Functional linear regression from sparse to dense designs: a pooling-ridge method and minimax optimality

• URL: http://arxiv.org/abs/2608.25468v1
• Date: Wed, 26 Aug 2026 07:36:39 GMT
• Title: Functional linear regression from sparse to dense designs: a pooling-ridge method and minimax optimality
• Abstract summary: Functional data analysis is an important statistical field that treats data as random functions.<n>In practice, the random functions are often not fully observed but instead measured at discrete times.<n>We propose a novel approach, referred to as pooling ridge estimation, which combines the advantages of pooling strategy and RKHS-based method.
• Score: 0.962543698736491
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Functional data analysis is an important statistical field that treats data as random functions. In practice, the random functions are often not fully observed but instead measured at discrete times. While simpler problems, such as mean and covariance estimation, have been widely studied for discretely observed data, optimal estimation of linear regression for this data type has remained unsolved for over two decades. To tackle this fundamental challenge, we propose a novel approach, referred to as pooling ridge estimation, which combines the advantages of pooling strategy and RKHS-based method by incorporating the unbiased estimation of operators based on discretely observed measurements from all subjects. This unified estimation framework enables us to achieve minimax optimality in prediction risk in arbitrary sampling schemes ranging from sparse to dense designs, for both scalar-on-function and function-on-function regression models. Such methodological and theoretical advances are obtained for the first time and accurately reveal the influence of discrete sampling. For scalar-on-function regression, the phase transition occurs once, separating the convergence behavior into two distinct regimes. Remarkably, for function-on-function regression, up to three phase transitions may occur, determined by the sampling frequencies of the predictor/response functions. Finally, simulation experiments and two real data examples provide empirical support for the proposed methods.

Related papers

• Optimal estimation for Functional Linear Regression with Noisy Discretized Data [0.0]
  The underlying curves are reconstructed from the discrete noisy observations.<n>The slope function is estimated by a penalized least-squares criterion over finite-dimensional trigonometric spaces.<n>The proposed method is illustrated on simulated data and on a real meteorological dataset.
  arXiv  Detail & Related papers  (2026-09-08T12:37:10Z)
• An Elementary Approach to Scheduling in Generative Diffusion Models [55.171367482496755]
  An elementary approach to characterizing the impact of noise scheduling and time discretization in generative diffusion models is developed.<n> Experiments across different datasets and pretrained models demonstrate that the time discretization strategy selected by our approach consistently outperforms baseline and search-based strategies.
  arXiv  Detail & Related papers  (2026-01-20T05:06:26Z)
• Efficient Covariance Estimation for Sparsified Functional Data [51.69796254617083]
  proposed Random-knots (Random-knots-Spatial) and B-spline (Bspline-Spatial) estimators of the covariance function are computationally efficient.<n>Asymptotic pointwise of the covariance are obtained for sparsified individual trajectories under some regularity conditions.
  arXiv  Detail & Related papers  (2025-11-23T00:50:33Z)
• Learning from Summarized Data: Gaussian Process Regression with Sample Quasi-Likelihood [9.13755431537592]
  This study tackles learning and inference using only summarized data within the framework of Gaussian process regression.<n>We introduce the concept of sample quasi-likelihood, which facilitates learning and inference using only summarized data.
  arXiv  Detail & Related papers  (2024-12-23T10:21:38Z)
• A Provably Accurate Randomized Sampling Algorithm for Logistic Regression [2.7930955543692817]
  We present a simple, randomized sampling-based algorithm for logistic regression problem. We prove that accurate approximations can be achieved with a sample whose size is much smaller than the total number of observations. Overall, our work sheds light on the potential of using randomized sampling approaches to efficiently approximate the estimated probabilities in logistic regression.
  arXiv  Detail & Related papers  (2024-02-26T06:20:28Z)
• Structured Radial Basis Function Network: Modelling Diversity for Multiple Hypotheses Prediction [51.82628081279621]
  Multi-modal regression is important in forecasting nonstationary processes or with a complex mixture of distributions. A Structured Radial Basis Function Network is presented as an ensemble of multiple hypotheses predictors for regression problems. It is proved that this structured model can efficiently interpolate this tessellation and approximate the multiple hypotheses target distribution.
  arXiv  Detail & Related papers  (2023-09-02T01:27:53Z)
• Semi-Parametric Inference for Doubly Stochastic Spatial Point Processes: An Approximate Penalized Poisson Likelihood Approach [3.085995273374333]
  Doubly-stochastic point processes model the occurrence of events over a spatial domain as an inhomogeneous process conditioned on the realization of a random intensity function. Existing implementations of doubly-stochastic spatial models are computationally demanding, often have limited theoretical guarantee, and/or rely on restrictive assumptions.
  arXiv  Detail & Related papers  (2023-06-11T19:48:39Z)
• Kernel-based off-policy estimation without overlap: Instance optimality beyond semiparametric efficiency [53.90687548731265]
  We study optimal procedures for estimating a linear functional based on observational data. For any convex and symmetric function class $mathcalF$, we derive a non-asymptotic local minimax bound on the mean-squared error.
  arXiv  Detail & Related papers  (2023-01-16T02:57:37Z)
• Coordinated Double Machine Learning [8.808993671472349]
  This paper argues that a carefully coordinated learning algorithm for deep neural networks may reduce the estimation bias. The improved empirical performance of the proposed method is demonstrated through numerical experiments on both simulated and real data.
  arXiv  Detail & Related papers  (2022-06-02T05:56:21Z)
• On the Double Descent of Random Features Models Trained with SGD [78.0918823643911]
  We study properties of random features (RF) regression in high dimensions optimized by gradient descent (SGD) We derive precise non-asymptotic error bounds of RF regression under both constant and adaptive step-size SGD setting. We observe the double descent phenomenon both theoretically and empirically.
  arXiv  Detail & Related papers  (2021-10-13T17:47:39Z)
• Leveraging Global Parameters for Flow-based Neural Posterior Estimation [90.21090932619695]
  Inferring the parameters of a model based on experimental observations is central to the scientific method. A particularly challenging setting is when the model is strongly indeterminate, i.e., when distinct sets of parameters yield identical observations. We present a method for cracking such indeterminacy by exploiting additional information conveyed by an auxiliary set of observations sharing global parameters.
  arXiv  Detail & Related papers  (2021-02-12T12:23:13Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.