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稀疏向量模型中的自适应稳健估计AI 解读

For the sparse vector model, we consider estimation of the target vector, of its ℓ2-norm and of the noise variance. We construct adaptive estimators and establish the optimal rates of adaptive estimation when adaptation is considered with respect to the triplet “noise level—noise distribution—sparsity.” We consider classes of noise distributions with polynomially and exponentially decreasing tails as well as the case of Gaussian noise. The obtained rates turn out to be…

通过镜像平均学习AI 解读

Given a finite collection of estimators or classifiers, we study the problem of model selection type aggregation, that is, we construct a new estimator or classifier, called aggregate, which is nearly as good as the best among them with respect to a given risk criterion. We define our aggregate by a simple recursive procedure which solves an auxiliary stochastic linear programming problem related to the original nonlinear one and constitutes a special case of the mirror…

讨论：局部 Rademacher 复杂度与风险最小化的预言不等式AI 解读

The paper of Vladimir Koltchinskii has been circulating around for several years and already has become an important reference in statistical learning theory.One of the main achievements of the paper (further abbreviated as [VK]) is to propose very general techniques of proving oracle inequalities for excess risk under a control of the variance, that is, for example, under conditions (6.1) or (6.2) (often called margin or low noise conditions) or similar assumptions in terms…

惩罚最大似然与半参数二阶效率AI 解读

We consider the problem of estimation of a shift parameter of an unknown symmetric function in Gaussian white noise. We introduce a notion of semiparametric second-order efficiency and propose estimators that are semiparametrically efficient and second-order efficient in our model. These estimators are of a penalized maximum likelihood type with an appropriately chosen penalty. We argue that second-order efficiency is crucial in semiparametric problems since only the…

平方根惩罚：分类与边界估计中的间隔自适应AI 解读

We consider the problem of adaptation to the margin in binary classification. We suggest a penalized empirical risk minimization classifier that adaptively attains, up to a logarithmic factor, fast optimal rates of convergence for the excess risk, that is, rates that can be faster than n−1/2, where n is the sample size. We show that our method also gives adaptive estimators for the problem of edge estimation.

逆问题的 Oracle 不等式AI 解读

We consider a sequence space model of statistical linear inverse problems where we need to estimate a function $f$ from indirect noisy observations. Let a finite set $\Lambda$ of linear estimators be given. Our aim is to mimic the estimator in $\Lambda$ that has the smallest risk on the true $f$. Under general conditions, we show that this can be achieved by simple minimization of an unbiased risk estimator, provided the singular values of the operator of the inverse problem…

Sobolev 类函数的逐点与上确界范数精确自适应估计AI 解读

The problem of nonparametric function estimation in the Gaussian white noise model is considered. It is assumed that the unknown function belongs to one of the Sobolev classes, with an unknown regularity parameter. Asymptotically exact adaptive estimators of functions are proposed on the scale of Sobolev classes, with respect to pointwise and sup-norm risks. It is shown that, unlike the case of $L_2$-risk, a loss of efficiency under adaptation is inevitable here. Bounds on…

密度水平集的非参数估计AI 解读

Let $X_1, \dots, X_n$ be independent identically distributed observations from an unknown probability density $f(\cdot)$. Consider the problem of estimating the level set $G = G_f(\lambda) = {x \epsilon\mathbb{R}^2: f(x) \geq \lambda}$ from the sample $X_1, \dots, X_n$, under the assumption that the boundary of G has a certain smoothness. We propose piecewise-polynomial estimators of G based on the maximization of local empirical excess masses. We show that the estimators…

定性平滑的视觉误差准则AI 解读

An important gap, between the classical mathematical theory and the practice and implementation of nonparametric curve estimation, is due to the fact that the usual norms on function spaces measure something different from what the eye can see visually in a graphical presentation. Mathematical error criteria that more closely follow “visual impression” are developed and analyzed from both graphical and mathematical viewpoints. Examples from wavelet regression and kernel…

光滑边界集合的渐近极小极大恢复AI 解读

In this paper optimal rates of convergence are derived for estimates of sets in $N$-dimensional "black and white" pictures under smoothness conditions. It is assumed that the boundaries of the "black" regions have a smooth parameterisation, that is, that the boundaries are given by smooth functions from the sphere $S^{N-1}$ into $\mathbb{R}^N$. Furthermore, classes of convex regions are considered. Two models are studied: edge estimation models motivated by image…

单调边界的有效估计AI 解读

Let $g: \lbrack 0, 1\rbrack \rightarrow \lbrack 0, 1\rbrack$ be a monotone nondecreasing function and let $G$ be the closure of the set $\{(x, y) \in \lbrack 0, 1\rbrack \times \lbrack 0, 1\rbrack: 0 \leq y \leq g (x)\}$. We consider the problem of estimating the set $G$ from a sample of i.i.d. observations uniformly distributed in $G$. The estimation error is measured in the Hausdorff metric. We propose the estimator which is asymptotically efficient in the minimax sense.

平均导数估计的带宽选择AI 解读

The average derivative is the expected value of the derivative of a regression function. Kernel methods have been proposed as a means of estimating this quantity. The problem of bandwidth selection for these kernel estimators is addressed here. Asymptotic representations are found for the variance and squared bias. These are compared with each other to find an insightful representation for a bandwidth optimizing terms of lower order than n-1. It is interesting that, for…

同时估计尺度曲线的稳健非参数回归AI 解读

Let $\{X_i, Y_i\}^n_{i=1} \subset \mathbb{R}^d \times \mathbb{R}$ be independent identically distributed random variables. If the conditional distribution $F(y \mid x)$ can be parametrized by $F(y \mid x) = F_0((y - m(x))/\sigma(x))$ with a fixed and known distribution $F_0$, the regression curve $m(x)$ and scale curve $\sigma(x)$ could be estimated by some parametric method. More generally, we assume that $F$ is unknown and consider nonparametric simultaneous $M$-type…