arXiv subjects

Oscar Hernan Madrid Padilla

Publications and source records attributed to Oscar Hernan Madrid Padilla.

At least 19 recordsLinked to original sources

arXiv2026

Two-Stage Score Estimation for Group-Specific Diffusion Models

Generative modeling across related distributions raises a question: how can shared structure support accurate generation for a target distribution with limited observations? We develop a statistical framework for diffusion transfer that analyzes a shared pooled score together with a target-specific offset. Under Besov regularity of the pooled density and a stable hierarchical composition model for the offset, we derive convergence rates that separate the costs of learning these components and identify when a simpler offset yields a faster risk-bound rate than separate fitting. Under suitable boundary and regularity assumptions, we prove that the offset remains bounded and jointly regular as diffusion time approaches zero. These results allow the lower time cutoff to vanish polynomially and provide total-variation guarantees for diffusion models as density estimators. Synthetic simulations and image benchmarks provide a proof of concept for the framework.

stat.ML↗

arXiv2026

Distributional Regression for Arbitrary Response Variables Using ReLU Neural Networks

This paper develops a unified framework for distributional regression with arbitrary response variables. Rather than relying on scalar thresholds or Euclidean structure, we model conditional event probabilities indexed by measurable sets, which extends distributional regression to general response spaces while retaining a direct connection to classical event-based formulations. Within this framework, we study two complementary estimators: a preliminary estimator based on separate regression of event indicators, and a constrained estimator that enforces inclusion-consistent predictions through a directed acyclic graph (DAG) over a finite set family. We establish general error bounds for both estimators and further specialize the results to dense ReLU networks, yielding explicit convergence rates overcoming the curse of dimensionality. Simulation studies and real-data experiments further demonstrate the practical effectiveness of the proposed framework.

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arXiv2026

Transfer Learning in Nonparametric Regression with Deep ReLU Networks

This paper develops a general transfer learning framework for nonparametric regression with data consisting of multiple groups. Under the assumption that groups share a common structure along with group-specific deviations in additive form, the proposed method employs a two-stage offset learning procedure: the first stage pools data from all groups to estimate an overall mean function, and the second stage estimates offsets for each group, yielding final group-level estimators through additive combination. Upper bounds on the $\mathcal L_2$ error are established for the proposed framework, covering a broad class of nonparametric estimators under mild complexity and noise conditions. When instantiated with deep ReLU networks, explicit convergence rates are derived under hierarchical composition models, demonstrating the ability to overcome the curse of dimensionality. Conditions that enable positive transfer with faster rates are considered, including learning with simpler functions and data augmentation through pooling samples across groups. Various simulations and real-data experiments further validate the effectiveness of the proposed method.

stat.ML↗

arXiv2026

Bayesian Transfer Learning for Enhanced Estimation and Inference

Transfer learning enhances model performance in a target population with limited samples by leveraging knowledge from related studies. While many works focus on improving predictive performance, challenges of statistical inference persist. Bayesian approaches naturally offer uncertainty quantification for parameter estimates, yet existing Bayesian transfer learning methods are typically limited to single-source scenarios or require individual-level data. We introduce TRansfer leArning via guideD horseshoE prioR (TRADER), a novel approach enabling multi-source transfer through pre-trained models in high-dimensional linear regression. TRADER shrinks target parameters towards a weighted average of source estimates, accommodating sources with different scales. Theoretical investigation shows that TRADER achieves faster posterior contraction rates than standard continuous shrinkage priors when sources align well with the target while preventing negative transfer from heterogeneous sources. The analysis of finite-sample marginal posterior behavior reveals that TRADER achieves desired frequentist coverage probabilities, even for coefficients with moderate signal strength--a scenario where standard continuous shrinkage priors struggle. Extensive numerical studies and a real-data application estimating the association between blood glucose and insulin use in the Hispanic diabetic population demonstrate that TRADER improves estimation and inference accuracy over continuous shrinkage priors using target data alone, while outperforming a state-of-the-art transfer learning method that requires individual-level data.

stat.ME↗

arXiv2026

Spatio-temporal model via Locally Adaptive Regression Splines

This paper focuses on the estimation of a non-parametric regression function in the presence of data with spatio-temporal dependencies. In such a context, we study Locally Adaptive Regression Splines, a nonparametric estimator introduced by Mammen and Van De Geer (1997) and Rudin et al. (1992). To the best of our knowledge, this estimator has not previously been examined in a similar context. For univariate settings, the signals we consider are assumed to have a kth weak derivative with bounded total variation, allowing for a general degree of smoothness. In the multivariate setting, we study a variant of the K-Nearest Neighbor fused lasso estimator studied by Padilla et al. (2018a). For this case, the function is required to have bounded variation and satisfy a property that extends a piecewise Lipschitz continuity criterion, or the function is assumed to be piecewise Lipschitz. We develop an ADMM algorithm for practical computation. By aligning with lower bounds, the minimax optimality of our univariate and multivariate estimators is shown. A unique phase transition phenomenon, previously unprecedented in Locally Adaptive Regression Splines studies, emerges through our analysis. Both simulation studies and real data applications underscore the superior performance of our method when compared with established techniques in the existing literature.

