arXiv subjects

Billy Fang

Publications and source records attributed to Billy Fang.

4 recordsLinked to original sources

arXiv2021

MARS via LASSO

Multivariate adaptive regression splines (MARS) is a popular method for nonparametric regression introduced by Friedman in 1991. MARS fits simple nonlinear and non-additive functions to regression data. We propose and study a natural lasso variant of the MARS method. Our method is based on least squares estimation over a convex class of functions obtained by considering infinite-dimensional linear combinations of functions in the MARS basis and imposing a variation based complexity constraint. Our estimator can be computed via finite-dimensional convex optimization, although it is defined as a solution to an infinite-dimensional optimization problem. Under a few standard design assumptions, we prove that our estimator achieves a rate of convergence that depends only logarithmically on dimension and thus avoids the usual curse of dimensionality to some extent. We also show that our method is naturally connected to nonparametric estimation techniques based on smoothness constraints. We implement our method with a cross-validation scheme for the selection of the involved tuning parameter and compare it to the usual MARS method in various simulation and real data settings.

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arXiv2019

Multivariate extensions of isotonic regression and total variation denoising via entire monotonicity and Hardy-Krause variation

We consider the problem of nonparametric regression when the covariate is $d$-dimensional, where $d \geq 1$. In this paper we introduce and study two nonparametric least squares estimators (LSEs) in this setting---the entirely monotonic LSE and the constrained Hardy-Krause variation LSE. We show that these two LSEs are natural generalizations of univariate isotonic regression and univariate total variation denoising, respectively, to multiple dimensions. We discuss the characterization and computation of these two LSEs obtained from $n$ data points. We provide a detailed study of their risk properties under the squared error loss and fixed uniform lattice design. We show that the finite sample risk of these LSEs is always bounded from above by $n^{-2/3}$ modulo logarithmic factors depending on $d$; thus these nonparametric LSEs avoid the curse of dimensionality to some extent. We also prove nearly matching minimax lower bounds. Further, we illustrate that these LSEs are particularly useful in fitting rectangular piecewise constant functions. Specifically, we show that the risk of the entirely monotonic LSE is almost parametric (at most $1/n$ up to logarithmic factors) when the true function is well-approximable by a rectangular piecewise constant entirely monotone function with not too many constant pieces. A similar result is also shown to hold for the constrained Hardy-Krause variation LSE for a simple subclass of rectangular piecewise constant functions. We believe that the proposed LSEs yield a novel approach to estimating multivariate functions using convex optimization that avoid the curse of dimensionality to some extent.

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arXiv2018

From Gauss to Kolmogorov: Localized Measures of Complexity for Ellipses

The Gaussian width is a fundamental quantity in probability, statistics and geometry, known to underlie the intrinsic difficulty of estimation and hypothesis testing. In this work, we show how the Gaussian width, when localized to any given point of an ellipse, can be controlled by the Kolmogorov width of a set similarly localized. This connection leads to an explicit characterization of the estimation error of least-squares regression as a function of the true regression vector within the ellipse. The rate of error decay varies substantially as a function of location: as a concrete example, in Sobolev ellipses of smoothness $\alpha$, we exhibit rates that vary from $(\sigma^2)^{\frac{2 \alpha}{2 \alpha + 1}}$, corresponding to the classical global rate, to the faster rate $(\sigma^2)^{\frac{4 \alpha}{4 \alpha + 1}}$. We also show how the local Kolmogorov width can be related to local metric entropy.

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arXiv2017

On the risk of convex-constrained least squares estimators under misspecification

We consider the problem of estimating the mean of a noisy vector. When the mean lies in a convex constraint set, the least squares projection of the random vector onto the set is a natural estimator. Properties of the risk of this estimator, such as its asymptotic behavior as the noise tends to zero, have been well studied. We instead study the behavior of this estimator under misspecification, that is, without the assumption that the mean lies in the constraint set. For appropriately defined notions of risk in the misspecified setting, we prove a generalization of a low noise characterization of the risk due to Oymak and Hassibi in the case of a polyhedral constraint set. An interesting consequence of our results is that the risk can be much smaller in the misspecified setting than in the well-specified setting. We also discuss consequences of our result for isotonic regression.

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