct regimes: one at low compression rate where the signal can be recovered exactly, and one at high compression rate where the signal cannot be recovered accurately. In the second part, we present two message-passing algorithms based on our first results for the l0-norm optimization problem. The proposed algorithms are able to recover the signal at compression rates higher than the ones achieved by LASSO while being computationally efficient.

The time continuous Volterra equations valued in $\mathbb{R}$ with nonnegative resolvent kernels have two basic monotone properties. The first is that any two solution curves do not intersect with suitable given signals. The second is that the solutions to the autonomous equations are monotone. The so-called CM-preserving schemes (Comm. Math. Sci., 2021,19(5), 1301-1336) have been proposed to preserve the complete monotonicity property and thus these monotonicity properties but they are restricted to uniform meshes. In this work, through an analogue of the convolution on nonuniform meshes, we introduce the concept of ``right complementary monotone'' (R-CMM) kernels in the discrete level for nonuniform meshes, which is an analogue of the CM-preserving property but much more flexible. We prove that the discrete solutions preserve these two monotone properties if the discretized kernel satisfies R-CMM property. Technically, we highly rely on the resolvent kernels to achieve this.

The idea of embedding optimization problems into deep neural networks as optimization layers to encode constraints and inductive priors has taken hold in recent years. Most existing methods focus on implicitly differentiating Karush-Kuhn-Tucker (KKT) conditions in a way that requires expensive computations on the Jacobian matrix, which can be slow and memory-intensive. In this paper, we developed a new framework, named Alternating Differentiation (Alt-Diff), that differentiates optimization problems (here, specifically in the form of convex optimization problems with polyhedral constraints) in a fast and recursive way. Alt-Diff decouples the differentiation procedure into a primal update and a dual update in an alternating way. Accordingly, Alt-Diff substantially decreases the dimensions of the Jacobian matrix especially for optimization with large-scale constraints and thus increases the computational speed of implicit differentiation. We show that the gradients obtained by Alt-Diff are consistent with those obtained by differentiating KKT conditions. In addition, we propose to truncate Alt-Diff to further accelerate the computational speed. Under some standard assumptions, we show that the truncation error of gradients is upper bounded by the same order of variables' estimation error. Therefore, Alt-Diff can be truncated to further increase computational speed without sacrificing much accuracy. A series of comprehensive experiments validate the superiority of Alt-Diff.

Trajectory data has the potential to greatly benefit a wide-range of real-world applications, such as tracking the spread of the disease through people's movement patterns and providing personalized location-based services based on travel preference. However, privay concerns and data protection regulations have limited the extent to which this data is shared and utilized. To overcome this challenge, local differential privacy provides a solution by allowing people to share a perturbed version of their data, ensuring privacy as only the data owners have access to the original information. Despite its potential, existing point-based perturbation mechanisms are not suitable for real-world scenarios due to poor utility, dependence on external knowledge, high computational overhead, and vulnerability to attacks. To address these limitations, we introduce LDPTrace, a novel locally differentially private trajectory synthesis framework. Our framework takes into account three crucial patterns inferred from users' trajectories in the local setting, allowing us to synthesize trajectories that closely resemble real ones with minimal computational cost. Additionally, we present a new method for selecting a proper grid granularity without compromising privacy. Our extensive experiments using real-world data, various utility metrics and attacks, demonstrate the efficacy and efficiency of LDPTrace.

PCA-Net is a recently proposed neural operator architecture which combines principal component analysis (PCA) with neural networks to approximate operators between infinite-dimensional function spaces. The present work develops approximation theory for this approach, improving and significantly extending previous work in this direction: First, a novel universal approximation result is derived, under minimal assumptions on the underlying operator and the data-generating distribution. Then, two potential obstacles to efficient operator learning with PCA-Net are identified, and made precise through lower complexity bounds; the first relates to the complexity of the output distribution, measured by a slow decay of the PCA eigenvalues. The other obstacle relates to the inherent complexity of the space of operators between infinite-dimensional input and output spaces, resulting in a rigorous and quantifiable statement of the curse of dimensionality. In addition to these lower bounds, upper complexity bounds are derived. A suitable smoothness criterion is shown to ensure an algebraic decay of the PCA eigenvalues. Furthermore, it is shown that PCA-Net can overcome the general curse of dimensionality for specific operators of interest, arising from the Darcy flow and the Navier-Stokes equations.

Recently, more and more attention has been focused on the intellectual property protection of deep neural networks (DNNs), promoting DNN watermarking to become a hot research topic. Compared with embedding watermarks directly into DNN parameters, inserting trigger-set watermarks enables us to verify the ownership without knowing the internal details of the DNN, which is more suitable for application scenarios. The cost is we have to carefully craft the trigger samples. Mainstream methods construct the trigger samples by inserting a noticeable pattern to the clean samples in the spatial domain, which does not consider sample imperceptibility, sample robustness and model robustness, and therefore has limited the watermarking performance and the model generalization. It has motivated the authors in this paper to propose a novel DNN watermarking method based on Fourier perturbation analysis and frequency sensitivity clustering. First, we analyze the perturbation impact of different frequency components of the input sample on the task functionality of the DNN by applying random perturbation. Then, by K-means clustering, we determine the frequency components that result in superior watermarking performance for crafting the trigger samples. Our experiments show that the proposed work not only maintains the performance of the DNN on its original task, but also provides better watermarking performance compared with related works.

Under-approximations of reachable sets and tubes have been receiving growing research attention due to their important roles in control synthesis and verification. Available under-approximation methods applicable to continuous-time linear systems typically assume the ability to compute transition matrices and their integrals exactly, which is not feasible in general, and/or suffer from high computational costs. In this note, we attempt to overcome these drawbacks for a class of linear time-invariant (LTI) systems, where we propose a novel method to under-approximate finite-time forward reachable sets and tubes, utilizing approximations of the matrix exponential and its integral. In particular, we consider the class of continuous-time LTI systems with an identity input matrix and initial and input values belonging to full dimensional sets that are affine transformations of closed unit balls. The proposed method yields computationally efficient under-approximations of reachable sets and tubes, when implemented using zonotopes, with first-order convergence guarantees in the sense of the Hausdorff distance. To illustrate its performance, we implement our approach in three numerical examples, where linear systems of dimensions ranging between 2 and 200 are considered.