Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: In high dimensional settings, sparse structures are crucial for eciency, both in term of memory, computation and performance. It is customary to consider ‘1 penalty to enforce sparsity in such scenarios. Sparsity enforcing methods, the Lasso being a canonical example, are popular candidates to address high dimension. For eciency, they rely on tuning a parameter trading data fitting versus sparsity. For the Lasso theory to hold this tuning parameter should be proportional to the noise level, yet the latter is often unknown in practice. A possible remedy is to jointly optimize over the regression parameter as ll as over the noise level. This has been considered under several names in the literature: Scaled-Lasso, Square-root Lasso, Concomitant Lasso estimation for instance, and could be of interest for confidence sets or uncertainty quantification. In this work, after illustrating numerical diculties for the Smoothed Concomitant Lasso formulation,  propose a modification  coined Smoothed Concomitant Lasso, aimed at increasing numerical stability.  propose an ecient and accurate solver leading to a computational cost no more expansive than the one for the Lasso.  leverage on standard ingredients behind the success of fast Lasso solvers: a coordinate descent algorithm, combined with safe screening rules to achieve speed eciency, by eliminating early irrelevant features.
['Eugene Ndiaye', 'Olivier Fercoq', 'Alexandre Gramfort', 'Vincent Leclère', 'Joseph Salmon']
I am writing a research paper on a idea Efficient Smoothed Concomitant Lasso Estimation for High Dimensional Regression
[{"from": "human", "value": "I am writing a research paper on a idea Efficient Smoothed Concomitant Lasso Estimation for High Dimensional Regression"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: In high dimensional settings, sparse structures are crucial for eciency, both in term of memory, computation and performance. It is customary to consider \u20181 penalty to enforce sparsity in such scenarios. Sparsity enforcing methods, the Lasso being a canonical example, are popular candidates to address high dimension. For eciency, they rely on tuning a parameter trading data fitting versus sparsity. For the Lasso theory to hold this tuning parameter should be proportional to the noise level, yet the latter is often unknown in practice. A possible remedy is to jointly optimize over the regression parameter as ll as over the noise level. This has been considered under several names in the literature: Scaled-Lasso, Square-root Lasso, Concomitant Lasso estimation for instance, and could be of interest for confidence sets or uncertainty quantification. In this work, after illustrating numerical diculties for the Smoothed Concomitant Lasso formulation,  propose a modification  coined Smoothed Concomitant Lasso, aimed at increasing numerical stability.  propose an ecient and accurate solver leading to a computational cost no more expansive than the one for the Lasso.  leverage on standard ingredients behind the success of fast Lasso solvers: a coordinate descent algorithm, combined with safe screening rules to achieve speed eciency, by eliminating early irrelevant features."}]