Related papers: Random noise increases Kolmogorov complexity and H…
Define $||n||$ to be the complexity of $n$, the smallest number of ones needed to write $n$ using an arbitrary combination of addition and multiplication. The set $\mathscr{D}$ of defects, differences $\delta(n):=||n||-3\log_3 n$, is known…
Various specifiable combinatorial structures, with d extensive parameters, can be exactly sampled both by the recursive method, with linear arithmetic complexity if a heavy preprocessing is performed, or by the Boltzmann method, with…
Utilizing the hyperspace of noise-based logic, we show two string verification methods with low communication complexity. One of them is based on continuum noise-based logic. The other one utilizes noise-based logic with random telegraph…
In the trace reconstruction problem our goal is to learn an unknown string $x\in \{0,1\}^n$ given independent traces of $x$. A trace is obtained by independently deleting each bit of $x$ with some probability $\delta$ and concatenating the…
Rademacher complexity is often used to characterize the learnability of a hypothesis class and is known to be related to the class size. We leverage this observation and introduce a new technique for estimating the size of an arbitrary…
The notion of incremental learning is to train an ANN algorithm in stages, as and when newer training data arrives. Incremental learning is becoming widespread in recent times with the advent of deep learning. Noise in the training data…
We perform numerical simulations of cosmic string evolution with intercommuting probability $P$ in the range $5\times 10^{-3}\le P\le 1$, both in the matter and radiation eras, using a modified version of the Allen-Shellard code. We find…
Let $\{u_t(x),t\ge 0, x\in {\mathbb{R}}\}$ be a random string taking values in ${\mathbb{R}}^d$, specified by the following stochastic partial differential equation [Funaki (1983)]: \[\frac{\partial u_t(x)}{\partial…
We survey the diverse approaches to the notion of information content: from Shannon entropy to Kolmogorov complexity. The main applications of Kolmogorov complexity are presented namely, the mathematical notion of randomness (which goes…
In this paper, we redesign and simplify an algorithm due to Remy et al. for the generation of rooted planar trees that satisfies a given partition of degrees. This new version is now optimal in terms of random bit complexity, up to a…
We consider the random $k$-SAT problem with $n$ variables, $m=m(n)$ clauses, and clause density $\alpha=\lim_{n\to\infty}m/n$ for $k=2,3$. It is known that if $\alpha$ is small enough, then the random $k$-SAT problem admits a solution with…
We study the largest component of a random (multi)graph on n vertices with a given degree sequence. We let n tend to infinity. Then, under some regularity conditions on the degree sequences, we give conditions on the asymptotic shape of the…
We prove a new lower bound on the algorithmic information content of points lying on a line in $\mathbb{R}^n$. More precisely, we show that a typical point $z$ on any line $\ell$ satisfies \begin{equation*} K_r(z)\geq \frac{K_r(\ell)}{2} +…
We obtain a recurrence relation in $d$ for the average singular value $% \alpha (d)$ of a complex valued $d\times d$\ matrix $\frac{1}{\sqrt{d}}X$ with random i.i.d., N( 0,1) entries, and use it to show that $\alpha (d)$ decreases…
Andersson and Nilsson introduced in 1993 a level-compressed trie (in short: LC trie) in which a full subtree of a node is compressed to a single node of degree being the size of the subtree. Recent experimental results indicated a 'dramatic…
Shallit and Wang studied deterministic automatic complexity of words. They showed that the automatic Hausdorff dimension $I(\mathbf t)$ of the infinite Thue word satisfies $1/3\le I(\mathbf t)\le 2/3$. We improve that result by showing that…
We consider the asymptotic behaviour of the fluctuation process for large stochastic systems of interacting particles driven by both idiosyncratic and common noise with an interaction kernel \(k \in L^2(\R^d) \cap L^\infty(\R^d)\). Our…
Accurate inference on a scalar interest parameter in the presence of a nuisance parameter may be obtained using an adjusted version of the signed root likelihood ratio statistic, in particular Barndorff-Nielsen's $R^*$ statistic. The…
The problem of the order of the fluctuation of the Longest Common Subsequence (LCS) of two independent sequences has been open for decades. There exist contradicting conjectures on the topic, due to Chvatal - Sankoff in 1975 and Waterman in…
For 0 < x < 1, take the binary expansion with infinitely many 0's, replace each 0 with -1, this gives the polarized binary expansion of x. Let R_i(x) be the ith "polarized bit" and let S_n(x) be the sum of the first n R_i(x). {S_n} is the…