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Regularizing random points by deleting a few

Dmitriy Bilyk, Stefan Steinerberger

TL;DR

The paper addresses the problem of improving the uniformity of random points on $[0,1]$ by deleting a small subset. It introduces a constructive, online binning–thinning strategy that reduces the Kolmogorov–Smirnov discrepancy from the typical $\sqrt{\log n}/\sqrt{n}$ scale to $\log n/m$ when deleting at most $m$ points (up to $0.001n$). In the regime $m=cn$, the bound becomes essentially $\log n / n$, matching the optimal order up to constants, and extends to any absolutely continuous density via a change of variables. The method relies on concentration within bins, uniform thinning to equalize counts, and a Dvoretzky–Kiefer–Wolfowitz bound combined with a union bound; the results are argued to be near-optimal and are amenable to online implementation.

Abstract

It is well understood that if one is given a set $X \subset [0,1]$ of $n$ independent uniformly distributed random variables, then $$ \sup_{0 \leq x \leq 1} \left| \frac{\# X \cap [0,x]}{\# X} - x \right| \lesssim \frac{\sqrt{\log{n}}}{ \sqrt{n}} \qquad \mbox{with very high probability.} $$ We show that one can improve the error term by removing a few of the points. For any $m \leq 0.001n$ there exists a subset $Y \subset X$ obtained by deleting at most $m$ points, so that the error term drops from $\sim \sqrt{\log{n}}/\sqrt{n}$ to $ \log{(n)}/m$ with high probability. When $m=cn$ for a small $0 \leq c \leq 0.001$, this achieves the essentially optimal asymptotic order of discrepancy $\log(n)/n$. The proof is constructive and works in an online setting (where one is given the points sequentially, one at a time, and has to decide whether to keep or discard it). A change of variables shows the same result for any random variables on the real line with absolutely continuous density.

Regularizing random points by deleting a few

TL;DR

The paper addresses the problem of improving the uniformity of random points on by deleting a small subset. It introduces a constructive, online binning–thinning strategy that reduces the Kolmogorov–Smirnov discrepancy from the typical scale to when deleting at most points (up to ). In the regime , the bound becomes essentially , matching the optimal order up to constants, and extends to any absolutely continuous density via a change of variables. The method relies on concentration within bins, uniform thinning to equalize counts, and a Dvoretzky–Kiefer–Wolfowitz bound combined with a union bound; the results are argued to be near-optimal and are amenable to online implementation.

Abstract

It is well understood that if one is given a set of independent uniformly distributed random variables, then We show that one can improve the error term by removing a few of the points. For any there exists a subset obtained by deleting at most points, so that the error term drops from to with high probability. When for a small , this achieves the essentially optimal asymptotic order of discrepancy . The proof is constructive and works in an online setting (where one is given the points sequentially, one at a time, and has to decide whether to keep or discard it). A change of variables shows the same result for any random variables on the real line with absolutely continuous density.
Paper Structure (9 sections, 2 theorems, 40 equations, 2 figures)

This paper contains 9 sections, 2 theorems, 40 equations, 2 figures.

Table of Contents

  1. Introduction and Result
  2. Problem.
  3. Result.
  4. Related results.
  5. Proof
  6. Binning.
  7. Thinning.
  8. Bounding.
  9. Conclusion.

Key Result

Theorem 1

Let $X = \left\{x_1, \dots, x_n \right\}$ be independent uniformly distributed random variables on $[0,1]$. For any $0 \leq m \leq 0.001n$, there exists $Y \subset X$ with such that, with high likelihood, If $m=cn$ for some $c\in (0,0.001]$, this bound is

Figures (2)

  • Figure 1: Left: $n=10^5$ points and $\sqrt{\# X} \cdot (x_i - i/\# X)$. Right: $\sqrt{\# Y} \cdot (y_i - i/\# Y)$ where $Y \subset X$ and $\# Y \geq 0.95 \cdot \# X$. By dropping $5\%$ of the points, one can dramatically increase regularity.
  • Figure 2: The idea behind the proof visualized: by binning and thinning, the distribution function can be rewritten as the concatenation of independent distribution function that achieve discrepancy 0 at equispaced intervals. The continuum limit are concatenated Brownian bridges.

Theorems & Definitions (4)

  • Theorem
  • Corollary
  • proof
  • proof