{"ID":6138391,"CreatedAt":"2026-07-09T01:07:32.349475501Z","UpdatedAt":"2026-07-11T17:47:58.155493336Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.07694","arxiv_id":"2607.07694","title":"Minimum Norm Interpolation via The Local Theory of Banach Spaces: The Role of Gaussianity","abstract":"We study minimum-norm interpolation (MNI) in overparameterized linear regression with isotropic Gaussian covariates, in settings where the MNI has no closed-form formula. Whereas most prior work relied on Gaussian comparison tools such as the convex Gaussian min--max theorem (CGMT), our approach uses tools from high-dimensional geometry and probability. First, when the norm is in isotropic position, we obtain an ``offset'' bound that controls the amount by which the MNI shrinks the ground truth. Second, we show that the ``intrinsic'' variance of the $\\ell_1$-MNI is at most $O(\\tfrac{1}{n\\log(d/n)^2})$, using a variant of Talagrand's $L_1$--$L_2$ inequality due to Cordero-Erausquin and Ledoux [2012], together with a classical result of Gluskin [1988]. We recover the sharp mean-squared error (MSE) bound for the $\\ell_1$-MNI obtained by Wang et al. [2022], using the work of Fleury [2012] on the symmetric Gaussian polytope, which is defined via \\[ P_{n,d} := \\mathrm{conv}\\{\\pm X_i\\}_{i=1}^{d} \\text{ where } X_i \\overset{\\mathrm{i.i.d.}}{\\sim} N(0,\\mathrm{I}_{n \\times n}), \\] rather than CGMT. Our methods also imply improvements on previous results in high-dimensional geometry that may be of independent interest. First, we show that with overwhelming probability, the ratio between the isotropic constant of $P_{n,d}$ and that of the Euclidean ball in $\\mathbb{R}^n$ is at most $1+O((\\log(d/n))^{-2})$, improving a result of Klartag and Kozma [2009]. We also establish a refined weighted thin-shell estimate on $P_{n,d}$, and provide an elementary proof of the main theorem of Fleury [2012].","short_abstract":"We study minimum-norm interpolation (MNI) in overparameterized linear regression with isotropic Gaussian covariates, in settings where the MNI has no closed-form formula. Whereas most prior work relied on Gaussian comparison tools such as the convex Gaussian min--max theorem (CGMT), our approach uses tools from high-di...","url_abs":"https://arxiv.org/abs/2607.07694","url_pdf":"https://arxiv.org/pdf/2607.07694v1","authors":"[\"Gil Kur\",\"Reese Pathak\"]","published":"2026-07-08T17:50:15Z","proceeding":"math.ST","tasks":"[\"math.ST\",\"math.MG\",\"math.PR\"]","methods":"[]","has_code":false}