Functional Regression Models in Hilbert Spaces

Functional regression models in Hilbert spaces constitute a class of statistical methods that link functional predictors and/or responses via operators defined on infinite-dimensional inner-product spaces. They extend classical scalar or multivariate regression by treating entire curves, surfaces or other functional objects as covariates or outcomes. A general model may express a functional response as the action of a linear operator on a functional predictor, or relate scalar responses to functional inputs through integral transforms. Estimation typically proceeds via basis expansions—such as splines, wavelets or functional principal components—together with penalisation or regularisation to address ill-posedness and to control smoothness. Reproducing Kernel Hilbert Spaces provide a unifying framework for nonparametric functional regression, allowing the direct estimation of operators via kernel methods. Recent advances have clarified asymptotic properties under various sampling designs (dense, sparse or irregular), delivered minimax rates of convergence, and introduced high-dimensional penalties for variable selection. Computational strategies now include iterative algorithms for large datasets and efficient implementations in statistical software. Applications span environmental monitoring, biomedicine, genomics and finance, wherever predictors or responses are observed over a continuum. This body of work underscores the blend of functional data analysis, operator theory and regularisation techniques at the heart of modern functional regression in Hilbert spaces.

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A recent methodological development extends the LASSO penalty to function-on-scalar regression with ultra-high-dimensional predictors. By formulating a functional LASSO in a dense or sparse functional setting, the approach simultaneously selects important scalar covariates and estimates smooth coefficient functions, with theoretical guarantees that accommodate an exponential number of predictors. This framework has been applied to genome-wide association studies, yielding interpretable genetic insights into longitudinal health outcomes. Another line of work introduces an operator-based procedure for simultaneous predictor selection and smoothing in high-dimensional function-on-scalar models. Using subspaces defined by desired properties (such as smoothness or periodicity), this method generalises to arbitrary separable Hilbert spaces and offers a fast coordinate-descent algorithm. Empirical comparisons demonstrate superior statistical and computational performance in genetic and clinical applications. More recently, distribution-on-distribution regression models have been proposed for multivariate Gaussian measures under the Wasserstein metric. By exploiting the geometry of optimal transport, these models map Gaussian distributions into a linear matrix space, enabling intuitive linear regression frameworks and simplified computation, with extensions to non-Gaussian settings and theoretical guarantees on prediction error convergence.

Functional Regression Models in Hilbert Spaces publication trend

The graph below shows the total number of articles in functional regression models in hilbert spaces across all publications each year (not limited to Nature Index journals).

Technical terms

Hilbert space: A complete vector space with an inner product, enabling geometric concepts such as orthogonality and projection in infinite dimensions.

Reproducing Kernel Hilbert Space (RKHS): A Hilbert space of functions equipped with a kernel function that allows evaluation and inner products to be computed directly, facilitating nonparametric estimation.

Functional principal component analysis (FPCA): A dimension-reduction technique that represents functional data through orthogonal basis functions capturing the dominant modes of variability.

Operator: A mapping between function spaces, often linear, that parameterises the transformation of functional predictors into responses in regression models.

Penalisation: A regularisation approach that adds a penalty term to the estimation objective to enforce smoothness or sparsity, improving stability and interpretability.

References

1. The function-on-scalar LASSO with applications to longitudinal GWAS. Electronic Journal of Statistics (2017).
2. Simultaneous variable selection and smoothing for high-dimensional function-on-scalar regression. Electronic Journal of Statistics (2018).
3. Distribution-on-distribution regression with Wasserstein metric: Multivariate Gaussian case. Journal of Multivariate Analysis (2024).