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Adaptive nuclear-norm penalization is proposed for low-rank matrix approximation, by which we develop a new reduced-rank estimation method for the general high-dimensional multivariate regression problems. The adaptive nuclear norm of a…

We study the sample complexity of the best-case Empirical Risk Minimizer in the setting of stochastic convex optimization. We show that there exists an instance in which the sample size is linear in the dimension, learning is possible, but…

An important objective of the classical processing of stationary random sequences under nonparametric uncertainty is the problem of filtering in case when the distribution of the underlying signal is unknown. In this paper it is assumed…

We consider stationary hidden Markov models with finite state space and nonparametric modeling of the emission distributions. It has remained unknown until very recently that such models are identifiable. In this paper, we propose a new…

We consider robust low rank matrix estimation as a trace regression when outputs are contaminated by adversaries. The adversaries are allowed to add arbitrary values to arbitrary outputs. Such values can depend on any samples. We deal with…

This work provides closed-form solutions and minimum achievable errors for a large class of low-rank approximation problems in Hilbert spaces. The proposed theorem generalizes to the case of bounded linear operators the previous results…

We consider the most common variants of linear regression, including Ridge, Lasso and Support-vector regression, in a setting where the learner is allowed to observe only a fixed number of attributes of each example at training time. We…

We consider estimation in a sparse additive regression model with the design points on a regular lattice. We establish the minimax convergence rates over Sobolev classes and propose a Fourier-based rate-optimal estimator which is adaptive…

Empirical research typically involves a robustness-efficiency tradeoff. A researcher seeking to estimate a scalar parameter can invoke strong assumptions to motivate a restricted estimator that is precise but may be heavily biased, or they…

In the framework of matrix valued observables with low rank means, Stein's unbiased risk estimate (SURE) can be useful for risk estimation and for tuning the amount of shrinkage towards low rank matrices. This was demonstrated by Cand\`es…

We consider the problem of estimating self-exciting generalized linear models from limited binary observations, where the history of the process serves as the covariate. We analyze the performance of two classes of estimators, namely the…

Estimating the matrix of connections probabilities is one of the key questions when studying sparse networks. In this work, we consider networks generated under the sparse graphon model and the in-homogeneous random graph model with missing…

First, we extend the results of approximate matrix multiplication from the Frobenius norm to the spectral norm. Second, We develop a class of fast approximate generalized linear regression algorithms with respect to the spectral norm.…

This paper concerns the problem of matrix completion, which is to estimate a matrix from observations in a small subset of indices. We propose a calibrated spectrum elastic net method with a sum of the nuclear and Frobenius penalties and…

This work deals with the ill-posed inverse problem of reconstructing a function $f$ given implicitly as the solution of $g = Af$, where $A$ is a compact linear operator with unknown singular values and known eigenfunctions. We observe the…

In this paper, we investigate the generalized low rank approximation to the symmetric positive semidefinite matrix in the Frobenius norm: $$\underset{ rank(X)\leq k}{\min} \sum^m_{i=1}\left \Vert A_i - B_i XB_i^T \right \Vert^2_F,$$ where…

An effective two-stage method for an estimation of parameters of the linear regression is considered. For this purpose we introduce a certain quasi-estimator that, in contrast to usual estimator, produces two alternative estimates. It is…

The advent of large-scale inference has spurred reexamination of conventional statistical thinking. In a Gaussian model for $n$ many $z$-scores with at most $k < \frac{n}{2}$ nonnulls, Efron suggests estimating the location and scale…

An optimum solution free from degeneration is found to the system of linear algebraic equations with empirical coefficients and right-hand sides. The quadratic risk of estimators of the unknown solution vector is minimized over a class of…

We propose methodology for estimation of sparse precision matrices and statistical inference for their low-dimensional parameters in a high-dimensional setting where the number of parameters $p$ can be much larger than the sample size. We…