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A Unified Framework for High-Dimensional Analysis of $M$-Estimators with Decomposable Regularizers
Open accessAbstract
High-dimensional statistical inference deals with models in which the the number of parameters $p$ is comparable to or larger than the sample size $n$. Since it is usually impossible to obtain consistent procedures unless $p/n\rightarrow0$, a line of recent work has studied models with various types of low-dimensional structure, including sparse vectors, sparse and structured matrices, low-rank matrices and combinations thereof. In such settings, a general approach to estimation is to solve a regularized optimization problem, which combines a loss function measuring how well the model fits the data with some regularization function that encourages the assumed structure. This paper provides a unified framework for establishing consistency and convergence rates for such regularized $M$-estimators under high-dimensional scaling. We state one main theorem and show how it can be used to re-derive some existing results, and also to obtain a number of new results on consistency and convergence rates, in both $\ell_{2}$-error and related norms. Our analysis also identifies two key properties of loss and regularization functions, referred to as restricted strong convexity and decomposability, that ensure corresponding regularized $M$-estimators have fast convergence rates and which are optimal in many well-studied cases.
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Metadata
- Authors
- Negahban, S. N., Ravikumar, P., Wainwright, M. J., & Yu, B.
- Venue
- Statistical Science
- Publisher
- Institute of Mathematical Statistics
- Published
- 2012-11-01
- Volume
- 27
- Issue
- 4
- Pages
- 538-557
- ISSN
- 0883-4237 (ISSN-L)
- OpenAlex
- W2151825876
- Cited by
- 287
- References
- 90
- FWCI
- 32.06
- Type
- journal article
- Language
- en
- Open access
- bronze
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- Negahban, S. N., Ravikumar, P., Wainwright, M. J., & Yu, B. (2012). A Unified Framework for High-Dimensional Analysis of $M$-Estimators with Decomposable Regularizers. Statistical Science. https://doi.org/10.1214/12-sts400
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