Opuscula Math. 36, no. 5 (2016), 671-679
http://dx.doi.org/10.7494/OpMath.2016.36.5.671
Opuscula Mathematica
On a dense minimizer of empirical risk in inverse problems
Jacek Podlewski
Zbigniew Szkutnik
Abstract. Properties of estimators of a functional parameter in an inverse problem setup are studied. We focus on estimators obtained through dense minimization (as opposed to minimization over \(\delta\)-nets) of suitably defined empirical risk. At the cost of imposition of a sort of local finite-dimensionality assumption, we fill some gaps in the proofs of results published by Klemelä and Mammen [Ann. Statist. 38 (2010), 482-511]. We also give examples of functional classes that satisfy the modified assumptions.
Keywords: inverse problem, empirical risk minimization.
Mathematics Subject Classification: 62G07.
- L. Birgé, P. Massart, Rates of convergence for minimum contrast estimators, Probab. Theory Relat. Fields 97 (1993), 113-150.
- L. Birgé, P. Massart, An adaptive compression algorithm in Besov spaces, Const. Approx. 16 (2000), 1-36.
- A. Cohen, W. Dahmen, I. Daubechies, R. DeVore, Tree approximation and optimal encoding, Appl. and Comp. Harmonic Analysis 11 (2001), 192-226.
- F. Comte, M.-L. Taupine, Y. Rosenholc, Penalized contrast estimator for density deconvolution, Canad. J. Statist. 34 (2006), 431-452.
- R. DeVore, G. Lorentz, Constructive Approximation, Springer-Verlag, New York, 1993.
- E. Gassiat, R. van Handel, The local geometry of finite mixtures, Trans. Amer. Math. Soc. 366 (2014), 1047-1072.
- J. Heinonen, Lectures on Analysis on Metric Spaces, Springer-Verlag, New York, 2001.
- J. Klemelä, E. Mammen, Empirical risk minimization in inverse problems: Extended technical version, (2009), available at http://arxiv.org/abs/0904.2977v1
- J. Klemelä, E. Mammen, Empirical risk minimization in inverse problems, Ann. Statist. 38 (2010), 482-511.
- M. Kosorok, Introduction to Empirical Processes and Semiparametric Inference, Springer-Verlag, New York, 2008.
- L.M. Le Cam, Asymptotic Methods in Statistical Decision Theory, Springer-Verlag, New York, 1986.
- T. Nicoleris, Y.G. Yatracos, Rates of convergence of estimates, Kolmogorov's entropy and the dimensionality reduction principle in regression, Ann. Statist. 25 (1997), 2493-2511.
- M.J. van der Laan, S. Dudoit, A.W. van der Vaart, The cross-validated adaptive epsilon-net estimator, Statist. Decisions 24 (2006), 373-395.
- V.N. Vapnik, The Nature of Statistical Learning Theory, Springer-Verlag, New York, 1995.
- Y.G. Yatracos, Rates of convergence of minimum distance estimators and Kolmogorov's entropy, Ann. Statist. 13 (1985), 768-774.
- Jacek Podlewski
- StatSoft Poland, ul. Kraszewskiego 36, 30-110 Krakow, Poland
- Zbigniew Szkutnik
- AGH University of Science and Technology, Faculty of Applied Mathematics, al. A. Mickiewicza 30, 30-059 Krakow, Poland
- Communicated by Andrzej Kozek.
- Received: 2015-10-15.
- Revised: 2016-03-10.
- Accepted: 2016-03-10.
- Published online: 2016-06-29.