stat.ME↗

arXiv2026

Online Change Point Detection for Multivariate Inhomogeneous Poisson Processes Time Series

We study online change point detection for multivariate inhomogeneous Poisson point process time series. This setting arises commonly in applications such as earthquake seismology, climate monitoring, and epidemic surveillance, yet remains underexplored in the machine learning and statistics literature. We propose a method that uses low-rank matrices to represent the multivariate Poisson intensity functions, resulting in an adaptive nonparametric detection procedure. Our algorithm is single-pass and requires only constant computational cost per new observation, independent of the elapsed length of the time series. We provide theoretical guarantees to control the overall false alarm probability and characterize the detection delay under temporal dependence. We also develop a new Matrix Bernstein inequality for temporally dependent Poisson point process time series, which may be of independent interest. Numerical experiments demonstrate that our method is both statistically robust and computationally efficient.

stat.ME↗

arXiv2026

Multivariate Poisson intensity estimation via low-rank tensor decomposition

In this work, we propose new matrix- and tensor-based methodologies for estimating multivariate intensity functions of inhomogeneous point processes. By viewing multivariate intensity functions as infinite-dimensional matrices or tensors within function spaces, our algorithms attain the optimal bias-variance trade-off, yielding rate-optimal estimation error, with model complexity governed by matrix or tensor ranks. They substantially improve estimation accuracy, while simultaneously reducing computational cost. To illustrate the adaptivity of the proposed framework, we show that many fundamental classes of multivariate functions, including additive and mean-field models, admit finite-rank tensor representations. We apply our method to a four-dimensional U.S. Geological Survey earthquake dataset, comprising features such as latitude, longitude, depth, and magnitude. Our tensor estimator recovers localized seismicity patterns (California, Oklahoma, Pacific Northwest, north-central U.S.), whereas the kernel baseline oversmooths them.

stat.ME↗

arXiv2026

Decoder-only Clustering in Attributed Graphs

This manuscript studies nodal clustering in graphs having multivariate attributes at each node. The framework includes node-specific priors for low-dimensional representations, coupled with a neural decoder that bridges observed attributes with latent variables. Structural and attribute information are incorporated through a graph-fused LASSO regularization on the prior means, promoting nodal clustering. The optimization problem is solved via alternating direction method of multipliers, with Langevin dynamics for posterior inference. Simulation studies on grid graphs, and applications to real data with complex settings, demonstrate the effectiveness of the proposed clustering method.

stat.ME↗

arXiv2026

Optimal Bias-variance Tradeoff in Matrix and Tensor Estimation

We study matrix and tensor denoising when the underlying signal is \textbf{not} necessarily low-rank. In the tensor setting, we observe \[ Y = X^\ast + Z \in \mathbb{R}^{p_1 \times p_2 \times p_3}, \] where $X^\ast$ is an unknown signal tensor and $Z$ is a noise tensor. We propose a one-step variant of the higher-order SVD (HOSVD) estimator, denoted $\widetilde X$, and show that, uniformly over any user-specified Tucker ranks $(r_1,r_2,r_3)$, with high probability, \[ \|\widetilde X - X^\ast\|_{\mathrm F}^2 = O\Big( κ^2\Big\{r_1r_2r_3 + \sum_{k=1}^3 p_k r_k\Big\} + ξ_{(r_1,r_2,r_3)}^2 \Big). \] Here, $ξ_{(r_1,r_2,r_3)}$ is the best achievable Tucker rank-$(r_1,r_2,r_3)$ approximation error of $X^\ast$ (bias), $κ^2$ quantifies the noise level, and $κ^2\{r_1r_2r_3+\sum_{k=1}^3 p_k r_k\}$ is the variance term scaling with the effective degrees of freedom of $\widetilde X$. This yields a rank-adaptive bias-variance tradeoff: increasing $(r_1,r_2,r_3)$ decreases the bias $ξ_{(r_1,r_2,r_3)}$ while increasing variance. In the matrix setting, we show that truncated SVD achieves an analogous bias-variance tradeoff for arbitrary signal matrices. Notably, our matrix result requires \textbf{no} assumptions on the signal matrix, such as finite rank or spectral gaps. Finally, we complement our upper bounds with matching information-theoretic lower bounds, showing that the resulting bias-variance tradeoff is minimax optimal up to universal constants in both the matrix and tensor settings.

