# Geometric Inference for General High-Dimensional Linear Inverse Problems

@article{Cai2014GeometricIF, title={Geometric Inference for General High-Dimensional Linear Inverse Problems}, author={T. Tony Cai and Tengyuan Liang and Alexander Rakhlin}, journal={arXiv: Statistics Theory}, year={2014} }

This paper presents a unified geometric framework for the statistical analysis of a general ill-posed linear inverse model which includes as special cases noisy compressed sensing, sign vector recovery, trace regression, orthogonal matrix estimation, and noisy matrix completion. We propose computationally feasible convex programs for statistical inference including estimation, confidence intervals and hypothesis testing. A theoretical framework is developed to characterize the local estimation…

## 23 Citations

A Unified Theory of Confidence Regions and Testing for High-Dimensional Estimating Equations

- Computer Science, MathematicsStatistical Science
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A new inferential framework for constructing confidence regions and testing hypotheses in statistical models specified by a system of high dimensional estimating equations is proposed, which is likelihood-free and provides valid inference for a broad class of highdimensional constrained estimating equation problems, which are not covered by existing methods.

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- Mathematics, Computer ScienceThe Annals of Statistics
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In the Tucker low-rank tensor PCA or regression model, provided with any estimates achieving some attainable error rate, the data-driven confidence regions for the singular subspace of the parameter tensor are developed based on the asymptotic distribution of an updated estimate by two-iteration alternating minimization.

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- Mathematics, Computer ScienceIEEE Transactions on Information Theory
- 2018

The results demonstrate that a linear convergence rate is attainable even though the least squares objective is not strongly convex in these settings, and present a unified convergence analysis of the gradient projection algorithm applied to such problems.

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- Computer ScienceICML
- 2020

The findings unveil the statistical optimality of nonconvex tensor completion: it attains un-improvable $\ell_{2}$ accuracy when estimating both the unknown tensor and the underlying tensor factors.

New Computational and Statistical Aspects of Regularized Regression with Application to Rare Feature Selection and Aggregation

- Computer Science, MathematicsArXiv
- 2019

This work provides a unified computational framework for defining norms that promote structures and develops associated tools for optimization involving such norms given only the orthogonal projection oracle onto the non-convex set of desired models.

A lava attack on the recovery of sums of dense and sparse signals

- Computer ScienceArXiv
- 2015

This work proposes a new penalization-based method, called lava, which is computationally efficient and strictly dominates both lasso and ridge estimation, and derives analytic expressions for the finite-sample risk function of the lava estimator in the Gaussian sequence model.

Inference and uncertainty quantification for noisy matrix completion

- Computer Science, MathematicsProceedings of the National Academy of Sciences
- 2019

A simple procedure to compensate for the bias of the widely used convex and nonconvex estimators and derive distributional characterizations for the resulting debiased estimators, which enable optimal construction of confidence intervals/regions for the missing entries and the low-rank factors.

Rate Optimal Estimation and Confidence Intervals for High-dimensional Regression with Missing Covariates

- Mathematics, Computer ScienceJ. Multivar. Anal.
- 2019

Confidence Region of Singular Subspaces for Low-Rank Matrix Regression

- Computer ScienceIEEE Transactions on Information Theory
- 2019

This paper revisits the low-rank matrix regression model and introduces a two-step procedure to construct confidence regions of the singular subspaces, and proves asymptotical normality of the joint projection distance with data-dependent centering and normalization.

On the robustness of minimum-norm interpolators

- Mathematics, Computer Science
- 2021

A quantitative bound for the prediction error is given, relating it to the Rademacher complexity of the covariates, the norm of the minimum norm interpolator of the errors and the shape of the subdifferential around the true parameter.

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