Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2309.15115

Planted Random Number Partitioning Problem

Eren C. Kızıldağ
View original source ↗Read source document ↗Export metadata ↓

Abstract

We consider the random number partitioning problem (\texttt{NPP}): given a list $X\sim \mathcal{N}(0,I_n)$ of numbers, find a partition $σ\in\{-1,1\}^n$ with a small objective value $H(σ)=\frac{1}{\sqrt{n}}\left|\langle σ,X\rangle\right|$. The \texttt{NPP} is widely studied in computer science; it is also closely related to the design of randomized controlled trials. In this paper, we propose a planted version of the \texttt{NPP}: fix a $σ^*$ and generate $X\sim \mathcal{N}(0,I_n)$ conditional on $H(σ^*)\le 3^{-n}$. The \texttt{NPP} and its planted counterpart are statistically distinguishable as the smallest objective value under the former is $Θ(\sqrt{n}2^{-n})$ w.h.p. Our first focus is on the values of $H(σ)$. We show that, perhaps surprisingly, planting does not induce partitions with an objective value substantially smaller than $2^{-n}$: $\min_{σ\ne \pm σ^*}H(σ) = \widetildeΘ(2^{-n})$ w.h.p. Furthermore, we completely characterize the smallest $H(σ)$ achieved at any fixed distance from $σ^*$. Our second focus is on the algorithmic problem of efficiently finding a partition $σ$, not necessarily equal to $\pmσ^*$, with a small $H(σ)$. We show that planted \texttt{NPP} exhibits an intricate geometrical property known as the multi Overlap Gap Property ($m$-OGP) for values $2^{-Θ(n)}$. We then leverage the $m$-OGP to show that stable algorithms satisfying a certain anti-concentration property fail to find a $σ$ with $H(σ)=2^{-Θ(n)}$. Our results are the first instance of the $m$-OGP being established and leveraged to rule out stable algorithms for a planted model. More importantly, they show that the $m$-OGP framework can also apply to planted models, if the algorithmic goal is to return a solution with a small objective value.

Explore related subjects

math.STcs.CCcs.DSmath.PRstat.TH

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Eren C. Kızıldağ. 2023-09-26. Planted Random Number Partitioning Problem. https://arxiv.org/abs/2309.15115

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Statistical models as natural transformations: meaningfulness, coherence and priors as states in Markov categories

We show that a statistical model in the sense of McCullagh, in the form given by Brøns, is a natural transformation between two functors from the category of designs to the Kleisli category Stoch of the Giry monad, provided that its components are measurable in the parameter. The condition is empty for finite models. A design-indexed quantity is a family of morphisms of Stoch defined on the parameter objects, called meaningful if it is natural. We prove that Tjur's criterion, imposed on parameter functions indexed by finite samples with multiplicities, forces the indexing by the support and then coincides with naturality over the insertions. For finite designs we show that a quantity can be corrected to a natural one within a given class of corrections if and only if a class vanishes in the first cohomology group of a Baues-Wirsching complex relative to that class, while its image in the absolute group is always zero. In the one-way layout, marginal dispersion is not meaningful, and within-group dispersion is the unique correction that leaves the merged design unchanged. A prior is a family of states on the parameter objects, called coherent over a class of design morphisms if it is natural over that class. We show that coherence at a merge confines the prior to the image of the corresponding parameter map, that coherence over the insertions is Kolmogorov consistency, and that coherence over the injections adds the exchangeability assumed by the categorical de Finetti theorem. In the finite one-way scheme, the coherent priors form polytopes of known dimension. The analogue of Jeffreys' general rule is not coherent, while the analogue for location-scale families is. Finally, we show that ridge regression is the Bayesian inversion of the Gaussian linear model with respect to a Gaussian prior, which is coherent over the insertions and never over the injections.

math.ST↗
arXiv2026

Sample complexity and weak limits of nonsmooth multimarginal Schrödinger system with application to optimal transport barycenter

Multimarginal optimal transport (MOT) has emerged as a useful framework for many applied problems. However, compared to the well-studied classical two-marginal optimal transport theory, analysis of MOT is far more challenging and remains much less developed. In this paper, we study the statistical estimation and inference problems for the entropic MOT (EMOT), whose optimal solution is characterized by the multimarginal Schrödinger system. Assuming only boundedness of the cost function, we derive sharp sample complexity for estimating several key quantities pertaining to EMOT (cost functional and Schrödinger coupling) from point clouds that are randomly sampled from the input marginal distributions. Moreover, with substantially weaker smoothness assumption on the cost function than the existing literature, we derive distributional limits and bootstrap validity of various key EMOT objects. As an application, we propose the multimarginal Schrödinger barycenter as a new and natural way to regularize the exact Wasserstein barycenter and demonstrate its statistical optimality.

math.ST↗
arXiv2026

Nonparametric spectral density estimation using interactive mechanisms under local differential privacy

We study the problem of estimating the spectral density of a centered stationary Gaussian time series under local differential privacy constraints. Specifically, we propose new interactive privacy mechanisms for three tasks: recovering a single covariance coefficient, recovering the spectral density at a fixed frequency, and global recovery. Our approach achieves faster rates through a two-stage process: we first apply the Laplace mechanism to the truncated value, and then use the resulting privatized sample to learn about the dependence mechanism in the time series. For spectral densities belonging to Hölder and Sobolev smoothness classes, we demonstrate that our algorithms improve upon the non-interactive mechanism of Kroll (2024) for small privacy parameter $α$, since the pointwise rates depend on $nα^2$ instead of $nα^4$. Moreover, we show that the rate $(nα^4)^{-1}$ is optimal for estimating a covariance coefficient with non-interactive mechanisms. However, the $L_2$ rate of our interactive estimator is slower than the pointwise rate. We show how to use these procedures to provide a bona fide locally differentially private estimator of the entire covariance matrix. A simulation study validates our findings.

math.ST↗