stat.ML↗

arXiv2025

Conditional Mean and Variance Estimation via \textit{k}-NN Algorithm with Automated Variance Selection

We introduce a novel \textit{k}-nearest neighbor (\textit{k}-NN) regression method for joint estimation of the conditional mean and variance. The proposed algorithm preserves the computational efficiency and manifold-learning capabilities of classical non-parametric \textit{k}-NN models, while integrating a data-driven variable selection step that improves empirical performance. By accurately estimating both conditional mean and variance regression functions, the method effectively reconstructs the conditional distribution and density functions for multiple families of scale-and-localization generative models. We show that our estimator can achieve fast convergence rates, and we derive practical rules for selecting the smoothing parameter~$k$ that enhance the precision of the algorithm in finite sample regimes. Extensive simulations for low, moderate and large-dimensional covariate spaces, together with a real-world biomedical application, demonstrate that the proposed method can consistently outperform the conventional \textit{k-NN} regression algorithm while being more interpretable in the model output.

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arXiv2025

Change Point Detection on A Separable Model for Dynamic Networks

This paper studies the unsupervised change point detection problem in time series of networks using the Separable Temporal Exponential-family Random Graph Model (STERGM). Inherently, dynamic network patterns are complex due to dyadic and temporal dependence, and change points detection can identify the discrepancies in the underlying data generating processes to facilitate downstream analysis. In particular, the STERGM that utilizes network statistics and nodal attributes to represent the structural patterns is a flexible and parsimonious model to fit dynamic networks. We propose a new estimator derived from the Alternating Direction Method of Multipliers (ADMM) procedure and Group Fused Lasso (GFL) regularization to simultaneously detect multiple time points where the parameters of a time-heterogeneous STERGM have shifted. Experiments on both simulated and real data show good performance of the proposed framework, and an R package CPDstergm is developed to implement the method.

stat.ME↗

arXiv2025

A causal fused lasso for interpretable heterogeneous treatment effects estimation

We propose a novel method for estimating heterogeneous treatment effects based on the fused lasso. By first ordering samples based on the propensity or prognostic score, we match units from the treatment and control groups. We then run the fused lasso to obtain piecewise constant treatment effects with respect to the ordering defined by the score. Similar to the existing methods based on discretizing the score, our methods yield interpretable subgroup effects. However, existing methods fixed the subgroup a priori, but our causal fused lasso forms data-adaptive subgroups. We show that the estimator consistently estimates the treatment effects conditional on the score under very general conditions on the covariates and treatment. We demonstrate the performance of our procedure using extensive experiments that show that it can be interpretable and competitive with state-of-the-art methods.

stat.ME↗

arXiv2025

Risk Bounds For Distributional Regression

This work examines risk bounds for nonparametric distributional regression estimators. For convex-constrained distributional regression, general upper bounds are established for the continuous ranked probability score (CRPS) and the worst-case mean squared error (MSE) across the domain. These theoretical results are applied to isotonic and trend filtering distributional regression, yielding convergence rates consistent with those for mean estimation. Furthermore, a general upper bound is derived for distributional regression under non-convex constraints, with a specific application to neural network-based estimators. Comprehensive experiments on both simulated and real data validate the theoretical contributions, demonstrating their practical effectiveness.

stat.ML↗

arXiv2025

A Bayesian framework for change-point detection with uncertainty quantification

We introduce a novel Bayesian method that can detect multiple structural breaks in the mean and variance of a length $T$ time-series. Our method quantifies uncertainty by returning $α$-level credible sets around the estimated locations of the breaks. In the case of a single change in the mean and/or the variance of an independent sub-Gaussian sequence, we prove that our method attains a localization rate that is minimax optimal up to a $\log T$ factor. For an $α$-mixing sequence with dependence, we prove this optimality holds up to $\log^2 T$ factor. For $d$-dimensional mean changes, we show that if $d \gg \log T$ and the mean signal is dense, then our method exactly recovers the location of the change. Our method detects multiple change-points by modularly ``stacking'' single change-point models and searching for a variational approximation to the posterior distribution. This approach is applicable to both continuous and count data. Extensive simulation studies demonstrate that our method is competitive with the state-of-the-art in terms of speed and performance, and we produce credible sets that are an order of magnitude smaller than our competitors without sacrificing nominal coverage guarantees. We apply our method to real data by detecting i) the gating of an ion channel in the outer membrane of a bacterial cell, and ii) changes in the lithological structure of an oil well.

