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We obtain the upper error bounds of robust estimators for mean vector, using the median-of-means (MOM) method. The method is designed to handle data with heavy tails and contamination, with only a finite second moment, which is weaker than many others, relying on the VC dimension rather than the Rademacher complexity to measure statistical complexity. This allows us to implement MOM in covariance estimation, without imposing conditions such as $L$-sub-Gaussian or $L_{4}-L_{2}$ norm equivalence. In particular, we derive a new robust estimator, the MOM version of the halfspace depth, along with error bounds for mean estimation in any norm.

In this paper, we obtain a $p$-th moment bound for the suprema of a log-concave-tailed nonhomogeneous chaos process, which is optimal in some special cases. A crucial ingredient of the proof is a novel decoupling inequality, which may be of independent interest. With this $p$-th moment bound, we show two uniform Hanson-Wright type deviation inequalities for $\alpha$-subexponential entries ($1\le \alpha\le 2$), which recover some known results. As applications, we prove the restricted isometry property of partial random circulant matrices and time-frequency structured random matrices induced by standard $\alpha$-subexponential vectors ($1\le \alpha\le 2$), which extends the previously known results for the subgaussian case.

In the study of the supremum of stochastic processes, Talagrand's chaining functionals and his generic chaining method are heavily related to the distribution of stochastic processes. In the present paper, we construct Talagrand's type functionals in the general distribution case and obtain the upper bound for the suprema of all $p$-th moments of the stochastic process using the generic chaining method. As applications, we obtained the Johnson-Lindenstrauss lemma, the upper bound for the supremum of all $p$-th moment of order 2 Gaussian chaos, and convex signal recovery in our setting.

This paper studies the moments for the spectral norm of random matrices with dependent entries. In particular, we consider a random matrix $BA$, where $A$ is a random matrix with independent mean zero subexponential entries, and $B$ is a deterministic matrix. We show a sharp moment bound for the spectral norm of an $m\times n$ matrix $BA$ based on a comparison theorem due to Lata{\l}a, van Handel and Youssef. Applying this result, we prove an estimate of the smallest singular value of an $N\times n$ random subexponential matrix., Comment: arXiv admin note: text overlap with arXiv:0812.2432 by other authors

In this paper, we consider estimating spot/instantaneous volatility matrices of high-frequency data collected for a large number of assets. We first combine classic nonparametric kernel-based smoothing with a generalised shrinkage technique in the matrix estimation for noise-free data under a uniform sparsity assumption, a natural extension of the approximate sparsity commonly used in the literature. The uniform consistency property is derived for the proposed spot volatility matrix estimator with convergence rates comparable to the optimal minimax one. For the high-frequency data contaminated by microstructure noise, we introduce a localised pre-averaging estimation method that reduces the effective magnitude of the noise. We then use the estimation tool developed in the noise-free scenario, and derive the uniform convergence rates for the developed spot volatility matrix estimator. We further combine the kernel smoothing with the shrinkage technique to estimate the time-varying volatility matrix of the high-dimensional noise vector. In addition, we consider large spot volatility matrix estimation in time-varying factor models with observable risk factors and derive the uniform convergence property. We provide numerical studies including simulation and empirical application to examine the performance of the proposed estimation methods in finite samples.

With the rapid development of classical and quantum machine learning, a large number of machine learning frameworks have been proposed. However, existing machine learning frameworks usually only focus on classical or quantum, rather than both. Therefore, based on VQNet 1.0, we further propose VQNet 2.0, a new generation of unified classical and quantum machine learning framework that supports hybrid optimization. The core library of the framework is implemented in C++, and the user level is implemented in Python, and it supports deployment on quantum and classical hardware. In this article, we analyze the development trend of the new generation machine learning framework and introduce the design principles of VQNet 2.0 in detail: unity, practicality, efficiency, and compatibility, as well as full particulars of implementation. We illustrate the functions of VQNet 2.0 through several basic applications, including classical convolutional neural networks, quantum autoencoders, hybrid classical-quantum networks, etc. After that, through extensive experiments, we demonstrate that the operation speed of VQNet 2.0 is higher than the comparison method. Finally, through extensive experiments, we demonstrate that VQNet 2.0 can deploy on different hardware platforms, the overall calculation speed is faster than the comparison method. It also can be mixed and optimized with quantum circuits composed of multiple quantum computing libraries.

