Related papers: Smooth numbers are orthogonal to nilsequences

We introduce a geometrically transparent strict saddle property for nonsmooth functions. This property guarantees that simple proximal algorithms on weakly convex problems converge only to local minimizers, when randomly initialized. We…

Random integers, sampled uniformly from $[1,x]$, share similarities with random permutations, sampled uniformly from $S_n$. These similarities include the Erd\H{o}s--Kac theorem on the distribution of the number of prime factors of a random…

In this paper we address the convergence of stochastic approximation when the functions to be minimized are not convex and nonsmooth. We show that the "mean-limit" approach to the convergence which leads, for smooth problems, to the ODE…

In this paper we consider a location model of the form $Y = m(X) + \varepsilon$, where $m(\cdot)$ is the unknown regression function, the error $\varepsilon$ is independent of the $p$-dimensional covariate $X$ and $E(\varepsilon)=0$. Given…

This paper considers approximate smoothing for discretely observed non-linear stochastic differential equations. The problem is tackled by developing methods for linearising stochastic differential equations with respect to an arbitrary…

Integer partitions have fascinated people for centuries, from Ramanujan's groundbreaking congruences to the modern theory of modular forms. This paper investigates the statistical properties of odd unimodal sequences--a natural refinement…

Stochastic gradient descent type methods are ubiquitous in machine learning, but they are only applicable to the optimization of differentiable functions. Proximal algorithms are more general and applicable to nonsmooth functions. We…

In this note we improve an algorithm from a recent paper by Bauer and Bennett for computing a function of Erd\"os that measures the minimal gap size $f(k)$ in the sequence of integers at least one of whose prime factors exceeds $k$. This…

We derive limiting distributions of symmetrized estimators of scatter, where instead of all $n(n-1)/2$ pairs of the $n$ observations we only consider $nd$ suitably chosen pairs, $1 \le d < \lfloor n/2\rfloor$. It turns out that the…

We introduce the notions of strong asymptotic uniform smoothness and convexity. We show that the injective tensor product of strongly asymptotically uniformly smooth spaces is asymptotically uniformly smooth. This applies in particular to…

We explore the asymptotic distributions of sequences of integer-valued additive functions defined on the symmetric group endowed with the Ewens probability measure as the order of the group increases. Applying the method of factorial…

In this paper, we consider the problem of minimizing the average of a large number of nonsmooth and convex functions. Such problems often arise in typical machine learning problems as empirical risk minimization, but are computationally…

Let $\{\Lambda_n=\{\lambda_{1,n},\ldots,\lambda_{d_n,n}\}\}_n$ be a sequence of finite multisets of real numbers such that $d_n\to\infty$ as $n\to\infty$, and let $f:\Omega\subset\mathbb R^d\to\mathbb R$ be a Lebesgue measurable function…

Let $X^N$ be a family of $N\times N$ independent GUE random matrices, $Z^N$ a family of deterministic matrices, $P$ a self-adjoint non-commutative polynomial, that is for any $N$, $P(X^N)$ is self-adjoint, $f$ a smooth function. We prove…

Information retrieval (IR) systems traditionally aim to maximize metrics built on rankings, such as precision or NDCG. However, the non-differentiability of the ranking operation prevents direct optimization of such metrics in…

We investigate exponential sums over those numbers $\leq x$ all of whose prime factors are $\leq y$. We prove fairly good minor arc estimates, valid whenever $\log^{3}x \leq y \leq x^{1/3}$. Then we prove sharp upper bounds for the $p$-th…

For any finitely generated group G, let n ---> \Phi_G(n) be the function that describes the rough asymptotic behavior of the probability of return to the identity element at time 2n of a symmetric simple random walk on G (this is an…

Let $\xi_1,\xi_2,\ldots$ be an iid sequence with negative mean. The $(m,n)$-segment is the subsequence $\xi_{m+1},\ldots,\xi_n$ and its \textit{score} is given by $\max\{\sum_{m+1}^n\xi_i,0\}$. Let $R_n$ be the largest score of any segment…

The happy function $H: \mathbb{N} \rightarrow \mathbb{N}$ sends a positive integer to the sum of the squares of its digits. A number $x$ is said to be happy if the sequence $\{H^n(x)\}^\infty_{n=1}$ eventually reaches one. A basic open…

We establish a prime number theorem for all uniquely ergodic, analytic skew products on the $2$-torus $\mathbb{T}^2$. More precisely, for every irrational $\alpha$ and every $1$-periodic real analytic $g:\mathbb{R}\to\mathbb{R}$ of zero…