相关论文: Geometric View of Measurement Errors
The multi-level Monte Carlo method proposed by M. Giles (2008) approximates the expectation of some functionals applied to a stochastic process with optimal order of convergence for the mean-square error. In this paper, a modified…
A new risk bound is presented for the problem of convex/concave function estimation, using the least squares estimator. The best known risk bound, as had appeared in \citet{GSvex}, scaled like $\log(en) n^{-4/5}$ under the mean squared…
In the 1990's exponential-type error bounds appeared in the theory of radial basis functions. This kind of error bounds is very powerful. However it only measures the difference between the approximant and approximand. Mathematicians and…
A framework previously introduced in [3] for solving a sequence of stochastic optimization problems with bounded changes in the minimizers is extended and applied to machine learning problems such as regression and classification. The…
An adaptive algorithm, based on residual type a posteriori indicators of errors measured in $L^{\infty}(L^2)$ and $L^2(L^2)$ norms, for a numerical scheme consisting of implicit Euler method in time and discontinuous Galerkin method in…
We introduce the concept of coverage risk as an error measure for density ridge estimation. The coverage risk generalizes the mean integrated square error to set estimation. We propose two risk estimators for the coverage risk and we show…
The investigation of samples with a spatial resolution in the nanometer range relies on the precise and stable positioning of the sample. Due to inherent mechanical instabilities of typical sample stages in optical microscopes, it is…
The square of a skew-symmetric matrix is a symmetric matrix whose eigenvalues have even multiplicities. When the matrices have rank two, they represent the Grassmannian of lines, and the squaring operation takes Pl\"ucker coordinates to…
The commonly used radial distortion model for camera calibration is in fact an assumption or a restriction. In practice, camera distortion could happen in a general geometrical manner that is not limited to the radial sense. This paper…
The Morse-Smale complex is a standard tool in visual data analysis. The classic definition is based on a continuous view of the gradient of a scalar function where its zeros are the critical points. These points are connected via gradient…
Stochastic gradient descent (SGD) is the workhorse of modern machine learning. Sometimes, there are many different potential gradient estimators that can be used. When so, choosing the one with the best tradeoff between cost and variance is…
For strongly convex objectives that are smooth, the classical theory of gradient descent ensures linear convergence relative to the number of gradient evaluations. An analogous nonsmooth theory is challenging. Even when the objective is…
We consider least squares estimation in a general nonparametric regression model. The rate of convergence of the least squares estimator (LSE) for the unknown regression function is well studied when the errors are sub-Gaussian. We find…
Minimizing a convex function of a measure with a sparsity-inducing penalty is a typical problem arising, e.g., in sparse spikes deconvolution or two-layer neural networks training. We show that this problem can be solved by discretizing the…
During the past two decades there has been a lot of interest in developing statistical depth notions that generalize the univariate concept of ranking to multivariate data. The notion of depth has also been extended to regression models and…
This paper defines a distance function that measures the dissimilarity between planar geometric figures formed with straight lines. This function can in turn be used in partial matching of different geometric figures. For a given pair of…
We consider the problem of estimating the distance, or range, between two locations by measuring the phase of multiple sinusoidal signals transmitted between the locations. Traditional estimators developed for optical interferometry include…
We consider the problem of estimating the covariance matrix of a random vector by observing i.i.d samples and each entry of the sampled vector is missed with probability $p$. Under the standard $L_4-L_2$ moment equivalence assumption, we…
This paper proposes an original Riemmanian geometry for low-rank structured elliptical models, i.e., when samples are elliptically distributed with a covariance matrix that has a low-rank plus identity structure. The considered geometry is…
For the pure biharmonic equation and a biharmonic singular perturbation problem, a residual-based error estimator is introduced which applies to many existing nonconforming finite elements. The error estimator involves the local…