# The Geometry of Over-parameterized Regression and Adversarial Perturbations

arXiv: https://arxiv.org/abs/2103.14108 · HTML: https://mlsanity.com/paper/2103.14108/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 36 references that are in the map.
- First posted on arXiv 25 Mar 2021.

## Builds on (most cited first, 30 of 30)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [Reconciling modern machine learning practice and the bias-variance trade-off](https://mlsanity.com/paper/1812.11118/) (2018)
- [Deep Double Descent: Where Bigger Models and More Data Hurt](https://mlsanity.com/paper/1912.02292/) (2019)
- [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)
- [High-dimensional dynamics of generalization error in neural networks](https://mlsanity.com/paper/1710.03667/) (2017)
- [Two models of double descent for weak features](https://mlsanity.com/paper/1903.07571/) (2019)
- [Just Interpolate: Kernel "Ridgeless" Regression Can Generalize](https://mlsanity.com/paper/1808.00387/) (2018)
- [Generalisation error in learning with random features and the hidden manifold model](https://mlsanity.com/paper/2002.09339/) (2020)
- [Harmless interpolation of noisy data in regression](https://mlsanity.com/paper/1903.09139/) (2019)
- [Optimal Errors and Phase Transitions in High-Dimensional Generalized Linear Models](https://mlsanity.com/paper/1708.03395/) (2017)
- [Rethinking Bias-Variance Trade-off for Generalization of Neural Networks](https://mlsanity.com/paper/2002.11328/) (2020)
- [Double Trouble in Double Descent : Bias and Variance(s) in the Lazy Regime](https://mlsanity.com/paper/2003.01054/) (2020)
- [An analytic theory of generalization dynamics and transfer learning in deep linear networks](https://mlsanity.com/paper/1809.10374/) (2018)
- [A Model of Double Descent for High-dimensional Binary Linear Classification](https://mlsanity.com/paper/1911.05822/) (2019)
- [Understanding Double Descent Requires a Fine-Grained Bias-Variance Decomposition](https://mlsanity.com/paper/2011.03321/) (2020)
- [A Random Matrix Analysis of Random Fourier Features: Beyond the Gaussian Kernel, a Precise Phase Transition, and the Corresponding Double Descent](https://mlsanity.com/paper/2006.05013/) (2020)
- [Triple descent and the two kinds of overfitting: Where & why do they appear?](https://mlsanity.com/paper/2006.03509/) (2020)
- [Implicit Regularization of Random Feature Models](https://mlsanity.com/paper/2002.08404/) (2020)
- [A Brief Prehistory of Double Descent](https://mlsanity.com/paper/2004.04328/) (2020)
- [More Data Can Hurt for Linear Regression: Sample-wise Double Descent](https://mlsanity.com/paper/1912.07242/) (2019)
- [Exact expressions for double descent and implicit regularization via surrogate random design](https://mlsanity.com/paper/1912.04533/) (2019)
- [Analytic Study of Double Descent in Binary Classification: The Impact of Loss](https://mlsanity.com/paper/2001.11572/) (2020)
- [What causes the test error? Going beyond bias-variance via ANOVA](https://mlsanity.com/paper/2010.05170/) (2020)
- [Memorizing without overfitting: Bias, variance, and interpolation in over-parameterized models](https://mlsanity.com/paper/2010.13933/) (2020)
- [Understanding overfitting peaks in generalization error: Analytical risk curves for l2 and l1 penalized interpolation](https://mlsanity.com/paper/1906.03667/) (2019)
- [Benign Overfitting and Noisy Features](https://mlsanity.com/paper/2008.02901/) (2020)
- [A New Look at an Old Problem: A Universal Learning Approach to Linear Regression](https://mlsanity.com/paper/1905.04708/) (2019)

## Built on by (most cited first, 4 of 4)

- [Double Descent Demystified: Identifying, Interpreting & Ablating the Sources of a Deep Learning Puzzle](https://mlsanity.com/paper/2303.14151/) (2023)
- [Towards an Improved Understanding and Utilization of Maximum Manifold Capacity Representations](https://mlsanity.com/paper/2406.09366/) (2024)
- [Correlating and Predicting Human Evaluations of Language Models from Natural Language Processing Benchmarks](https://mlsanity.com/paper/2502.18339/) (2025)
- [No Free Lunch From Random Feature Ensembles: Scaling Laws and Near-Optimality Conditions](https://mlsanity.com/paper/2412.05418/) (2024)

## Abstract (arXiv)

Classical regression has a simple geometric description in terms of a projection of the training labels onto the column space of the design matrix. However, for over-parameterized models -- where the number of fit parameters is large enough to perfectly fit the training data -- this picture becomes uninformative. Here, we present an alternative geometric interpretation of regression that applies to both under- and over-parameterized models. Unlike the classical picture which takes place in the space of training labels, our new picture resides in the space of input features. This new feature-based perspective provides a natural geometric interpretation of the double-descent phenomenon in the context of bias and variance, explaining why it can occur even in the absence of label noise. Furthermore, we show that adversarial perturbations -- small perturbations to the input features that result in large changes in label values -- are a generic feature of biased models, arising from the underlying geometry. We demonstrate these ideas by analyzing three minimal models for over-parameterized linear least squares regression: without basis functions (input features equal model features) and with linear or nonlinear basis functions (two-layer neural networks with linear or nonlinear activation functions, respectively).