Adaptive piecewise polynomial estimation via trend filtering (1304.2986v2)

Abstract: We study trend filtering, a recently proposed tool of Kim et al. [SIAM Rev. 51 (2009) 339-360] for nonparametric regression. The trend filtering estimate is defined as the minimizer of a penalized least squares criterion, in which the penalty term sums the absolute $k$th order discrete derivatives over the input points. Perhaps not surprisingly, trend filtering estimates appear to have the structure of $k$th degree spline functions, with adaptively chosen knot points (we say ``appear'' here as trend filtering estimates are not really functions over continuous domains, and are only defined over the discrete set of inputs). This brings to mind comparisons to other nonparametric regression tools that also produce adaptive splines; in particular, we compare trend filtering to smoothing splines, which penalize the sum of squared derivatives across input points, and to locally adaptive regression splines [Ann. Statist. 25 (1997) 387-413], which penalize the total variation of the $k$th derivative. Empirically, we discover that trend filtering estimates adapt to the local level of smoothness much better than smoothing splines, and further, they exhibit a remarkable similarity to locally adaptive regression splines. We also provide theoretical support for these empirical findings; most notably, we prove that (with the right choice of tuning parameter) the trend filtering estimate converges to the true underlying function at the minimax rate for functions whose $k$th derivative is of bounded variation. This is done via an asymptotic pairing of trend filtering and locally adaptive regression splines, which have already been shown to converge at the minimax rate [Ann. Statist. 25 (1997) 387-413]. At the core of this argument is a new result tying together the fitted values of two lasso problems that share the same outcome vector, but have different predictor matrices.

• The paper presents trend filtering as an adaptive spline technique that outperforms smoothing splines in capturing local variations.
• It establishes theoretical guarantees by achieving the minimax convergence rate for functions with bounded variation in their derivatives.
• The method offers computational efficiency and robustness, making it ideal for large-scale nonparametric regression and signal processing applications.

Overview of "Adaptive Piecewise Polynomial Estimation via Trend Filtering"

The paper "Adaptive Piecewise Polynomial Estimation via Trend Filtering" by Ryan J. Tibshirani tackles the problem of nonparametric regression through trend filtering, a method initially proposed by Kim et al. in 2009. Trend filtering estimates the underlying function of observed data by minimizing a penalized least squares criterion, where the penalty is the total variation of the $k$-th discrete derivative. The work provides comprehensive insights into the comparative performance and theoretical guarantees of trend filtering against other spline-based methods, such as smoothing splines and locally adaptive regression splines.

Core Contributions

1. Trend Filtering as Adaptive Splines: The paper positions trend filtering as an effective tool for piecewise polynomial estimation. The estimates appear to function as high-degree splines with adaptively chosen knot points. The method's adaptivity allows it to maintain an advantageous position compared to other spline techniques, in terms of both computational efficiency and adaptivity in handling varying smoothness over input domains.
2. Comparison with Smoothing Splines and Locally Adaptive Regression Splines: Tibshirani provides empirical evidence and theoretical support illustrating that trend filtering can outperform smoothing splines in terms of local adaptiveness. Moreover, trend filtering shows remarkable empirical similarity to locally adaptive regression splines, even though theoretically distinct, providing the same convergence rates under specific conditions.
3. Theoretical Underpinnings: One of the paper's notable achievements is the theoretical demonstration that trend filtering achieves the minimax rate of convergence in estimating functions whose $k$-th derivative is of bounded variation. This rate matches the best rate previously achieved by locally adaptive regression splines but is not feasible for smoothing splines.
4. Algorithmic Efficiency: The discussion emphasizes the computational efficacy of trend filtering, which leverages the sparsity in the discrete derivative operator. This characteristic renders it considerably faster than locally adaptive regression splines, particularly in large-scale problems.

Implications and Future Directions

• Practical Implications: For practitioners, trend filtering presents as a powerful tool for applications demanding adaptive sensitivity to local variations, such as in signal processing and time series analysis. The computational efficiency and theoretical robustness make it suitable for large-scale data environments.
• Theoretical Extensions: Future work could delve into extending trend filtering beyond its current univariate framework to handle multivariate inputs or to explore further the synthesis versus analysis paradigms in the context of $\ell_1$-penalized estimation.
• Methodological Developments: The ability of trend filtering to generate accurate estimates with minimal tuning parameters or theoretical knowledge positions it as a candidate for default choice in nonparametric regression tasks, potentially furthering its applicability through practical software implementations.

In conclusion, Tibshirani's paper on trend filtering provides a rigorous foundation for a method that balances theoretical robustness with computational efficiency. Its place in the landscape of nonparametric regression is well-justified, filling a niche that neither smoothing splines nor locally adaptive regression splines can singularly satisfy as effectively. The work opens avenues for further methodological advancements and wider adoption in practical settings.