# ${L}_{1}$-penalization in functional linear regression with subgaussian design

Vladimir Koltchinskii^{[1]}; Stanislav Minsker^{[2]}

- [1] School of Mathematics, Georgia Institute of Technology 686 Cherry Street, Atlanta, GA 30332-0160 USA
- [2] Department of Mathematics, Duke University Box 90320, Durham NC 27708-0320,

Journal de l’École polytechnique — Mathématiques (2014)

- Volume: 1, page 269-330
- ISSN: 2270-518X

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topKoltchinskii, Vladimir, and Minsker, Stanislav. "$L_1$-penalization in functional linear regression with subgaussian design." Journal de l’École polytechnique — Mathématiques 1 (2014): 269-330. <http://eudml.org/doc/275615>.

@article{Koltchinskii2014,

abstract = {We study functional regression with random subgaussian design and real-valued response. The focus is on the problems in which the regression function can be well approximated by a functional linear model with the slope function being “sparse” in the sense that it can be represented as a sum of a small number of well separated “spikes”. This can be viewed as an extension of now classical sparse estimation problems to the case of infinite dictionaries. We study an estimator of the regression function based on penalized empirical risk minimization with quadratic loss and the complexity penalty defined in terms of $L_1$-norm (a continuous version of LASSO). The main goal is to introduce several important parameters characterizing sparsity in this class of problems and to prove sharp oracle inequalities showing how the $L_2$-error of the continuous LASSO estimator depends on the underlying sparsity of the problem.},

affiliation = {School of Mathematics, Georgia Institute of Technology 686 Cherry Street, Atlanta, GA 30332-0160 USA; Department of Mathematics, Duke University Box 90320, Durham NC 27708-0320,},

author = {Koltchinskii, Vladimir, Minsker, Stanislav},

journal = {Journal de l’École polytechnique — Mathématiques},

keywords = {Functional regression; sparse recovery; LASSO; oracle inequality; infinite dictionaries; functional regression},

language = {eng},

pages = {269-330},

publisher = {École polytechnique},

title = {$L_1$-penalization in functional linear regression with subgaussian design},

url = {http://eudml.org/doc/275615},

volume = {1},

year = {2014},

}

TY - JOUR

AU - Koltchinskii, Vladimir

AU - Minsker, Stanislav

TI - $L_1$-penalization in functional linear regression with subgaussian design

JO - Journal de l’École polytechnique — Mathématiques

PY - 2014

PB - École polytechnique

VL - 1

SP - 269

EP - 330

AB - We study functional regression with random subgaussian design and real-valued response. The focus is on the problems in which the regression function can be well approximated by a functional linear model with the slope function being “sparse” in the sense that it can be represented as a sum of a small number of well separated “spikes”. This can be viewed as an extension of now classical sparse estimation problems to the case of infinite dictionaries. We study an estimator of the regression function based on penalized empirical risk minimization with quadratic loss and the complexity penalty defined in terms of $L_1$-norm (a continuous version of LASSO). The main goal is to introduce several important parameters characterizing sparsity in this class of problems and to prove sharp oracle inequalities showing how the $L_2$-error of the continuous LASSO estimator depends on the underlying sparsity of the problem.

LA - eng

KW - Functional regression; sparse recovery; LASSO; oracle inequality; infinite dictionaries; functional regression

UR - http://eudml.org/doc/275615

ER -

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