Abstract
We study geometrical properties of the ridge function manifold \(\mathcal{R}_n\) consisting of all possible linear combinations of n functions of the form g(a· x), where a·x is the inner product in \({\mathbb R}^d\). We obtain an estimate for the ε-entropy numbers in terms of smaller ε-covering numbers of the compact class G n,s formed by the intersection of the class \(\mathcal{R}_n\) with the unit ball \(B\mathcal{P}_s^d\) in the space of polynomials on \({\mathbb R}^d\) of degree s. In particular we show that for n ≤ s d − 1 the ε-entropy number H ε (G n,s,L q ) of the class G n,s in the space L q is of order nslog1/ε (modulo a logarithmic factor). Note that the ε-entropy number \(H_\varepsilon(B\mathcal{P}_s^d,L_q)\) of the unit ball is of order s dlog1/ε. Moreover, we obtain an estimate for the pseudo-dimension of the ridge function class G n,s.
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Communicated by Juan Manuel Peña.
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Maiorov, V. Geometric properties of the ridge function manifold. Adv Comput Math 32, 239–253 (2010). https://doi.org/10.1007/s10444-008-9106-3
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DOI: https://doi.org/10.1007/s10444-008-9106-3