stat.ME↗

arXiv2025

Change Point Localization and Inference in Dynamic Multilayer Networks

We study offline change point localization and inference in dynamic multilayer random dot product graphs (D-MRDPGs), where at each time point, a multilayer network is observed with shared node latent positions and time-varying, layer-specific connectivity patterns. We propose a novel two-stage algorithm that combines seeded binary segmentation with low-rank tensor estimation, and establish its consistency in estimating both the number and locations of change points. Furthermore, we derive the limiting distributions of the refined estimators under both vanishing and non-vanishing jump regimes. To the best of our knowledge, this is the first result of its kind in the context of dynamic network data. We also develop a fully data-driven procedure for constructing confidence intervals. Extensive numerical experiments demonstrate the superior performance and practical utility of our methods compared to existing alternatives.

stat.ME↗

arXiv2025

Dense ReLU Neural Networks for Temporal-spatial Model

In this paper, we focus on fully connected deep neural networks utilizing the Rectified Linear Unit (ReLU) activation function for nonparametric estimation. We derive non-asymptotic bounds that lead to convergence rates, addressing both temporal and spatial dependence in the observed measurements. By accounting for dependencies across time and space, our models better reflect the complexities of real-world data, enhancing both predictive performance and theoretical robustness. We also tackle the curse of dimensionality by modeling the data on a manifold, exploring the intrinsic dimensionality of high-dimensional data. We broaden existing theoretical findings of temporal-spatial analysis by applying them to neural networks in more general contexts and demonstrate that our proof techniques are effective for models with short-range dependence. Our empirical simulations across various synthetic response functions underscore the superior performance of our method, outperforming established approaches in the existing literature. These findings provide valuable insights into the strong capabilities of dense neural networks (Dense NN) for temporal-spatial modeling across a broad range of function classes.

stat.ML↗

arXiv2025

Model-Free Kernel Conformal Depth Measures Algorithm for Uncertainty Quantification in Regression Models in Separable Hilbert Spaces

Depth measures are powerful tools for defining level sets in emerging, non--standard, and complex random objects such as high-dimensional multivariate data, functional data, and random graphs. Despite their favorable theoretical properties, the integration of depth measures into regression modeling to provide prediction regions remains a largely underexplored area of research. To address this gap, we propose a novel, model-free uncertainty quantification algorithm based on conditional depth measures--specifically, conditional kernel mean embeddings and an integrated depth measure. These new algorithms can be used to define prediction and tolerance regions when predictors and responses are defined in separable Hilbert spaces. The use of kernel mean embeddings ensures faster convergence rates in prediction region estimation. To enhance the practical utility of the algorithms with finite samples, we also introduce a conformal prediction variant that provides marginal, non-asymptotic guarantees for the derived prediction regions. Additionally, we establish both conditional and unconditional consistency results, as well as fast convergence rates in certain homoscedastic settings. We evaluate the finite--sample performance of our model in extensive simulation studies involving various types of functional data and traditional Euclidean scenarios. Finally, we demonstrate the practical relevance of our approach through a digital health application related to physical activity, aiming to provide personalized recommendations

stat.ML↗

arXiv2025

Quantile Additive Trend Filtering

This paper investigates risk bounds for quantile additive trend filtering, a method gaining increasing significance in the realms of additive trend filtering and quantile regression. We investigate the constrained version of quantile trend filtering within additive models, considering both fixed and growing input dimensions. In the fixed dimension case, we discover an error rate that mirrors the non-quantile minimax rate for additive trend filtering, featuring the main term $n^{-2r/(2r+1)}V^{2/(2r+1)}$, when the underlying quantile function is additive, with components whose $(r-1)$th derivatives are of bounded variation by $V$. In scenarios with a growing input dimension $d$, quantile additive trend filtering introduces a polynomial factor of $d^{(2r+2)/(2r+1)}$. This aligns with the non-quantile variant, featuring a linear factor $d$, particularly pronounced for larger $r$ values. Additionally, we propose a practical algorithm for implementing quantile trend filtering within additive models, using dimension-wise backfitting. We conduct experiments with evenly spaced data points or data that samples from a uniform distribution in the interval $[0,1]$, applying distinct component functions and introducing noise from normal and heavy-tailed distributions. Our findings confirm the estimator's convergence as $n$ increases and its superiority, particularly in heavy-tailed distribution scenarios. These results deepen our understanding of additive trend filtering models in quantile settings, offering valuable insights for practical applications and future research.

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