Related papers: Euclidean Partitions Optimizing Noise Stability
We study the fundamental problem of learning the parameters of a high-dimensional Gaussian in the presence of noise -- where an $\varepsilon$-fraction of our samples were chosen by an adversary. We give robust estimators that achieve…
All physical systems are affected by some noise that limits the resolution that can be attained in partitioning their state space. For chaotic, locally hyperbolic flows, this resolution depends on the interplay of the local…
We give a possible explanation for the mystery of a missing number in the statement of a problem that asks for the non-negative integers to be partitioned into three subsets. We interpret the missing number as one of the clues that can lead…
Pairwise Euclidean distance calculation is a fundamental step in many machine learning and data analysis algorithms. In real-world applications, however, these distances are frequently distorted by heteroskedastic noise$\unicode{x2014}$a…
For each prime $p\equiv 1\pmod{4}$ consider the Legendre character $\chi=(\frac{\cdot}{p})$. Let $p_\pm(n)$ be the number of partitions of $n$ into parts $\lambda>0$ such that $\chi(\lambda)=\pm 1$. Petersson proved a beautiful limit…
We find that a wide variety of families of partition statistics stabilize in a fashion similar to $p_k(n)$, the number of partitions of n with k parts, which satisfies $p_k(n) = p_{k+1}(n + 1), k \geq n/2$. We bound the regions of…
In this paper we aim at generalizing the results of A. K. Zvonkin and A. Y. Veretennikov on the construction of unique strong solutions of stochastic differential equations with singular drift vector field and additive noise in the…
The problem of clustering is considered, for the case when each data point is a sample generated by a stationary ergodic process. We propose a very natural asymptotic notion of consistency, and show that simple consistent algorithms exist,…
The problem of clustering is considered, for the case when each data point is a sample generated by a stationary ergodic process. We propose a very natural asymptotic notion of consistency, and show that simple consistent algorithms exist,…
Stochastic approximation is a framework unifying many random iterative algorithms occurring in a diverse range of applications. The stability of the process is often difficult to verify in practical applications and the process may even be…
Partition functions arise in statistical physics and probability theory as the normalizing constant of Gibbs measures and in combinatorics and graph theory as graph polynomials. For instance the partition functions of the hard-core model…
We prove the Murphy and Cohen's conjecture that the maximum number of collisions of n + 1 elastic particles moving freely on a line is n(n+1)/2 if no interior particle has mass less than the arithmetic mean of the masses of its immediate…
This paper investigates sharp stability estimates for the fractional Hardy-Sobolev inequality: $$\mu_{s,t}\left(\mathbb{R}^N\right) \left(\int_{\mathbb{R}^N} \frac{|u|^{2^*_s(t)}}{|x|^t} \,{\rm d}x \right)^{\frac{2}{2^*_s(t)}} \leq…
Consider a real-valued function that can only be observed with stochastic noise at a finite set of design points within a Euclidean space. We wish to determine whether there exists a convex function that goes through the true function…
We provide a crystal structure on the set of ordered multiset partitions, which recently arose in the pursuit of the Delta Conjecture. This conjecture was stated by Haglund, Remmel and Wilson as a generalization of the Shuffle Conjecture.…
Recovering a low-complexity signal from its noisy observations by regularization methods is a cornerstone of inverse problems and compressed sensing. Stable recovery ensures that the original signal can be approximated linearly by optimal…
The Delta Conjecture of Haglund, Remmel, and Wilson is a recent generalization of the Shuffle Conjecture in the field of diagonal harmonics. In this paper we give evidence for the Delta Conjecture by proving a pair of conjectures of Wilson…
We study the problems of covering or partitioning a polygon $P$ (possibly with holes) using a minimum number of small pieces, where a small piece is a connected sub-polygon contained in an axis-aligned unit square. For covering, we seek to…
Tverberg's theorem asserts that every (k-1)(d+1)+1 points in R^d can be partitioned into k parts, so that the convex hulls of the parts have a common intersection. Calder and Eckhoff asked whether there is a purely combinatorial deduction…
We investigate an example of noise-induced stabilization in the plane that was also considered in (Gawedzki, Herzog, Wehr 2010) and (Birrell, Herzog, Wehr 2011). We show that despite the deterministic system not being globally stable, the…