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In sparse linear regression, the SLOPE estimator generalizes LASSO by assigning magnitude-dependent regular- izations to different coordinates of the estimate. In this paper, we present an asymptotically exact characterization of the performance of SLOPE in the high-dimensional regime where the number of unknown parameters grows in proportion to the number of observations. Our asymptotic characterization enables us to derive optimal regularization sequences to either minimize the MSE or to maximize the power in variable selection under any given level of Type-I error. In both cases, we show that the optimal design can be recast as certain infinite-dimensional convex optimization problems, which have efficient and accurate finite-dimensional approximations. Numerical simulations verify our asymptotic predictions. They also demonstrate the superi- ority of our optimal design over LASSO and a regularization sequence previously proposed in the literature.
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Adaptive nonparametric regression with the K-nearest neighbour fused lasso
Summary The fused lasso, also known as total-variation denoising, is a locally adaptive function estimator over a regular grid of design points. In this article, we extend the fused lasso to settings in which the points do not occur on a regular grid, leading to a method for nonparametric regression. This approach, which we call the $$K$$-nearest-neighbours fused lasso, involves computing the $$K$$-nearest-neighbours graph of the design points and then performing the fused lasso over this graph. We show that this procedure has a number of theoretical advantages over competing methods: specifically, it inherits local adaptivity from its connection to the fused lasso, and it inherits manifold adaptivity from its connection to the $$K$$-nearest-neighbours approach. In a simulation study and an application to flu data, we show that excellent results are obtained. For completeness, we also study an estimator that makes use of an $$\epsilon$$-graph rather than a $$K$$-nearest-neighbours graph and contrast it with the $$K$$-nearest-neighbours fused lasso.
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- Award ID(s):
- 1712996
- PAR ID:
- 10383980
- Date Published:
- Journal Name:
- Biometrika
- Volume:
- 107
- Issue:
- 2
- ISSN:
- 0006-3444
- Page Range / eLocation ID:
- 293 to 310
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/biomet/asz071
