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Jun Fan

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4 papers

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JMLR Journal 2025 Journal Article

Nonlinear functional regression by functional deep neural network with kernel embedding

Zhongjie Shi
Jun Fan
Linhao Song
Ding-Xuan Zhou
Johan A.K. Suykens

Recently, deep learning has been widely applied in functional data analysis (FDA) with notable empirical success. However, the infinite dimensionality of functional data necessitates an effective dimension reduction approach for functional learning tasks, particularly in nonlinear functional regression. In this paper, we introduce a functional deep neural network with an adaptive and discretization-invariant dimension reduction method. Our functional network architecture consists of three parts: first, a kernel embedding step that features an integral transformation with an adaptive smooth kernel; next, a projection step that uses eigenfunction bases based on a projection Mercer kernel for the dimension reduction; and finally, a deep ReLU neural network is employed for the prediction. Explicit rates of approximating nonlinear smooth functionals across various input function spaces by our proposed functional network are derived. Additionally, we conduct a generalization analysis for the empirical risk minimization (ERM) algorithm applied to our functional net, by employing a novel two-stage oracle inequality and the established functional approximation results. Ultimately, we conduct numerical experiments on both simulated and real datasets to demonstrate the effectiveness and benefits of our functional net. [abs] [ pdf ][ bib ] &copy JMLR 2025. ( edit, beta )

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JMLR Journal 2020 Journal Article

A Statistical Learning Approach to Modal Regression

Yunlong Feng
Jun Fan
Johan A.K. Suykens

This paper studies the nonparametric modal regression problem systematically from a statistical learning viewpoint. Originally motivated by pursuing a theoretical understanding of the maximum correntropy criterion based regression (MCCR), our study reveals that MCCR with a tending-to-zero scale parameter is essentially modal regression. We show that the nonparametric modal regression problem can be approached via the classical empirical risk minimization. Some efforts are then made to develop a framework for analyzing and implementing modal regression. For instance, the modal regression function is described, the modal regression risk is defined explicitly and its Bayes rule is characterized; for the sake of computational tractability, the surrogate modal regression risk, which is termed as the generalization risk in our study, is introduced. On the theoretical side, the excess modal regression risk, the excess generalization risk, the function estimation error, and the relations among the above three quantities are studied rigorously. It turns out that under mild conditions, function estimation consistency and convergence may be pursued in modal regression as in vanilla regression protocols such as mean regression, median regression, and quantile regression. On the practical side, the implementation issues of modal regression including the computational algorithm and the selection of the tuning parameters are discussed. Numerical validations on modal regression are also conducted to verify our findings. [abs] [ pdf ][ bib ] &copy JMLR 2020. ( edit, beta )

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JMLR Journal 2016 Journal Article

Sparsity and Error Analysis of Empirical Feature-Based Regularization Schemes

Xin Guo
Jun Fan
Ding-Xuan Zhou

We consider a learning algorithm generated by a regularization scheme with a concave regularizer for the purpose of achieving sparsity and good learning rates in a least squares regression setting. The regularization is induced for linear combinations of empirical features, constructed in the literatures of kernel principal component analysis and kernel projection machines, based on kernels and samples. In addition to the separability of the involved optimization problem caused by the empirical features, we carry out sparsity and error analysis, giving bounds in the norm of the reproducing kernel Hilbert space, based on a priori conditions which do not require assumptions on sparsity in terms of any basis or system. In particular, we show that as the concave exponent $q$ of the concave regularizer increases to $1$, the learning ability of the algorithm improves. Some numerical simulations for both artificial and real MHC-peptide binding data involving the $\ell^q$ regularizer and the SCAD penalty are presented to demonstrate the sparsity and error analysis. [abs] [ pdf ][ bib ] &copy JMLR 2016. ( edit, beta )

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JMLR Journal 2016 Journal Article

Structure-Leveraged Methods in Breast Cancer Risk Prediction

Jun Fan
Yirong Wu
Ming Yuan
David Page
Jie Liu
Irene M. Ong
Peggy Peissig
Elizabeth Burnside

Predicting breast cancer risk has long been a goal of medical research in the pursuit of precision medicine. The goal of this study is to develop novel penalized methods to improve breast cancer risk prediction by leveraging structure information in electronic health records. We conducted a retrospective case- control study, garnering 49 mammography descriptors and 77 high- frequency/low-penetrance single-nucleotide polymorphisms (SNPs) from an existing personalized medicine data repository. Structured mammography reports and breast imaging features have long been part of a standard electronic health record (EHR), and genetic markers likely will be in the near future. Lasso and its variants are widely used approaches to integrated learning and feature selection, and our methodological contribution is to incorporate the dependence structure among the features into these approaches. More specifically, we propose a new methodology by combining group penalty and $\ell^p$ ($1\leq p\leq2$) fusion penalty to improve breast cancer risk prediction, taking into account structure information in mammography descriptors and SNPs. We demonstrate that our method provides benefits that are both statistically significant and potentially significant to people's lives. [abs] [ pdf ][ bib ] &copy JMLR 2016. ( edit, beta )

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