Related papers: A link between error terms when counting smooth an…
A fundamental tool in network information theory is the covering lemma, which lower bounds the probability that there exists a pair of random variables, among a give number of independently generated candidates, falling within a given set.…
We give efficient algorithms for ranking Lyndon words of length $n$ over an alphabet of size $\sigma$. The rank of a Lyndon word is its position in the sequence of lexicographically ordered Lyndon words of the same length. The outputs are…
We consider methods for finding high-precision approximations to simple zeros of smooth functions. As an application, we give fast methods for evaluating the elementary functions log(x), exp(x), sin(x) etc. to high precision. For example,…
We present a bound for value-prediction error with respect to model misspecification that is tight, including constant factors. This is a direct improvement of the "simulation lemma," a foundational result in reinforcement learning. We…
Computing the probability of a formula given the probabilities or weights associated with other formulas is a natural extension of logical inference to the probabilistic setting. Surprisingly, this problem has received little attention in…
Word embedding, specially with its recent developments, promises a quantification of the similarity between terms. However, it is not clear to which extent this similarity value can be genuinely meaningful and useful for subsequent tasks.…
Let $g(n)$ be the largest positive integer $k$ such that there are distinct primes $p_i$ for $1\leq i\leq k$ so that $p_i |n+i$. This function is related to a celebrated conjecture of C.A. Grimm. We establish upper and lower bounds for…
Easily computable lower and upper bounds are found for the sum of Catalan numbers. The lower bound is proven to be tighter than the upper bound, which previously was declared to be only an asymptotic. The average of these bounds is proven…
We consider upper bounds on the error probability in channel coding. We derive an improved maximum-likelihood union bound, which takes into account events where the likelihood of the correct codeword is tied with that of some competitors.…
In a paper entitled Binary lambda calculus and combinatory logic, John Tromp presents a simple way of encoding lambda calculus terms as binary sequences. In what follows, we study the numbers of binary strings of a given size that represent…
In a posteriori error analysis, the relationship between error and estimator is usually spoiled by so-called oscillation terms, which cannot be bounded by the error. In order to remedy, we devise a new approach where the oscillation has the…
Modern large-scale statistical models require to estimate thousands to millions of parameters. This is often accomplished by iterative algorithms such as gradient descent, projected gradient descent or their accelerated versions. What are…
An information-theoretic upper bound on the generalization error of supervised learning algorithms is derived. The bound is constructed in terms of the mutual information between each individual training sample and the output of the…
This paper is concerned with the analysis of the randomized subspace iteration for the computation of low-rank approximations. We present three different kinds of bounds. First, we derive both bounds for the canonical angles between the…
In this short note, we obtain error estimates for Riemann sums of some singular functions.
We mainly study numerical integration of real valued functions defined on the $d$-dimensional unit cube with all partial derivatives up to some finite order $r\ge1$ bounded by one. It is well known that optimal algorithms that use $n$…
In this paper, we consider certain finite sums related to the "largest odd divisor", and we obtain, using simple ideas and recurrence relations, sharp upper and lower bounds for these sums.
We analyze the forward error in the floating point summation of real numbers, for computations in low precision or extreme-scale problem dimensions that push the limits of the precision. We present a systematic recurrence for a martingale…
Let $X$ denote a nonnegative random variable with $\mathsf{E} X<\infty$. Upper and lower bounds on $\mathsf{E} X-\exp\mathsf{E}\ln X$ are obtained, which are exact, in terms of $V_X$ and $E_X$ for the upper bound and in terms of $V_X$ and…
We examine how the measure and the number of vertices of the convex hull of a random sample of $n$ points from an arbitrary probability measure in $\mathbf{R}^d$ relates to the wet part of that measure. This extends classical results for…