Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  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 \({\mathbb R}^d\).  obtain an estimate for the e-entropy numbers in terms of smaller e-covering numbers of the compact class Gn,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  show that for n ≤ sd − 1 the e-entropy number He(Gn,s,Lq) of the class Gn,s in the space Lq is of order nslog1/e (modulo a logarithmic factor). Note that the e-entropy number \(H_\varepsilon(B\mathcal{P}_s^d,L_q)\) of the unit ball is of order sdlog1/e. Moreover,  obtain an estimate for the pseudo-dimension of the ridge function class Gn,s.
['Vitaly Maiorov']
I am writing a research paper on a idea Geometric properties of the ridge function manifold
[{"from": "human", "value": "I am writing a research paper on a idea Geometric properties of the ridge function manifold"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  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\u00b7 x), where a\u00b7x is the inner product in \\({\\mathbb R}^d\\).  obtain an estimate for the e-entropy numbers in terms of smaller e-covering numbers of the compact class Gn,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  show that for n \u2264 sd \u2212 1 the e-entropy number He(Gn,s,Lq) of the class Gn,s in the space Lq is of order nslog1/e (modulo a logarithmic factor). Note that the e-entropy number \\(H_\\varepsilon(B\\mathcal{P}_s^d,L_q)\\) of the unit ball is of order sdlog1/e. Moreover,  obtain an estimate for the pseudo-dimension of the ridge function class Gn,s."}]