A New Perspective on Minimum-Norm Interpolation Under Gaussian Covariates

Gil Kur, Zong Shang, Paul Simanjuntak, Guillaume Lecué, Reese Pathak
Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, PMLR 300:4483-4491, 2026.

Abstract

Minimum-Norm Interpolators (MNI) in overparameterized linear models have gained attention as a tractable framework for studying interpolation phenomena that resemble empirical observations in neural networks. Most prior work on these interpolators either exploits closed-form solutions when available or relies heavily on Gaussian comparison results, such as the convex Gaussian Min-Max Theorem (CGMT). In this paper, we introduce a new perspective on MNI under isotropic Gaussian covariates by leveraging tools from high-dimensional geometry. First, we obtain a “localized” bound on the MNI’s shrinkage of the original ground truth that occurs under isotropic Gaussian covariates when the norm is in an isotropic position. Then, we prove a sharp bound on the Mean Squared Error (MSE) of the $\ell_1$-MNI, as obtained by Wang 22’ via a geometric proof, which avoids invoking the CGMT and instead relies on the work of Fleury 12’ on Gaussian polytopes.

Cite this Paper

BibTeX
@InProceedings{pmlr-v300-kur26a, title = { A New Perspective on Minimum-Norm Interpolation Under Gaussian Covariates }, author = {Kur, Gil and Shang, Zong and Simanjuntak, Paul and Lecu{\'e}, Guillaume and Pathak, Reese}, booktitle = {Proceedings of The 29th International Conference on Artificial Intelligence and Statistics}, pages = {4483--4491}, year = {2026}, editor = {Khan, Emtiyaz and Li, Yingzhen and Solin, Arno and Ramdas, Aaditya}, volume = {300}, series = {Proceedings of Machine Learning Research}, month = {02--05 May}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v300/main/assets/kur26a/kur26a.pdf}, url = {https://proceedings.mlr.press/v300/kur26a.html}, abstract = { Minimum-Norm Interpolators (MNI) in overparameterized linear models have gained attention as a tractable framework for studying interpolation phenomena that resemble empirical observations in neural networks. Most prior work on these interpolators either exploits closed-form solutions when available or relies heavily on Gaussian comparison results, such as the convex Gaussian Min-Max Theorem (CGMT). In this paper, we introduce a new perspective on MNI under isotropic Gaussian covariates by leveraging tools from high-dimensional geometry. First, we obtain a “localized” bound on the MNI’s shrinkage of the original ground truth that occurs under isotropic Gaussian covariates when the norm is in an isotropic position. Then, we prove a sharp bound on the Mean Squared Error (MSE) of the $\ell_1$-MNI, as obtained by Wang 22’ via a geometric proof, which avoids invoking the CGMT and instead relies on the work of Fleury 12’ on Gaussian polytopes. } }
Endnote
%0 Conference Paper %T A New Perspective on Minimum-Norm Interpolation Under Gaussian Covariates %A Gil Kur %A Zong Shang %A Paul Simanjuntak %A Guillaume Lecué %A Reese Pathak %B Proceedings of The 29th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2026 %E Emtiyaz Khan %E Yingzhen Li %E Arno Solin %E Aaditya Ramdas %F pmlr-v300-kur26a %I PMLR %P 4483--4491 %U https://proceedings.mlr.press/v300/kur26a.html %V 300 %X Minimum-Norm Interpolators (MNI) in overparameterized linear models have gained attention as a tractable framework for studying interpolation phenomena that resemble empirical observations in neural networks. Most prior work on these interpolators either exploits closed-form solutions when available or relies heavily on Gaussian comparison results, such as the convex Gaussian Min-Max Theorem (CGMT). In this paper, we introduce a new perspective on MNI under isotropic Gaussian covariates by leveraging tools from high-dimensional geometry. First, we obtain a “localized” bound on the MNI’s shrinkage of the original ground truth that occurs under isotropic Gaussian covariates when the norm is in an isotropic position. Then, we prove a sharp bound on the Mean Squared Error (MSE) of the $\ell_1$-MNI, as obtained by Wang 22’ via a geometric proof, which avoids invoking the CGMT and instead relies on the work of Fleury 12’ on Gaussian polytopes.
APA
Kur, G., Shang, Z., Simanjuntak, P., Lecué, G. & Pathak, R.. (2026). A New Perspective on Minimum-Norm Interpolation Under Gaussian Covariates . Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 300:4483-4491 Available from https://proceedings.mlr.press/v300/kur26a.html.

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