In model selection, several types of cross-validation are commonly used and many variants have been introduced. While consistency of some of these methods has been proven, their rate of convergence to the oracle is generally still unknown. Until now, an asymptotic analysis of crossvalidation able to answer this question has been lacking. Existing results focus on the ''pointwise'' estimation of the risk of a single estimator, whereas analysing model selection requires understanding how the CV risk varies with the model. In this article, we investigate the asymptotics of the CV risk in the neighbourhood of the optimal model, for trigonometric series estimators in density estimation. Asymptotically, simple validation and ''incomplete'' V --fold CV behave like the sum of a convex function fn and a symmetrized Brownian changed in time W gn/V. We argue that this is the right asymptotic framework for studying model selection.
相關內容
In this study, we generalize a problem of sampling a scalar Gauss Markov Process, namely, the Ornstein-Uhlenbeck (OU) process, where the samples are sent to a remote estimator and the estimator makes a causal estimate of the observed realtime signal. In recent years, the problem is solved for stable OU processes. We present solutions for the optimal sampling policy that exhibits a smaller estimation error for both stable and unstable cases of the OU process along with a special case when the OU process turns to a Wiener process. The obtained optimal sampling policy is a threshold policy. However, the thresholds are different for all three cases. Later, we consider additional noise with the sample when the sampling decision is made beforehand. The estimator utilizes noisy samples to make an estimate of the current signal value. The mean-square error (mse) is changed from previous due to noise and the additional term in the mse is solved which provides performance upper bound and room for a pursuing further investigation on this problem to find an optimal sampling strategy that minimizes the estimation error when the observed samples are noisy. Numerical results show performance degradation caused by the additive noise.
We propose an energy-stable parametric finite element method (ES-PFEM) to discretize the motion of a closed curve under surface diffusion with an anisotropic surface energy $\gamma(\theta)$ -- anisotropic surface diffusion -- in two dimensions, while $\theta$ is the angle between the outward unit normal vector and the vertical axis. By introducing a positive definite surface energy (density) matrix $G(\theta)$, we present a new and simple variational formulation for the anisotropic surface diffusion and prove that it satisfies area/mass conservation and energy dissipation. The variational problem is discretized in space by the parametric finite element method and area/mass conservation and energy dissipation are established for the semi-discretization. Then the problem is further discretized in time by a (semi-implicit) backward Euler method so that only a linear system is to be solved at each time step for the full-discretization and thus it is efficient. We establish well-posedness of the full-discretization and identify some simple conditions on $\gamma(\theta)$ such that the full-discretization keeps energy dissipation and thus it is unconditionally energy-stable. Finally the ES-PFEM is applied to simulate solid-state dewetting of thin films with anisotropic surface energies, i.e. the motion of an open curve under anisotropic surface diffusion with proper boundary conditions at the two triple points moving along the horizontal substrate. Numerical results are reported to demonstrate the efficiency and accuracy as well as energy dissipation of the proposed ES-PFEM.
Extropy and its properties are explored to quantify the uncertainty. In this paper, we obtain alternative expressions for cumulative residual extropy and negative cumulative extropy. We obtain simple estimators of cumulative (residual) extropy. Asymptotic properties of the proposed estimators are studied. We also present new estimators of cumulative (residual) extropy when the data is right censored. The finite sample performance of the estimators is evaluated through Monte Carlo simulation studies. We use the proposed estimators to analyse different real data sets. Finally, we obtain the relationship between different dynamic and weighted extropy measures and reliability concepts, which leads to several open problems associated with these measures.
We consider the problem of estimating high-dimensional covariance matrices of $K$-populations or classes in the setting where the samples sizes are comparable to the data dimension. We propose estimating each class covariance matrix as a distinct linear combination of all class sample covariance matrices. This approach is shown to reduce the estimation error when the sample sizes are limited, and the true class covariance matrices share a somewhat similar structure. We develop an effective method for estimating the coefficients in the linear combination that minimize the mean squared error under the general assumption that the samples are drawn from (unspecified) elliptically symmetric distributions possessing finite fourth-order moments. To this end, we utilize the spatial sign covariance matrix, which we show (under rather general conditions) to be an unbiased estimator of the normalized covariance matrix as the dimension grows to infinity. We also show how the proposed method can be used in choosing the regularization parameters for multiple target matrices in a single class covariance matrix estimation problem. We assess the proposed method via numerical simulation studies including an application in global minimum variance portfolio optimization using real stock data.
The performance of a quantum information processing protocol is ultimately judged by distinguishability measures that quantify how distinguishable the actual result of the protocol is from the ideal case. The most prominent distinguishability measures are those based on the fidelity and trace distance, due to their physical interpretations. In this paper, we propose and review several algorithms for estimating distinguishability measures based on trace distance and fidelity, and we evaluate their performance using simulators of quantum computers. The algorithms can be used for distinguishing quantum states, channels, and strategies (the last also known in the literature as "quantum combs"). The fidelity-based algorithms offer novel physical interpretations of these distinguishability measures in terms of the maximum probability with which a single prover (or competing provers) can convince a verifier to accept the outcome of an associated computation. We simulate these algorithms by using a variational approach with parameterized quantum circuits and find that they converge well for the examples that we consider.
