Localized Lasso for High-Dimensional Regression
This addresses the problem of interpretability and prediction in high-dimensional data for fields like personalized medicine, though it appears incremental as an extension of Lasso with localized and regularization techniques.
The paper tackles the challenge of building interpretable and predictive models in high-dimensional, small-sample settings by introducing the localized Lasso, which uses local sparse models with network regularization and exclusive group sparsity. It empirically outperforms alternatives on simulated and genomic data, offering a convex, tunable-free optimization method.
We introduce the localized Lasso, which is suited for learning models that are both interpretable and have a high predictive power in problems with high dimensionality $d$ and small sample size $n$. More specifically, we consider a function defined by local sparse models, one at each data point. We introduce sample-wise network regularization to borrow strength across the models, and sample-wise exclusive group sparsity (a.k.a., $\ell_{1,2}$ norm) to introduce diversity into the choice of feature sets in the local models. The local models are interpretable in terms of similarity of their sparsity patterns. The cost function is convex, and thus has a globally optimal solution. Moreover, we propose a simple yet efficient iterative least-squares based optimization procedure for the localized Lasso, which does not need a tuning parameter, and is guaranteed to converge to a globally optimal solution. The solution is empirically shown to outperform alternatives for both simulated and genomic personalized medicine data.