# De-floored Principal Component Regression: When Rank Selection Alone Is Insufficient for Prediction

arXiv: https://arxiv.org/abs/2607.16638 · HTML: https://mlsanity.com/paper/2607.16638/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 35 references that are in the map.
- First posted on arXiv 18 Jul 2026.

## Builds on (most cited first, 14 of 14)

- [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)
- [Benign overfitting in ridge regression](https://mlsanity.com/paper/2009.14286/) (2020)
- [Harmless interpolation of noisy data in regression](https://mlsanity.com/paper/1903.09139/) (2019)
- [On the Optimal Weighted ℓ2 Regularization in Overparameterized Linear Regression](https://mlsanity.com/paper/2006.05800/) (2020)
- [Benign Overfitting of Constant-Stepsize SGD for Linear Regression](https://mlsanity.com/paper/2103.12692/) (2021)
- [The Implicit Bias of Benign Overfitting](https://mlsanity.com/paper/2201.11489/) (2022)
- [On Robustness of Principal Component Regression](https://mlsanity.com/paper/1902.10920/) (2019)
- [Foolish Crowds Support Benign Overfitting](https://mlsanity.com/paper/2110.02914/) (2021)
- [Dimensionality reduction, regularization, and generalization in overparameterized regressions](https://mlsanity.com/paper/2011.11477/) (2020)
- [Principal Component Projection Without Principal Component Analysis](https://mlsanity.com/paper/1602.06872/) (2016)
- [Adaptive Principal Component Regression with Applications to Panel Data](https://mlsanity.com/paper/2307.01357/) (2023)
- [Faster Principal Component Regression and Stable Matrix Chebyshev Approximation](https://mlsanity.com/paper/1608.04773/) (2016)
- [Sketching for Principal Component Regression](https://mlsanity.com/paper/1803.02661/) (2018)

## Built on by (most cited first, 1 of 1)

- [Principal Component Regression Dominates all Monotone Spectral Filters for Linear Regression](https://mlsanity.com/paper/2609.39440/) (2026)

## Abstract (arXiv)

Principal component regression (PCR) regularizes high-dimensional prediction by choosing a spectral cutoff, but rank selection cannot correct systematic inflation of the retained empirical eigenvalues. We study clean Gaussian random designs in which the aggregate covariance tail creates a nearly scalar sample-space floor comparable to the predictive head scale. De-floored principal component regression (dPCR) retains the cutoff and subtracts an estimated floor from the retained denominators. We prove an ordinary-PCR prediction-risk lower bound uniform over all ranks and a high-probability dPCR upper bound. When the floor is sharp and inexpensive to remove in population prediction risk, the conditional risk of dPCR is asymptotically negligible relative to that of the best ordinary PCR rank. An exact risk decomposition explains the separation: denominator inflation is governed by first spectral mass, whereas the clean prediction cost of correction is governed by squared spectral mass. A same-sample trimmed-mean floor estimate attains the oracle dPCR upper-bound rate at a prespecified rank, and the separation persists under approximate predictive alignment when the tail prediction-energy fraction vanishes. Separate pointwise fixed-aspect formulas show that the risk-optimal positive scalar correction improves rank-1 PCR, whereas mean-floor subtraction is generally not optimal for a broad Marchenko--Pastur bulk.