Let $X$ be an $n\times n$ symmetric random matrix with independent but non-identically distributed entries. The deviation inequalities of the spectral norm of $X$ with Gaussian entries have been obtained by using the standard concentration of Gaussian measure results. This paper establishes an upper tail bound of the spectral norm of $X$ with sub-Exponential entries. Our method relies upon a crucial ingredient of a novel chaining argument that essentially involves both the particular structure of the sets used for the chaining and the distribution of coordinates of a point on the unit sphere., Comment: 18pages

Catoni proposed a robust M-estimator and gave the deviation inequality for one fixed test function. The present paper is devoted to the uniform concentration inequality for a family of test functions. As an application, we consider empirical risk minimization for heavy-tailed losses.

There has been a renewed interest in exponential concentration inequalities for stochastic processes in probability and statistics over the last three decades. De la Pe\~{n}a \cite{d} establishes a nice exponential inequality for discrete time locally square integrable martingale . In this paper, we obtain de la Pe\~{n}a's inequalities for stochastic integral of multivariate point processes. The proof is primarily based on Dol\'{e}ans-Dade exponential formula and the optional stopping theorem. As application, we obtain an exponential inequality for block counting process in $\Lambda-$coalescents.

In this paper, a new technique is introduced to obtain non-uniform Berry-Esseen bounds of normal and nonnormal approximation for unbounded exchangeable pairs. This technique does not rely on the concentration inequalities developed by Chen and Shao \cite{cls1, cls2} and can be applied to the quadratic forms, general Curie-Weiss model and an independence test. In particular, our non-uniform result about the independence test is under 6th moment condition, while the uniform bound in Chen and Shao \cite{cs2} requires 24th moment condition., Comment: 34pages, we reviewed the paper and corrected some typos. All comments are welcome

Recent progress in Generative Adversarial Networks (GANs) has shown promising signs of improving GAN training via architectural change. Despite some early success, at present the design of GAN architectures requires human expertise, laborious trial-and-error testings, and often draws inspiration from its image classification counterpart. In the current paper, we present the first neural architecture search algorithm, automated neural architecture search for deep generative models, or AGAN for abbreviation, that is specifically suited for GAN training. For unsupervised image generation tasks on CIFAR-10, our algorithm finds architecture that outperforms state-of-the-art models under same regularization techniques. For supervised tasks, the automatically searched architectures also achieve highly competitive performance, outperforming best human-invented architectures at resolution $32\times32$. Moreover, we empirically demonstrate that the modules learned by AGAN are transferable to other image generation tasks such as STL-10.

Transfer learning through fine-tuning a pre-trained neural network with an extremely large dataset, such as ImageNet, can significantly accelerate training while the accuracy is frequently bottlenecked by the limited dataset size of the new target task. To solve the problem, some regularization methods, constraining the outer layer weights of the target network using the starting point as references (SPAR), have been studied. In this paper, we propose a novel regularized transfer learning framework DELTA, namely DEep Learning Transfer using Feature Map with Attention. Instead of constraining the weights of neural network, DELTA aims to preserve the outer layer outputs of the target network. Specifically, in addition to minimizing the empirical loss, DELTA intends to align the outer layer outputs of two networks, through constraining a subset of feature maps that are precisely selected by attention that has been learned in an supervised learning manner. We evaluate DELTA with the state-of-the-art algorithms, including L2 and L2-SP. The experiment results show that our proposed method outperforms these baselines with higher accuracy for new tasks., Comment: Accepted at ICLR 2019

