arXiv subjects

Jean-Marc Brossier

Publications and source records attributed to Jean-Marc Brossier.

7 recordsLinked to original sources

arXiv2026

Analytical study of the optimal combination of binary classifiers based on classifiers-induced partitioning of the training set

This paper studies an optimal linear combination of binary classifiers based on a logical structuration of the dataset via truth tables. The given classifiers partition data into equivalence classes, allowing for a rigorous analysis of the convexified empirical risk through a multidimensional generalization of classification calibrated functions. We establish sufficient conditions for the existence and uniqueness of the (global) point of minimum of the convexified empirical risk for any list of classifiers (when the number of classifiers is large, there frequently could be no point of minimum). In the case of three classifiers, our analysis allows to list all the configurations leading to either a unique solution, infima or non-unique points of minimum. Furthermore, we derive explicit analytical formulae for optimal weights using Exponential (Boost) and Logistic (Logit) loss functions, bypassing iterative optimization. The stability of the resulting classifier and the analysis of data quality can be evaluated through the introduction of the notion of $\phi$-frontiers.

cs.LG↗

arXiv2024

When Analytic Calculus Cracks AdaBoost Code

The principle of boosting in supervised learning involves combining multiple weak classifiers to obtain a stronger classifier. AdaBoost has the reputation to be a perfect example of this approach. This study analyzes the (two classes) AdaBoost procedure implemented in scikit-learn. This paper shows that AdaBoost is an algorithm in name only, as the resulting combination of weak classifiers can be explicitly calculated using a truth table. Indeed, using a logical analysis of the training set with weak classifiers constructing a truth table, we recover, through an analytical formula, the weights of the combination of these weak classifiers obtained by the procedure. We observe that this formula does not give the point of minimum of the risk, we provide a system to compute the exact point of minimum and we check that the AdaBoost procedure in scikit-learn does not implement the algorithm described by Freund and Schapire.

cs.LG↗

arXiv2016

Generalization of the de Bruijn's identity to general $ϕ$-entropies and $ϕ$-Fisher informations

In this paper, we propose generalizations of the de Bruijn's identities based on extensions of the Shannon entropy, Fisher information and their associated divergences or relative measures. The foundation of these generalizations are the $ϕ$-entropies and divergences of the Csiszá's class (or Salicrú's class) considered within a multidimensional context, included the monodimensional case, and for several type of noisy channels characterized by a more general probability distribution beyond the well-known Gaussian noise. It is found that the gradient and/or the hessian of these entropies or divergences with respect to the noise parameters give naturally rise to generalized versions of the Fisher information or divergence, which are named as the $ϕ$-Fisher information (divergence). The obtained identities can be viewed as further extensions of the classical de Bruijn's identity. Analogously, it is shown that a similar relation holds between the $ϕ$-divergence and a extended mean-square error, named $ϕ$-mean square error, for the Gaussian channel.

cs.IT↗

arXiv2016

2D Time-frequency interference modelling using stochastic geometry for performance evaluation in Low-Power Wide-Area Networks

In wireless networks, interferences between trans- missions are modelled either in time or frequency domain. In this article, we jointly analyze interferences in the time- frequency domain using a stochastic geometry model assuming the total time-frequency resources to be a two-dimensional plane and transmissions from Internet of Things (IoT) devices time- frequency patterns on this plane. To evaluate the interference, we quantify the overlap between the information packets: provided that the overlap is not too strong, the packets are not necessarily lost due to capture effect. This flexible model can be used for multiple medium access scenarios and is especially adapted to the random time-frequency access schemes used in Low-Power Wide-Area Networks (LPWANs). By characterizing the outage probability and throughput, our approach permits to evaluate the performance of two representative LPWA technologies Sigfox{\textsuperscript \textregistered} and LoRaWA{\textsuperscript \textregistered}.

cs.NI↗

arXiv2013

Optimal Asymmetric Binary Quantization for Estimation Under Symmetrically Distributed Noise

Estimation of a location parameter based on noisy and binary quantized measurements is considered in this letter. We study the behavior of the Cramer-Rao bound as a function of the quantizer threshold for different symmetric unimodal noise distributions. We show that, in some cases, the intuitive choice of threshold position given by the symmetry of the problem, placing the threshold on the true parameter value, can lead to locally worst estimation performance.

cs.IT↗

arXiv2013

Optimal Scalar Quantization for Parameter Estimation

In this paper, we study an asymptotic approximation of the Fisher information for the estimation of a scalar parameter using quantized measurements. We show that, as the number of quantization intervals tends to infinity, the loss of Fisher information induced by quantization decreases exponentially as a function of the number of quantization bits. A characterization of the optimal quantizer through its interval density and an analytical expression for the Fisher information are obtained. A comparison between optimal uniform and non-uniform quantization for the location and scale estimation problems shows that non-uniform quantization is only slightly better. As the optimal quantization intervals are shown to depend on the unknown parameters, by applying adaptive algorithms that jointly estimate the parameter and set the thresholds in the location and scale estimation problems, we show that the asymptotic results can be approximately obtained in practice using only 4 or 5 quantization bits.

cs.IT↗

arXiv2012

Adaptive Quantizers for Estimation

In this paper, adaptive estimation based on noisy quantized observations is studied. A low complexity adaptive algorithm using a quantizer with adjustable input gain and offset is presented. Three possible scalar models for the parameter to be estimated are considered: constant, Wiener process and Wiener process with deterministic drift. After showing that the algorithm is asymptotically unbiased for estimating a constant, it is shown, in the three cases, that the asymptotic mean squared error depends on the Fisher information for the quantized measurements. It is also shown that the loss of performance due to quantization depends approximately on the ratio of the Fisher information for quantized and continuous measurements. At the end of the paper the theoretical results are validated through simulation under two different classes of noise, generalized Gaussian noise and Student's-t noise.

cs.IT↗