Trace Lasso: a trace norm regularization for correlated designs (1109.1990v1)

Abstract: Using the $\ell_1$-norm to regularize the estimation of the parameter vector of a linear model leads to an unstable estimator when covariates are highly correlated. In this paper, we introduce a new penalty function which takes into account the correlation of the design matrix to stabilize the estimation. This norm, called the trace Lasso, uses the trace norm, which is a convex surrogate of the rank, of the selected covariates as the criterion of model complexity. We analyze the properties of our norm, describe an optimization algorithm based on reweighted least-squares, and illustrate the behavior of this norm on synthetic data, showing that it is more adapted to strong correlations than competing methods such as the elastic net.

Trace Lasso: A Trace Norm Regularization for Correlated Designs

This paper presents the trace Lasso, a new regularization approach designed for linear models where covariate correlation can undermine estimator stability when using conventional regularization techniques such as the Lasso. Traditional Lasso methods, leveraging the $\ell_1$-norm, show instability in high correlation settings, as they might arbitrarily select among highly correlated variables, potentially leading to inconsistent model interpretations and parameter estimates. This work circumvents this issue through trace Lasso—a penalty function based on the trace norm, a convex surrogate of matrix rank, which takes advantage of the structure of correlations to produce more stable and interpretable models.

The paper articulates several key contributions:

1. Development and Analysis of Trace Lasso: The authors introduce trace Lasso by leveraging the trace norm, which provides a more refined treatment of predictor correlations. Unlike the Lasso or elastic net methods, trace Lasso adapts to the design structure by interpolating between the $\ell_1$-norm and the $\ell_2$-norm based on predictor correlations. This adaptability is not manually predetermined, allowing it to dynamically stabilize estimator selection.
2. Unique Solution Assurance: It is proven that the empirical risk minimization using trace Lasso, with a strongly convex loss function, guarantees a unique minimizer. This property addresses the unpredictability of traditional $\ell_1$-based regularizers in correlated settings, ensuring enhanced reliability in the model’s decision process.
3. Efficient Optimization Algorithm: The authors derive an optimization algorithm using reweighted least-squares, making the computationally intensive problem of trace norm computation tractable for practical use. They exploit a variational characterization of the trace norm to iteratively update both the parameters and the covariance structures efficiently.
4. Empirical Validation: Extensive synthetic experiments demonstrate that trace Lasso has superior performance in scenarios with high correlation. It consistently outperforms traditional methods like the elastic net, especially in scenarios that present 'strong-correlation' regimes.

The implications of this work are substantial in the domains of statistical modeling and machine learning, particularly for problems involving high-dimensional data structures with inherent correlation. The trace Lasso presents an automatic grouping mechanism akin to the group Lasso without requiring prior knowledge or explicit specification of groups, making it a flexible and robust choice for many real-world applications involving complex data designs.

Future Directions: The paper suggests further exploration of trace Lasso in inverse problem settings such as image deblurring, where correlated covariate designs are prevalent. Another avenue may include theoretical exploration of the conditions under which trace Lasso can offer guarantees similar to those of oracle-based methods like the group Lasso. Further work could also focus on extending the applicability of the trace Lasso to non-linear models or other machine learning frameworks, potentially broadening its utility and enhancing its adaptivity to a wider range of data patterns and structures.

Overall, the trace Lasso emerges as a powerful tool for researchers dealing with correlated design matrices, providing enhanced stability and interpretability in statistical modeling.