arXiv subjects

Jaehee Seo

Publications and source records attributed to Jaehee Seo.

3 recordsLinked to original sources

arXiv2026

Minimax Rate for Diffusion-Guided Local Intrinsic Dimension Estimation based on Fokker-Planck Equation

Diffusion-guided methods have recently emerged as effective tools for probing the intrinsic geometry of high-dimensional data. Understanding their accuracy requires quantifying both (i) finite-scale approximation error relative to the true local intrinsic dimension (LID) and (ii) estimation error from finite samples. We address these two questions for the FLIPD field (Kamkari et al., 2024; arXiv:2406.03537), defined at the population level through the logarithmic scale derivative of a Gaussian-smoothed density and admitting a Fokker--Planck representation. For $α$-smooth densities on a regular $d$-dimensional manifold, we establish a uniform $O(σ^2)$ approximation bound relative to the manifold dimension $d$. Under expected integrated squared loss, we prove a minimax lower bound of order $(nσ^d)^{-1}$ for estimating the FLIPD field from $n$ observations when $n^{-1/(2α+d)}\leσ\leσ_0$. A plug-in FLIPD estimator based on the empirical denoising score matching (DSM) solution attains a matching upper bound. Thus the minimax rate is of order $(nσ^d)^{-1}$ and it becomes $n^{-2α/(2α+d)}$ at its smallest scale.

stat.ML↗

arXiv2026

Nonparametric In-Context Learning under Growing Geometric Complexity: Minimax Optimality and Local Geometry-Adaptivity of Transformers

Transformers have become a central architecture for in-context learning (ICL), particularly through their state-of-the-art performance in large language models. This success motivates understanding how transformers exploit intrinsic geometry of real-world data. However, existing nonparametric ICL theory has largely focused on simplified settings, such as Euclidean domains or single manifold models, failing to fully explain its phenomenal empirical success in complex tasks. To address this gap, we study the prediction problem under unknown heterogeneous geometry, modeled by sample size-dependent mixtures of manifolds with heterogeneous dimensions, smoothness, and sampling masses. Under the growing geometric complexity model, we establish a minimax lower bound capturing the aggregate difficulty of the components and construct an oracle tangent local polynomial estimator with a matching upper bound. This estimator is connected to a structure-informed, two-stage softmax transformer with a geometric preconditioner and local polynomial solvers. The transformer achieves negligible approximation error relative to the minimax rate with logarithmic depth and polynomial size. Finally, we derive an in-context generalization bound for near empirical risk minimizers (ERM) over this class. Together, these results identify conditions under which the resulting predictor exploits local geometry and attains minimax rate.

stat.ML↗

arXiv2026

Semi-Supervised Kernel Ridge Fréchet Regression

Fréchet regression provides a framework for predicting random object-valued responses by minimizing the conditional expected squared distances. We propose semi-supervised kernel ridge Fréchet regression (SS-KRFR) to exploit unlabeled covariates when labeled samples are scarce. The method learns kernel spectral features from both labeled and unlabeled covariates, then utilizes labeled responses to estimate the conditional Fréchet function by kernel ridge regression. We derive prediction error bounds that distinguish the roles of labeled and unlabeled samples, and establish polynomial convergence rates under suitable conditions. Under additional kernel assumptions, a sharper analysis relaxes the sample size requirements for learning the features. We also develop hybrids that combine features, predictions, or both from models fitted with and without unlabeled covariates, establishing uniform stability over mixing weights in Hadamard spaces at common tuning and a separate rate guarantee for prediction mixing. Simulations and a real-data application show semi-supervised gains and improvements over competing Fréchet regression methods in several settings.

stat.ME↗