arXiv subjects

Matthieu Jonckheere

Publications and source records attributed to Matthieu Jonckheere.

At least 37 records · Page 2Linked to original sources

arXiv2022

Robust classification with flexible discriminant analysis in heterogeneous data

Linear and Quadratic Discriminant Analysis are well-known classical methods but can heavily suffer from non-Gaussian distributions and/or contaminated datasets, mainly because of the underlying Gaussian assumption that is not robust. To fill this gap, this paper presents a new robust discriminant analysis where each data point is drawn by its own arbitrary Elliptically Symmetrical (ES) distribution and its own arbitrary scale parameter. Such a model allows for possibly very heterogeneous, independent but non-identically distributed samples. After deriving a new decision rule, it is shown that maximum-likelihood parameter estimation and classification are very simple, fast and robust compared to state-of-the-art methods.

stat.ML↗

arXiv2021

Large deviations for the greedy exploration process on configuration models

We prove a large deviation principle for the greedy exploration of configuration models, building on a time-discretized version of the method proposed by Bermolen et al. and Brightwell et al. for jointly constructing a random graph from a given degree sequence and its exploration. The proof of this result follows the general strategy to study large deviations of processes proposed by Feng and Kurtz, based on the convergence of non-linear semigroups. We provide an intuitive interpretation of the LD cost function using Cramer's theorem for the average of random variables with appropriate distribution, depending on the degree distribution of explored nodes. The rate function can be expressed in a closed-form formula, and the large deviations trajectories can be obtained through explicit associated optimization problems. We then deduce large deviations results for the size of the independent set constructed by the algorithm. As a particular case, we analyze these results for d-regular graphs.

math.PR↗

arXiv2021

Persistence in a large network of locally interacting neurons

This article presents a biological neural network model driven by inhomogeneous Poisson processes accounting for the intrinsic randomness of synapses. The main novelty is the introduction of local interactions: each firing neuron triggers an instantaneous increase in electric potential to a fixed number of randomly chosen neurons. We prove that, as the number of neurons approaches infinity, the finite network converges to a nonlinear meanfield process characterised by a jump-type stochastic differential equation. We show that this process displays a phase transition: the activity of a typical neuron in the infinite network either rapidly dies out, or persists forever, depending on the global parameters describing the intensity of interconnection. This provides a way to understand the emergence of persistent activity triggered by weak input signals in large neural networks.

math.PR↗

arXiv2021

Exact asymptotic characterisation of running time for approximate gradient descent on random graphs

In this work we study the time complexity for the search of local minima in random graphs whose vertices have i.i.d. cost values. We show that, for Erdös-Rényi graphs with connection probability given by $λ/n^α$ (with $λ> 0$ and $0 < α< 1$), a family of local algorithms that approximate a gradient descent find local minima faster than the full gradient descent. Furthermore, we find a probabilistic representation for the running time of these algorithms leading to asymptotic estimates of the mean running times.

cs.DS↗

arXiv2021

A Survey of Stability Results for Redundancy Systems

Redundancy mechanisms consist in sending several copies of a same job to a subset of servers. It constitutes one of the most promising ways to exploit diversity in multiservers applications. However, its pros and cons are still not sufficiently understood in the context of realistic models with generic statistical properties of service-times distributions and correlation structures of copies. We aim at giving a survey of recent results concerning the stability-arguably the first benchmark of performance-of systems with cancel-oncompletion redundancy. We also point out open questions and conjectures.

cs.PF↗

arXiv2021

SIR dynamics with Vaccination in a large Configuration Model

We consider a SIR model with vaccination strategy on a sparse configuration model random graph. We show the convergence of the system when the number of nodes grows and characterize the scaling limits. Then, we prove the existence of optimal controls for the limiting equations formulated in the framework of game theory, both in the centralized and decentralized setting. We show how the characteristics of the graph (degree distribution) influence the vaccination efficiency for optimal strategies, and we compute the limiting final size of the epidemic depending on the degree distribution of the graph and the parameters of infection, recovery and vaccination. We also present several simulations for two types of vaccination, showing how the optimal controls allow to decrease the number of infections and underlining the crucial role of the network characteristics in the propagation of the disease and the vaccination program.

math.PR↗

arXiv2020

Uncovering differential equations from data with hidden variables

SINDy is a method for learning system of differential equations from data by solving a sparse linear regression optimization problem [Brunton et al., 2016]. In this article, we propose an extension of the SINDy method that learns systems of differential equations in cases where some of the variables are not observed. Our extension is based on regressing a higher order time derivative of a target variable onto a dictionary of functions that includes lower order time derivatives of the target variable. We evaluate our method by measuring the prediction accuracy of the learned dynamical systems on synthetic data and on a real data-set of temperature time series provided by the Réseau de Transport d'Électricité (RTE). Our method provides high quality short-term forecasts and it is orders of magnitude faster than competing methods for learning differential equations with latent variables.

