Loïck Lhote

• University of Caen Normandy

Closed-form expressions for generalized entropy rates of Markov chains are obtained through pertinent averaging. First, the rates are expressed in terms of Perron-Frobenius eigenvalues of perturbations of the transition matrices. This leads to a classification of generalized entropy functionals into five exclusive types. Then, a weighted expression...

For Markov chains with transition probabilities pij , the Shannon entropy rate is well-known to be equal to the sum of the −Σj pij log pij weighted by the stationary distribution. This expression derives from the chain rule specific to Shannon entropy. For an ergodic Markov chain, the stationary distribution is the limit of the marginal distributio...

We introduce and study an algorithm which computes the gcd of entries. This is a natural extension of the usual Euclid algorithm, and coincides with it for ; it performs Euclidean divisions, between the largest entry and the second largest entry, and then re-orderings. This is the discrete version of a multidimensional continued fraction algorithm...

We introduce and study a multiple gcd algorithm that is a natural extension of the usual Euclid algorithm, and coincides with it for two entries; it performs Euclidean divisions, between the largest entry and the second largest entry, and then re-orderings. This is the discrete version of a multidimensional continued fraction algorithm due to Brun....

Based on rescaling by some suitable sequence instead of the number of time units, the usual notion of divergence rate is here extended to define and determine meaningful generalized divergence rates. Rescaling entropy rates appears as a special case. Suitable rescaling is naturally induced by the asymptotic behavior of the marginal divergences. Clo...

Among multiple gcd algorithms on polynomials as on integers, one of the most natural ones performs a sequence of ℓ-1 phases (ℓ is the number of inputs), with each of them being the Euclid algorithm on two entries. We present here a complete probabilistic analysis of this algorithm, by providing both the average-case and the distributional analysis,...

In this paper, we study some average properties of hypergraphs and the average com-plexity of algorithms applied to hypergraphs under different probabilistic models. Our approach is both theoretical and experimental since our goal is to obtain a random model that is able to capture the real-data complexity. Starting from a model that generalizes th...

This paper provides a probabilistic analysis of an algorithm which computes the gcd of ℓ inputs (with ℓ ≥ 2), with a succession of ℓ - 1 phases, each of them being the Euclid algorithm on two entries. This algorithm is both basic and natural, and two kinds of inputs are studied: polynomials over the finite field Fq and integers. The analysis exhibi...

International audience Consider a countable alphabet $\mathcal{A}$. A multi-modular hidden pattern is an $r$-tuple $(w_1,\ldots , w_r)$, where each $w_i$ is a word over $\mathcal{A}$ called a module. The hidden pattern is said to occur in a text $t$ when the later admits the decomposition $t = v_0 w_1v_1 \cdots v_{r−1}w_r v_r$, for arbitrary words...

We study entropy rates of random sequences for general entropy functionals including the classical Shannon and Rényi entropies and the more recent Tsallis and Sharma-Mittal ones. In the first part, we obtain an explicit formula for the entropy rate for a large class of entropy functionals, as soon as the process satisfies a regularity property know...

In a tabular database, patterns that occur over a frequency threshold are called frequent patterns. They are central in numerous data processes and various efficient algorithms were recently designed for mining them. Unfortunately, very little is known about the real difficulty of this mining, which is closely related to the number of such patterns...

There exist fast variants of the gcd algorithm which are all based on principles due to Knuth and Schonhage. On inputs of sizen, these algorithms use a Divide and Conquer approach, perform FFT multiplications with complexity µ(n) and stop the recursion at a depth slightly smaller than lgn. A rough estimate of the worst-case complexity of these fast...

We provide sharp estimates for the probabilistic behaviour of the main parameters of the Euclid Algorithms, both on polynomials and on integer numbers. We study in particular the distribution of the bit-complexity which involves two main parameters: digit-costs and length of remainders. We first show here that an asymptotic Gaussian law holds for t...

This thesis deals with two main algorithmical domains: Data Mining and Arithmetical computations. In both, we are interested in the average-case analysis of algorithms, and, we adopt more precisely the dynamical analysis point of vue which is a mixed method between Analysis of Algorithms and Dynamical Systems. The Euclid algorithms compute the gcd...

We provide sharp estimates for the probabilistic behaviour of the main parameters of the Euclid algorithm, and we study in par- ticular the distribution of the bit-complexity which involves two main parameters : digit–costs and length of continuants. We perform a “dy- namical analysis” which heavily uses the dynamical system underlying the Euclidea...

In data mining, enumerate the frequent or the closed patterns is often the first difficult task leading to the association rules discovery. The number of these patterns represents a great interest. The lower bound is known to be constant whereas the upper bound is exponential, but both situations correspond to pathological cases. For the first time...

Résumé : Frequent and closed patterns are at the core of numerous Knowledge Discovery processes. Their mining is known to be difficult, because of the huge size of the search space, exponentially growing with the number of attributes. Unfortunately, most studies about pattern mining do not address the difficulty of the task, and provide their own a...

@inproceedings{CI-LHOTE-2005, author = {Lhote, L. and Rioult, F. and Soulet, A.}, title = {Average Number of Frequent (Closed) Patterns in Bernouilli and Markovian Databases}, booktitle = {5th IEEE International Conference on Data Mining (ICDM'05)}, publisher = {IEEE Computer Society Press}, address = {Houston, Texas, United States}, pages = {713-7...

There are numerous instances where mathematical constants do not admit a closed form. It is then of great interest to compute them, possibly in an e cient way. So the question is: does there exist an algorithm that computes the first d-digits of the constants and if so, what is the complexity in the number of arithmetic operations? We recall that a...