Smoothed Analysis of the Komlós Conjecture: Rademacher Noise

TL;DR

This work resolves the smoothed Komlós discrepancy problem for discrete Rademacher noise: for Komlós matrices $M$ and random perturbations $R$ with i.i.d. Rademacher entries, the discrepancy of $M+R/\sqrt{d}$ is $O(d^{-1/2})$ a.a.s. provided $d=\omega(1)$ and $n=\omega(d\log d)$. Extending the Gaussian analysis of Bansal et al., the authors develop a discretised counting framework based on a truncated Gram–Schmidt sampling distribution to construct a witness $\bold{x}\in\{-1,1\}^n$ with $\|(M+R/\sqrt{d})\bold{x}\|_\infty = O(d^{-1/2})$, while a Paley–Zygmund argument guarantees existence with high probability. Achieving the optimal $n=\omega(d\log d)$ regime for Rademacher noise requires a delicate combinatorial counting argument to control joint probabilities, illustrating a robust discrete analogue of the Gaussian smoothing method. The result closes the gap on discrete smoothed Komlós discrepancy in this noise model and motivates further study of other discrete perturbations (e.g., Bernoulli) in the same framework. The findings have significance for discrepancy theory under smoothed inputs and for understanding how discrete perturbations influence dimension-free bounds.

Abstract

The {\em discrepancy} of a matrix $M \in \mathbb{R}^{d \times n}$ is given by $\mathrm{DISC}(M) := \min_{\boldsymbol{x} \in \{-1,1\}^n} \|M\boldsymbol{x}\|_\infty$. An outstanding conjecture, attributed to Komlós, stipulates that $\mathrm{DISC}(M) = O(1)$, whenever $M$ is a Komlós matrix, that is, whenever every column of $M$ lies within the unit sphere. Our main result asserts that $\mathrm{DISC}(M + R/\sqrt{d}) = O(d^{-1/2})$ holds asymptotically almost surely, whenever $M \in \mathbb{R}^{d \times n}$ is Komlós, $R \in \mathbb{R}^{d \times n}$ is a Rademacher random matrix, $d = ω(1)$, and $n = ω(d \log d)$. The factor $d^{-1/2}$ normalising $R$ is essentially best possible and the dependency between $n$ and $d$ is asymptotically best possible. Our main source of inspiration is a result by Bansal, Jiang, Meka, Singla, and Sinha (ICALP 2022). They obtained an assertion similar to the one above in the case that the smoothing matrix is Gaussian. They asked whether their result can be attained with the optimal dependency $n = ω(d \log d)$ in the case of Bernoulli random noise or any other types of discretely distributed noise; the latter types being more conducive for Smoothed Analysis in other discrepancy settings such as the Beck-Fiala problem. For Bernoulli noise, their method works if $n = ω(d^2)$. In the case of Rademacher noise, we answer the question posed by Bansal, Jiang, Meka, Singla, and Sinha. Our proof builds upon their approach in a strong way and provides a discrete version of the latter. Breaking the $n = ω(d^2)$ barrier and reaching the optimal dependency $n = ω(d \log d)$ for Rademacher noise requires additional ideas expressed through a rather meticulous counting argument, incurred by the need to maintain a high level of precision all throughout the discretisation process.

Theorems & Definitions (27)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Remark 2.1
• Theorem 2.2
• Lemma 2.3
• Remark 2.4
• Lemma 2.5
• Claim 2.6