We consider a problem of manifold estimation from noisy observations. Many manifold learning procedures locally approximate a manifold by a weighted average over a small neighborhood. However, in the presence of large noise, the assigned weights become so corrupted that the averaged estimate shows very poor performance. We suggest a structure-adaptive procedure, which simultaneously reconstructs a smooth manifold and estimates projections of the point cloud onto this manifold. The proposed approach iteratively refines the weights on each step, using the structural information obtained at previous steps. After several iterations, we obtain nearly “oracle” weights, so that the final estimates are nearly efficient even in the presence of relatively large noise. In our theoretical study, we establish tight lower and upper bounds proving asymptotic optimality of the method for manifold estimation under the Hausdorff loss, provided that the noise degrades to zero fast enough.

This paper presents a new approach to non-parametric cluster analysis called adaptive weights' clustering. The method is fully adaptive and does not require to specify the number of clusters or their structure. The clustering results are not sensitive to noise and outliers, and the procedure is able to recover different clusters with sharp edges or manifold structure. The method is also scalable and computationally feasible. Our intensive numerical study shows a state-of-the-art performance of the method in various artificial examples and applications to text data. The idea of the method is to identify the clustering structure by checking at different points and for different scales on departure from local homogeneity. The proposed procedure describes the clustering structure in terms of weights wij, and each of them measures the degree of local inhomogeneity for two neighbor local clusters using statistical tests of “no gap” between them. The procedure starts from very local scale, and then, the parameter of locality grows by some factor at each step. We also provide a rigorous theoretical study of the procedure and state its optimal sensitivity to deviations from local homogeneity.

We derive two convergence results for a sequential alternating maximization procedure to approximate the maximizer of random functionals such as the realized log likelihood in MLE estimation. We manage to show that the sequence attains the same deviation properties as shown for the profile M-estimator by Andresen and Spokoiny (2013), that means a finite sample Wilks and Fisher theorem. Further under slightly stronger smoothness constraints on the random functional we can show nearly linear convergence to the global maximizer if the starting point for the procedure is well chosen.

Non-Gaussian component analysis (NGCA) introduced in offered a method for high-dimensional data analysis allowing for identifying a low-dimensional non-Gaussian component of the whole distribution in an iterative and structure adaptive way. An important step of the NGCA procedure is identification of the non-Gaussian subspace using principle component analysis (PCA) method. This article proposes a new approach to NGCA called sparse NGCA which replaces the PCA-based procedure with a new the algorithm we refer to as convex projection.

Obtaining the best linear unbiased estimator (BLUE) of noisy signals is a traditional but powerful approach to noise reduction. Explicitly computing BLUE usually requires the prior knowledge of the subspace to which the true signal belongs and the noise covariance matrix. However, such prior knowledge is often unavailable in reality, which prevents us from applying BLUE to real-world problems. In this paper, we therefore give a method for obtaining BLUE without such prior knowledge. Our additional assumption is that the true signal follows a non-Gaussian distribution while the noise is Gaussian

Finding non-Gaussian components of high-dimensional data is an important preprocessing step for efficient information processing. This article proposes a new linear method to identify the "non-Gaussian subspace" within a very general semi-parametric framework. Our proposed method, called NGCA (non-Gaussian component analysis), is based on a linear operator which, to any arbitrary nonlinear (smooth) function, associates a vector belonging to the low dimensional non-Gaussian target subspace, up to an estimation error. By applying this operator to a family of different nonlinear functions, one obtains a family of different vectors lying in a vicinity of the target space. As a final step, the target space itself is estimated by applying PCA to this family of vectors. We show that this procedure is consistent in the sense that the estimaton error tends to zero at a parametric rate, uniformly over the family, Numerical examples demonstrate the usefulness of our method.

We consider the problem of recovering edges of an image from noisy tomographic data. The original image is assumed to have a discontinuity jump (edge) along the boundary of a compact convex set. The Radon transform of the image is observed with noise, and the problem is to estimate the edge. We develop an estimation procedure which is based on recovering the support function of the edge. It is shown that the proposed estimator is nearly optimal in order in a minimax sense. Numerical examples illustrate reasonable practical behavior of the estimation procedure.

We propose a new linear method for dimension reduction to identify nonGaussian components in high dimensional data. Our method, NGCA (non-Gaussian component analysis), uses a very general semi-parametric framework. In contrast to existing projection methods we define what is uninteresting (Gaussian): by projecting out uninterestingness, we can estimate the relevant non-Gaussian subspace. We show that the estimation error of finding the non-Gaussian components tends to zero at a parametric rate. Once NGCA components are identified and extracted, various tasks can be applied in the data analysis process, like data visualization, clustering, denoising or classification. A numerical study demonstrates the usefulness of our method.