Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: It is known that for a certain class of single index models (SIMs) Y = f(Xp×1T β0, e), support recovery is impossible when X ∼ N(0; Ip×p) and a model complexity adjusted sample size is below a critical threshold. Recently, optimal algorithms based on Sliced Inverse Regression (SIR) re suggested. These algorithms work provably under the assumption that the design X comes from an i.i.d. Gaussian distribution. In the present paper  analyze algorithms based on covariance screening and least squares with L1 penalization (i.e. LASSO) and demonstrate that they can also enjoy optimal (up to a scalar) rescaled sample size in terms of support recovery, albeit under slightly different assumptions on f and e compared to the SIR based algorithms. Furthermore,  show more generally, that LASSO succeeds in recovering the signed support of β0 if X ∼ N(0, Σ), and the covariance Σ satisfies the irrepresentable condition. Our work extends existing results on the support recovery of LASSO for the linear model, to a more general class of SIMs.
['Matey Neykov', 'Jun S. Liu', 'Tianxi Cai']
I am writing a research paper on a idea L 1 -regularized least squares for support recovery of high dimensional single index models with Gaussian designs
[{"from": "human", "value": "I am writing a research paper on a idea L 1 -regularized least squares for support recovery of high dimensional single index models with Gaussian designs"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: It is known that for a certain class of single index models (SIMs) Y = f(Xp\u00d71T \u03b20, e), support recovery is impossible when X \u223c N(0; Ip\u00d7p) and a model complexity adjusted sample size is below a critical threshold. Recently, optimal algorithms based on Sliced Inverse Regression (SIR) re suggested. These algorithms work provably under the assumption that the design X comes from an i.i.d. Gaussian distribution. In the present paper  analyze algorithms based on covariance screening and least squares with L1 penalization (i.e. LASSO) and demonstrate that they can also enjoy optimal (up to a scalar) rescaled sample size in terms of support recovery, albeit under slightly different assumptions on f and e compared to the SIR based algorithms. Furthermore,  show more generally, that LASSO succeeds in recovering the signed support of \u03b20 if X \u223c N(0, \u03a3), and the covariance \u03a3 satisfies the irrepresentable condition. Our work extends existing results on the support recovery of LASSO for the linear model, to a more general class of SIMs."}]