stat.ML↗

arXiv2020

Improving the performance of heterogeneous data centers through redundancy

We analyze the performance of redundancy in a multi-type job and multi-type server system. We assume the job dispatcher is unaware of the servers' capacities, and we set out to study under which circumstances redundancy improves the performance. With redundancy an arriving job dispatches redundant copies to all its compatible servers, and departs as soon as one of its copies completes service. As a benchmark comparison, we take the non-redundant system in which a job arrival is routed to only one randomly selected compatible server. Service times are generally distributed and all copies of a job are identical, i.e., have the same service requirement. In our first main result, we characterize the sufficient and necessary stability conditions of the redundancy system. This condition coincides with that of a system where each job type only dispatches copies into its least-loaded servers, and those copies need to be fully served. In our second result, we compare the stability regions of the system under redundancy to that of no redundancy. We show that if the server's capacities are sufficiently heterogeneous, the stability region under redundancy can be much larger than that without redundancy. We apply the general solution to particular classes of systems, including redundancy-d and nested models, to derive simple conditions on the degree of heterogeneity required for redundancy to improve the stability. As such, our result is the first in showing that redundancy can improve the stability and hence performance of a system when copies are non-i.i.d..

cs.NI↗

arXiv2020

Clustering high dimensional meteorological scenarios: results and performance index

The Reseau de Transport d'Electricité (RTE) is the French main electricity network operational manager and dedicates large number of resources and efforts towards understanding climate time series data. We discuss here the problem and the methodology of grouping and selecting representatives of possible climate scenarios among a large number of climate simulations provided by RTE. The data used is composed of temperature times series for 200 different possible scenarios on a grid of geographical locations in France. These should be clustered in order to detect common patterns regarding temperatures curves and help to choose representative scenarios for network simulations, which in turn can be used for energy optimisation. We first show that the choice of the distance used for the clustering has a strong impact on the meaning of the results: depending on the type of distance used, either spatial or temporal patterns prevail. Then we discuss the difficulty of fine-tuning the distance choice (combined with a dimension reduction procedure) and we propose a methodology based on a carefully designed index.

stat.AP↗

arXiv2020

Generalized max-weight policies in stochastic matching

We consider a matching system where items arrive one by one at each node of a compatibility network according to Poisson processes and depart from it as soon as they are matched to a compatible item. The matching policy considered is a generalized max-weight policy where decisions can be noisy. Additionally, some of the nodes may have impatience, i.e. leave the system before being matched. Using specific properties of the max-weight policy, we construct several Lyapunov functions, including a simple quadratic one. This allows us to establish stability results, to construct bounds for the stationary mean and variances of the total amount of customers in the system, and to prove exponential convergence speed towards the stationary measure. We finally illustrate some of these results using simulations on toy examples.

math.PR↗

arXiv2020

A flexible EM-like clustering algorithm for noisy data

Though very popular, it is well known that the EM for GMM algorithm suffers from non-Gaussian distribution shapes, outliers and high-dimensionality. In this paper, we design a new robust clustering algorithm that can efficiently deal with noise and outliers in diverse data sets. As an EM-like algorithm, it is based on both estimations of clusters centers and covariances. In addition, using a semi-parametric paradigm, the method estimates an unknown scale parameter per data-point. This allows the algorithm to accommodate for heavier tails distributions and outliers without significantly loosing efficiency in various classical scenarios. We first derive and analyze the proposed algorithm in the context of elliptical distributions, showing in particular important insensitivity properties to the underlying data distributions. We then study the convergence and accuracy of the algorithm by considering first synthetic data. Then, we show that the proposed algorithm outperforms other classical unsupervised methods of the literature such as k-means, the EM for Gaussian mixture models and its recent modifications or spectral clustering when applied to real data sets as MNIST, NORB, and 20newsgroups.

stat.ML↗

arXiv2020

Sequential Algorithms and Independent Sets Discovering on Large Sparse Random Graphs

