Non-negative least squares for high-dimensional linear models: consistency and sparse recovery without regularization (1205.0953v2)

Abstract: Least squares fitting is in general not useful for high-dimensional linear models, in which the number of predictors is of the same or even larger order of magnitude than the number of samples. Theory developed in recent years has coined a paradigm according to which sparsity-promoting regularization is regarded as a necessity in such setting. Deviating from this paradigm, we show that non-negativity constraints on the regression coefficients may be similarly effective as explicit regularization if the design matrix has additional properties, which are met in several applications of non-negative least squares (NNLS). We show that for these designs, the performance of NNLS with regard to prediction and estimation is comparable to that of the lasso. We argue further that in specific cases, NNLS may have a better $\ell_{\infty}$-rate in estimation and hence also advantages with respect to support recovery when combined with thresholding. From a practical point of view, NNLS does not depend on a regularization parameter and is hence easier to use.

• The paper demonstrates that Non-negative Least Squares (NNLS) can achieve consistency and sparse recovery in high-dimensional linear models under specific design conditions, often without explicit regularization.
• Authors reveal that non-negativity constraints possess self-regularizing properties under certain designs, enabling NNLS to prevent overfitting and manage prediction tasks effectively in high dimensions.
• A key advantage highlighted is that NNLS avoids the need for regularization parameter tuning compared to methods like Lasso, suggesting potential for simplified high-dimensional models.

Non-negative Least Squares in High-dimensional Linear Models

The paper by Slawski and Hein offers a comprehensive paper of Non-negative Least Squares (NNLS) in the context of high-dimensional linear models, challenging the prevailing notion that regularization is necessary for such models. The authors propose that non-negativity constraints on regression coefficients can be effective in achieving consistency and sparse recovery without explicit regularization. This insight is particularly relevant for scenarios where the design matrix meets certain properties.

Key Findings and Contributions

The primary contribution of this paper is the demonstration that NNLS can perform comparably to the Lasso in terms of prediction accuracy and estimation under specific design conditions. The authors argue that, for certain designs, non-negativity constraints inherently possess self-regularizing properties. These properties allow NNLS to effectively manage prediction and estimation tasks in high-dimensional settings where $p \sim n$, or $p \gg n$.

Notably, the researchers highlight that NNLS exhibits favorable $\ell_{\infty}$-rate in estimation, which can present advantages in support recovery when combined with thresholding processes. They also assert practical benefits of NNLS, including its lack of dependence on regularization parameters, simplifying its application compared to regularization-based methods like Lasso.

Technical Insights

The authors analytically derive conditions under which NNLS can prevent overfitting, a significant concern in high-dimensional regimes. They introduce a "self-regularizing" design property defined by a positive separation or margin, $\tau_0 > 0$, between the origin and the convex hull of scaled columns of the design matrix. This property, along with restricted eigenvalue conditions, enables NNLS to achieve comparable $\ell_2$-prediction error rates to regularized approaches like Lasso.

Through rigorous theoretical development, the paper establishes bounds on estimation errors for NNLS, providing scenarios under which NNLS can achieve consistency even when the number of predictors $p$ is nearly exponential in the number of observations $n$.

Comparative Analysis

Slawski and Hein's comparative analysis of NNLS and the non-negative Lasso reveals nuances central to support recovery. While non-negative Lasso requires conditions like the non-negative irrepresentable condition for exact support recovery, NNLS's primary reliance on design conditions like self-regularization opens avenues for it to perform equivalently in certain regimes without exhaustive parameter tuning.

Future Implications

The findings and theoretical results of this paper hold significant implications for future developments in high-dimensional linear models and AI. This perspective could pave the way for more simplified models that reduce computational burden while maintaining robust predictive performance. The self-regularizing property framework might inspire new model designs and analytical techniques in the realms of machine learning and statistical theory — particularly in fields relying heavily on high-dimensional data such as genomics, proteomics, and image processing.

Conclusion

The paper sets the stage for reevaluating conventional approaches that necessitate regularization in high-dimensional settings. By meticulously analyzing the conditions under which NNLS thrives without explicit regularization, Slawski and Hein contribute valuable insights into the optimization and computation in sparse recovery, providing a foundation for future exploration of non-negativity constraints and their practical utility in linear models.