Time-to-event endpoints show an increasing popularity in phase II cancer trials. The standard statistical tool for such endpoints in one-armed trials is the one-sample log-rank test. It is widely known, that the asymptotic providing the correctness of this test does not come into effect to full extent for small sample sizes. There have already been some attempts to solve this problem. While some do not allow easy power and sample size calculations, others lack a clear theoretical motivation and require further considerations. The problem itself can partly be attributed to the dependence of the compensated counting process and its variance estimator. We provide a framework in which the variance estimator can be flexibly adopted to the present situation while maintaining its asymptotical properties. We exemplarily suggest a variance estimator which is uncorrelated to the compensated counting process. Furthermore, we provide sample size and power calculations for any approach fitting into our framework. Finally, we compare several methods via simulation studies and the hypothetical setup of a Phase II trial based on real world data.
We continue the investigation on the spectrum of operators arising from the discretization of partial differential equations. In this paper we consider a three field formulation recently introduced for the finite element least-squares approximation of linear elasticity. We discuss in particular the distribution of the discrete eigenvalues in the complex plane and how they approximate the positive real eigenvalues of the continuous problem. The dependence of the spectrum on the Lam\'e parameters is considered as well and its behavior when approaching the incompressible limit.
We propose a novel dimensionality reduction method for maximum inner product search (MIPS), named CEOs, based on the theory of concomitants of extreme order statistics. Utilizing the asymptotic behavior of these concomitants, we show that a few dimensions associated with the extreme values of the query signature are enough to estimate inner products. Since CEOs only uses the sign of a small subset of the query signature for estimation, we can precompute all inner product estimators accurately before querying. These properties yield a sublinear MIPS algorithm with an exponential indexing space complexity. We show that our exponential space is optimal for the $(1 + \epsilon)$-approximate MIPS on a unit sphere. The search recall of CEOs can be theoretically guaranteed under a mild condition. To deal with the exponential space complexity, we propose two practical variants, including sCEOs-TA and coCEOs, that use linear space for solving MIPS. sCEOs-TA exploits the threshold algorithm (TA) and provides superior search recalls to competitive MIPS solvers. coCEOs is a data and dimension co-reduction technique and outperforms sCEOs-TA on high recall requirements. Empirically, they are very simple to implement and achieve at least 100x speedup compared to the bruteforce search while returning top-10 MIPS with accuracy at least 90% on many large-scale data sets.
Leveraging biased click data for optimizing learning to rank systems has been a popular approach in information retrieval. Because click data is often noisy and biased, a variety of methods have been proposed to construct unbiased learning to rank (ULTR) algorithms for the learning of unbiased ranking models. Among them, automatic unbiased learning to rank (AutoULTR) algorithms that jointly learn user bias models (i.e., propensity models) with unbiased rankers have received a lot of attention due to their superior performance and low deployment cost in practice. Despite their differences in theories and algorithm design, existing studies on ULTR usually use uni-variate ranking functions to score each document or result independently. On the other hand, recent advances in context-aware learning-to-rank models have shown that multivariate scoring functions, which read multiple documents together and predict their ranking scores jointly, are more powerful than uni-variate ranking functions in ranking tasks with human-annotated relevance labels. Whether such superior performance would hold in ULTR with noisy data, however, is mostly unknown. In this paper, we investigate existing multivariate scoring functions and AutoULTR algorithms in theory and prove that permutation invariance is a crucial factor that determines whether a context-aware learning-to-rank model could be applied to existing AutoULTR framework. Our experiments with synthetic clicks on two large-scale benchmark datasets show that AutoULTR models with permutation-invariant multivariate scoring functions significantly outperform those with uni-variate scoring functions and permutation-variant multivariate scoring functions.
The availability of large microarray data has led to a growing interest in biclustering methods in the past decade. Several algorithms have been proposed to identify subsets of genes and conditions according to different similarity measures and under varying constraints. In this paper we focus on the exclusive row biclustering problem for gene expression data sets, in which each row can only be a member of a single bicluster while columns can participate in multiple ones. This type of biclustering may be adequate, for example, for clustering groups of cancer patients where each patient (row) is expected to be carrying only a single type of cancer, while each cancer type is associated with multiple (and possibly overlapping) genes (columns). We present a novel method to identify these exclusive row biclusters through a combination of existing biclustering algorithms and combinatorial auction techniques. We devise an approach for tuning the threshold for our algorithm based on comparison to a null model in the spirit of the Gap statistic approach. We demonstrate our approach on both synthetic and real-world gene expression data and show its power in identifying large span non-overlapping rows sub matrices, while considering their unique nature. The Gap statistic approach succeeds in identifying appropriate thresholds in all our examples.
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191