Computing the size of maximum independent sets is a NP-hard problem for fixed graphs. Characterizing and designing efficient algorithms to estimate this independence number for random graphs are notoriously difficult and still largely open issues. In a companion paper, we showed that a low complexity degree-greedy exploration is actually asymptotically optimal on a large class of sparse random graphs. Encouraged by this result, we present and study two variants of sequential exploration algorithms: static and dynamic degree-aware explorations. We derive hydrodynamic limits for both of them, which in turn allow us to compute the size of the resulting independent set. Whereas the former is simpler to compute, the latter may be used to arbitrarily approximate the degree-greedy algorithm. Both can be implemented in a distributed manner. The corresponding hydrodynamic limits constitute an efficient method to compute or bound the independence number for a large class of sparse random graphs. As an application, we then show how our method may be used to estimate the capacity of a large 802.11-based wireless network. We finally consider further indicators such as the fairness of the resulting configuration, and show how an unexpected trade-off between fairness and capacity can be achieved.

math.PR↗

arXiv2020

Stability of JSQ in queues with general server-job class compatibilities

We consider Poisson streams of exponentially distributed jobs arriving at each edge of a hypergraph of queues. Upon arrival, an incoming job is rooted to the shortest queue among the corresponding vertices. This generalizes many known models such as power-of-d load balancing and JSQ (join the shortest queue) on generic graphs. We provide a generic condition for stability of this model. We show that some graph topologies lead to a loss of capacity, implying more restrictive stability conditions than in, e.g., complete graphs.

math.PR↗

arXiv2019

Nonhomogeneous Euclidean first-passage percolation and distance learning

Consider an i.i.d. sample from an unknown density function supported on an unknown manifold embedded in a high dimensional Euclidean space. We tackle the problem of learning a distance between points, able to capture both the geometry of the manifold and the underlying density. We prove the convergence of this microscopic distance, as the sample size goes to infinity, to a macroscopic one that we call Fermat distance as it minimizes a path functional, resembling Fermat principle in optics. The proof boils down to the study of geodesics in Euclidean first-passage percolation for nonhomogeneous Poisson point processes.

math.PR↗

arXiv2019

F-KPP Scaling limit and selection principle for a Brunet-Derrida type particle system

We study a particle system with the following diffusion-branching-selection mechanism. Particles perform independent one dimensional Brownian motions and on top of that, at a constant rate, a pair of particles is chosen uniformly at random and both particles adopt the position of the rightmost one among them. We show that the cumulative distribution function of the empirical measure converges to a solution of the Fisher-Kolmogorov-Petrovskii-Piskunov (F-KPP) equation and use this fact to prove that the system selects the minimal macroscopic speed as the number of particles goes to infinity.

math.PR↗

arXiv2019

Asymptotic optimality of degree-greedy discovering of independent sets in Configuration Model graphs

Finding independent sets of maximum size in fixed graphs is well known to be an NP-hard task. Using scaling limits, we characterise the asymptotics of sequential degree-greedy explorations and provide sufficient conditions for this algorithm to find an independent set of asymptotically optimal size in large sparse random graphs with given degree sequences. In the special case of sparse Erdös-Rényi graphs, our results allow to give a simple proof of the so-called $e$-phenomenon identified by Karp and Sipser for matchings and to give an alternative characterisation of the asymptotic independence number.

math.PR↗

arXiv2018

Convergence properties of many parallel servers under power-of-D load balancing

We consider a system of N queues with decentralized load balancing such as power-of-D strategies(where D may depend on N) and generic scheduling disciplines. To measure the dependence of the queues, we use the clan of ancestors, a technique coming from interacting particle systems. Relying in that analysis we prove quantitative estimates on the queues correlations implying propagation of chaos for systems with Markovian arrivals and general service time distribution. This solves the conjecture posed by Bramsom et. al. in [*] concerning the asymptotic independence of the servers in the case of processor sharing policy. We then proceed to prove asymptotic insensitivity in the stationary regime for a wide class of scheduling disciplines and obtain speed of convergence estimates for light tailed service distribution. [*] M. BRAMSON, Y. LU AND B. PRABHAKAR, Asymptotic independence of queues under randomized load balancing, Queueing Syst., 71:247-292, 2012.

math.PR↗

arXiv2017

On laws of large numbers in $L^2$ for supercritical branching Markov processes beyond $λ$-positivity

We give necessary and sufficient conditions for laws of large numbers to hold in $L^2$ for the empirical measure of a large class of branching Markov processes, including $λ$-positive systems but also some $λ$-transient ones, such as the branching Brownian motion with drift and absorption at $0$. This is a significant improvement over previous results on this matter, which had only dealt so far with $λ$-positive systems. Our approach is purely probabilistic and is based on spinal decompositions and many-to-few lemmas. In addition, we characterize when the limit in question is always strictly positive on the event of survival, and use this characterization to derive a simple method for simulating (quasi-)stationary distributions.

math.PR↗