# Support vector machines and linear regression coincide with very high-dimensional features

arXiv: https://arxiv.org/abs/2105.14084 · HTML: https://mlsanity.com/paper/2105.14084/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 48 references that are in the map.
- First posted on arXiv 28 May 2021.

## Builds on (most cited first, 17 of 17)

- [The Implicit Bias of Gradient Descent on Separable Data](https://mlsanity.com/paper/1710.10345/) (2017)
- [Benign Overfitting in Linear Regression](https://mlsanity.com/paper/1906.11300/) (2019)
- [Surprises in High-Dimensional Ridgeless Least Squares Interpolation](https://mlsanity.com/paper/1903.08560/) (2019)
- [The generalization error of random features regression: Precise asymptotics and double descent curve](https://mlsanity.com/paper/1908.05355/) (2019)
- [Two models of double descent for weak features](https://mlsanity.com/paper/1903.07571/) (2019)
- [Harmless interpolation of noisy data in regression](https://mlsanity.com/paper/1903.09139/) (2019)
- [Classification vs regression in overparameterized regimes: Does the loss function matter?](https://mlsanity.com/paper/2005.08054/) (2020)
- [Finite-sample Analysis of Interpolating Linear Classifiers in the Overparameterized Regime](https://mlsanity.com/paper/2004.12019/) (2020)
- [Risk Bounds for Over-parameterized Maximum Margin Classification on Sub-Gaussian Mixtures](https://mlsanity.com/paper/2104.13628/) (2021)
- [The phase transition for the existence of the maximum likelihood estimate in high-dimensional logistic regression](https://mlsanity.com/paper/1804.09753/) (2018)
- [A Precise High-Dimensional Asymptotic Theory for Boosting and Minimum-ℓ1-Norm Interpolated Classifiers](https://mlsanity.com/paper/2002.01586/) (2020)
- [On the proliferation of support vectors in high dimensions](https://mlsanity.com/paper/2009.10670/) (2020)
- [Binary Classification of Gaussian Mixtures: Abundance of Support Vectors, Benign Overfitting and Regularization](https://mlsanity.com/paper/2011.09148/) (2020)
- [Understanding overfitting peaks in generalization error: Analytical risk curves for l2 and l1 penalized interpolation](https://mlsanity.com/paper/1906.03667/) (2019)
- [Dimensionality reduction, regularization, and generalization in overparameterized regressions](https://mlsanity.com/paper/2011.11477/) (2020)
- [Risk of the Least Squares Minimum Norm Estimator under the Spike Covariance Model](https://mlsanity.com/paper/1912.13421/) (2019)
- [Exact high-dimensional asymptotics for Support Vector Machine](https://mlsanity.com/paper/1905.05125/) (2019)

## Built on by (most cited first, 8 of 8)

- [A Farewell to the Bias-Variance Tradeoff? An Overview of the Theory of Overparameterized Machine Learning](https://mlsanity.com/paper/2109.02355/) (2021)
- [Benign Overfitting in Multiclass Classification: All Roads Lead to Interpolation](https://mlsanity.com/paper/2106.10865/) (2021)
- [Error Scaling Laws for Kernel Classification under Source and Capacity Conditions](https://mlsanity.com/paper/2201.12655/) (2022)
- [Minimax Supervised Clustering in the Anisotropic Gaussian Mixture Model: A new take on Robust Interpolation](https://mlsanity.com/paper/2111.07041/) (2021)
- [On how to avoid exacerbating spurious correlations when models are overparameterized](https://mlsanity.com/paper/2206.12739/) (2022)
- [General Loss Functions Lead to (Approximate) Interpolation in High Dimensions](https://mlsanity.com/paper/2303.07475/) (2023)
- [New Equivalences Between Interpolation and SVMs: Kernels and Structured Features](https://mlsanity.com/paper/2305.02304/) (2023)
- [Tight bounds for maximum ℓ1-margin classifiers](https://mlsanity.com/paper/2212.03783/) (2022)

## Abstract (arXiv)

The support vector machine (SVM) and minimum Euclidean norm least squares regression are two fundamentally different approaches to fitting linear models, but they have recently been connected in models for very high-dimensional data through a phenomenon of support vector proliferation, where every training example used to fit an SVM becomes a support vector. In this paper, we explore the generality of this phenomenon and make the following contributions. First, we prove a super-linear lower bound on the dimension (in terms of sample size) required for support vector proliferation in independent feature models, matching the upper bounds from previous works. We further identify a sharp phase transition in Gaussian feature models, bound the width of this transition, and give experimental support for its universality. Finally, we hypothesize that this phase transition occurs only in much higher-dimensional settings in the ℓ1 variant of the SVM, and we present a new geometric characterization of the problem that may elucidate this phenomenon for the general ℓp case.