The Annals of Statistics

Sparse recovery in convex hulls via entropy penalization

Vladimir Koltchinskii

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Let (X, Y) be a random couple in S×T with unknown distribution P and (X1, Y1), …, (Xn, Yn) be i.i.d. copies of (X, Y). Denote Pn the empirical distribution of (X1, Y1), …, (Xn, Yn). Let h1, …, hN: S↦[−1, 1] be a dictionary that consists of N functions. For λ∈ℝN, denote fλ:=∑j=1Nλjhj. Let : T×ℝ↦ℝ be a given loss function and suppose it is convex with respect to the second variable. Let (f)(x, y):=(y; f(x)). Finally, let Λ⊂ℝN be the simplex of all probability distributions on {1, …, N}. Consider the following penalized empirical risk minimization problem $$\begin{eqnarray*}\hat{\lambda}^{\varepsilon}:={\mathop{\textrm{argmin}}_{\lambda\in \Lambda}}\Biggl[P_{n}(\ell \bullet f_{\lambda})+\varepsilon \sum_{j=1}^{N}\lambda_{j}\log \lambda_{j}\Biggr]\end{eqnarray*} $$ along with its distribution dependent version $$\begin{eqnarray*}\lambda^{\varepsilon}:={\mathop{\textrm{argmin}}_{\lambda\in \Lambda}}\Biggl[P(\ell \bullet f_{\lambda})+\varepsilon \sum_{j=1}^{N}\lambda_{j}\log \lambda_{j}\Biggr],\end{eqnarray*}$$ where ɛ≥0 is a regularization parameter. It is proved that the “approximate sparsity” of λɛ implies the “approximate sparsity” of λ̂ɛ and the impact of “sparsity” on bounding the excess risk of the empirical solution is explored. Similar results are also discussed in the case of entropy penalized density estimation.

Article information

Ann. Statist., Volume 37, Number 3 (2009), 1332-1359.

First available in Project Euclid: 10 April 2009

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Zentralblatt MATH identifier

Primary: 62G07: Density estimation 62G08: Nonparametric regression 62H30: Classification and discrimination; cluster analysis [See also 68T10, 91C20]

Penalized empirical risk minimization sparsity entropy convex hulls


Koltchinskii, Vladimir. Sparse recovery in convex hulls via entropy penalization. Ann. Statist. 37 (2009), no. 3, 1332--1359. doi:10.1214/08-AOS621.

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