Semi-supervised learning with manifold regularization is a classical framework for jointly learning from both labeled and unlabeled data, where the key requirement is that the support of the unknown marginal distribution has the geometric structure of a Riemannian manifold. Typically, the Laplace-Beltrami operator-based manifold regularization can be approximated empirically by the Laplacian regularization associated with the entire training data and its corresponding graph Laplacian matrix. However, the graph Laplacian matrix depends heavily on the prespecified similarity metric and may lead to inappropriate penalties when dealing with redundant or noisy input variables. To address the above issues, this paper proposes a new \textit{Semi-Supervised Meta Additive Model (S$^2$MAM) based on a bilevel optimization scheme that automatically identifies informative variables, updates the similarity matrix, and simultaneously achieves interpretable predictions. Theoretical guarantees are provided for S$^2$MAM, including the computing convergence and the statistical generalization bound. Experimental assessments across 4 synthetic and 12 real-world datasets, with varying levels and categories of corruption, validate the robustness and interpretability of the proposed approach.

The additive model is one of the most popularly used models for high dimensional nonparametric regression analysis. However, its main drawback is that it neglects possible interactions between predictor variables. In this paper, we reexamine the group additive model proposed in the literature, and rigorously define the intrinsic group additive structure for the relationship between the response variable $Y$ and the predictor vector $\vect{X}$, and further develop an effective structure-penalized kernel method for simultaneous identification of the intrinsic group additive structure and nonparametric function estimation. The method utilizes a novel complexity measure we derive for group additive structures. We show that the proposed method is consistent in identifying the intrinsic group additive structure. Simulation study and real data applications demonstrate the effectiveness of the proposed method as a general tool for high dimensional nonparametric regression.

We present a novel approach for nonparametric regression using wavelet basis functions. Our proposal, waveMesh, can be applied to non-equispaced data with sample size not necessarily a power of 2. We develop an efficient proximal gradient descent algorithm for computing the estimator and establish adaptive minimax convergence rates. The main appeal of our approach is that it naturally extends to additive and sparse additive models for a potentially large number of covariates. We prove minimax optimal convergence rates under a weak compatibility condition for sparse additive models. The compatibility condition holds when we have a small number of covariates. Additionally, we establish convergence rates for when the condition is not met. We complement our theoretical results with empirical studies comparing waveMesh to existing methods.

It is desirable to restrict the number of components (i.e., encourage sparsity) for

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We present a novel approach for nonparametric regression using wavelet basis functions.

Bayesian Optimization (BO) is a versatile method for global optimization of a black-box function using noisy point-wise observations.

WepresentanewclassofGAMs thatusetensor rank decompositions of polynomials to learn powerful,inherently-interpretable models. Our approach, titled Scalable Polynomial Additive Models (SPAM) is effortlessly scalable and modelsall higher-order feature interactions without a combinatorial parameter explosion. SPAM outperforms allcurrent interpretable approaches, and matches DNN/XGBoost performance onaseries ofreal-world benchmarks with up to hundreds of thousands of features.