This paper discusses the local linear smoothing to estimate the unknown first and second infinitesimal moments in second-order jump-diffusion model based on Gamma asymmetric kernels. Under the mild conditions, we obtain the weak consistency and the asymptotic normality of these estimators for both interior and boundary design points. Besides the standard properties of the local linear estimation such as simple bias representation and boundary bias correction, the local linear smoothing using Gamma asymmetric kernels possess some extra advantages such as variable bandwidth, variance reduction and resistance to sparse design, which is validated through finite sample simulation study. Finally, we employ the estimators for the return of some high frequency financial data., Comment: 51 pages, 33 figures

In this paper, the weak convergence about the discretization error of stochastic iterated integrals in the Skorohod sense are studied, while the integrands and integrators of iterated integrals are supposed to be semimartingales with jumps. We explored the rate of convergence of its approximation based on the asymptotic behaviors of the associated normalized error and obtained that the rate is $1/n$ when the driving process is semimartingale with a nonvanishing continuous martingale component. As an application, we also studied the discretization of the Dol\'eans-Dade exponential., Comment: 19pages, all comments are welcome

In this paper, robust nonparametric estimators, instead of local linear estimators, are adapted for infinitesimal coefficients associated with integrated jump-diffusion models to avoid the impact of outliers on accuracy. Furthermore, consider the complexity of iteration of the solution for local M-estimator, we propose the one-step local M-estimators to release the computation burden. Under appropriate regularity conditions, we prove that one-step local M-estimators and the fully iterative M-estimators have the same performance in consistency and asymptotic normality. Through simulation, our method present advantages in bias reduction, robustness and reducing computation cost. In addition, the estimators are illustrated empirically through stock index under different sampling frequency., Comment: There are some lackness for simulation study and empirical analysis, some error correction to be done for the detailed proof

We consider the stochastic integrals of multivariate point processes and study their concentration phenomena. In particular, we obtain a Bernstein type of concentration inequality through Dol\'eans-Dade exponential formula and a uniform exponential inequality using a generic chaining argument. As applications, we obtain a upper bound for a sequence of discrete time martingales indexed by a class of functionals, and so derive the rate of convergence for nonparametric maximum likelihood estimators, which is an improvement of earlier work of van de Geer., Comment: 18 pages

Let $(\Omega, \mathcal{F}, (\mathcal{F})_{t\ge 0}, P)$ be a complete stochastic basis, $X$ a semimartingale with predictable compensator $(B, C, \nu)$. Consider a family of probability measures $\mathbf{P}=( {P}^{n, \psi}, \psi\in \Psi, n\ge 1)$, where $\Psi$ is an index set, $ {P}^{n, \psi}\stackrel {loc} \ll{P}$, and denote the likelihood ratio process by $Z_t^{n, \psi} =\frac{dP^{n, \psi}|_{\mathcal{F}_t}}{d P|_{\mathcal{F}_t}}$. Under some regularity conditions in terms of logarithm entropy and Hellinger processes, we prove that $\log Z_t^{n}$ converges weakly to a Gaussian process in $\ell^\infty(\Psi)$ as $n\rightarrow\infty$ for each fixed $t>0$.

Central limit theorems play an important role in the study of statistical inference for stochastic processes. However, when the nonparametric local polynomial threshold estimator, especially local linear case, is employed to estimate the diffusion coefficients of diffusion processes, the adaptive and predictable structure of the estimator conditionally on the $\sigma-$field generated by diffusion processes is destroyed, the classical central limit theorem for martingale difference sequences can not work. In this paper, we proved the central limit theorems of local polynomial threshold estimators for the volatility function in diffusion processes with jumps. We believe that our proof for local polynomial threshold estimators provides a new method in this fields, especially local linear case.

We use the martingale convergence method to get the weak convergence theorem on general functionals of partial sums of independent heavy-tailed random variables. The limiting process is the stochastic integral driven by $\alpha-$stable L\'evy process. Our method is very powerful to obtain the limit behavior of heavy-tailed random variables., Comment: 18pages