arXiv subjects

Philippe Robert

Publications and source records attributed to Philippe Robert.

At least 37 records · Page 2Linked to original sources

arXiv2018

Robust Point Light Source Estimation Using Differentiable Rendering

Illumination estimation is often used in mixed reality to re-render a scene from another point of view, to change the color/texture of an object, or to insert a virtual object consistently lit into a real video or photograph. Specifically, the estimation of a point light source is required for the shadows cast by the inserted object to be consistent with the real scene. We tackle the problem of illumination retrieval given an RGBD image of the scene as an inverse problem: we aim to find the illumination that minimizes the photometric error between the rendered image and the observation. In particular we propose a novel differentiable renderer based on the Blinn-Phong model with cast shadows. We compare our differentiable renderer to state-of-the-art methods and demonstrate its robustness to an incorrect reflectance estimation.

cs.CV↗

arXiv2018

Analysis of Large Urn Models with Local Mean-Field Interactions

The stochastic models investigated in this paper describe the evolution of a set of $F_N$ identical balls scattered into $N$ urns connected by an underlying symmetrical graph with constant degree $h_N$. After some random amount of time {\em all the balls} of any urn are redistributed locally, among the $h_N$ urns of its neighborhood. The allocation of balls is done at random according to a set of weights which depend on the state of the system. The main original features of this context is that the cardinality $h_N$ of the range of interaction is not necessarily linear with respect to $N$ as in a classical mean-field context and, also, that the number of simultaneous jumps of the process is not bounded due to the redistribution of all balls of an urn at the same time. The approach relies on the analysis of the evolution of the local empirical distributions associated to the state of urns located in the neighborhood of a given urn. Under convenient conditions, by taking an appropriate Wasserstein distance and by establishing several technical estimates for local empirical distributions, we are able to prove mean-field convergence results. When the load per node goes to infinity, a convergence result for the invariant distribution of the associated McKean-Vlasov process is obtained for several allocation policies. For the class of power of $d$ choices policies, we show that the associated invariant measure has an asymptotic finite support property under this regime. This result differs somewhat from the classical double exponential decay property usually encountered in the literature for power of $d$ choices policies.

math.PR↗

arXiv2018

On the Asymptotic Distribution of Nucleation Times of Polymerization Processes

In this paper, we investigate a stochastic model describing the time evolution of a polymerization process. A polymer is a macro-molecule resulting from the aggregation of several elementary sub-units called monomers. Polymers can grow by addition of monomers or can be split into several polymers. The initial state of the system consists mainly of monomers. We study the time evolution of the mass of polymers, in particular the asymptotic distribution of the first instant when the fraction of monomers used in polymers is above some positive threshold $δ$. A scaling approach is used by taking the mass $N$ as a scaling parameter. The mathematical model used in this paper includes {\em a nucleation property}: If $n_c$ is defined as the size of the nucleus, polymers with a size less than $n_c$ are quickly fragmented into smaller polymers, at a rate proportional to $Φ(N)$ for some non-decreasing and unbounded function $Φ$. For polymers of size greater than $n_c$, fragmentation still occurs but at bounded rates. If $T^N$ is the instant of creation of the first polymer whose size is $n_c$, it is shown that, under appropriate conditions, the variable $T^N{/}[Φ(N)^{n_c{-}2}{/}N]$ converges in distribution, and that the first instant $L_δ^N$ when a fraction $δ$ of monomers is polymerized has the same order of magnitude. An original feature proved for this model is the significant variability of the variable $T^N$. This is a well known phenomenon observed in biological experiments but few mathematical models of the literature have this property. The results are proved via a series of technical estimates for occupation measures of some functionals of the corresponding Markov processes on fast time scales and by using coupling techniques.

math.PR↗

arXiv2018

Alzheimer's Disease Modelling and Staging through Independent Gaussian Process Analysis of Spatio-Temporal Brain Changes

Alzheimer's disease (AD) is characterized by complex and largely unknown progression dynamics affecting the brain's morphology. Although the disease evolution spans decades, to date we cannot rely on long-term data to model the pathological progression, since most of the available measures are on a short-term scale. It is therefore difficult to understand and quantify the temporal progression patterns affecting the brain regions across the AD evolution. In this work, we tackle this problem by presenting a generative model based on probabilistic matrix factorization across temporal and spatial sources. The proposed method addresses the problem of disease progression modelling by introducing clinically-inspired statistical priors. To promote smoothness in time and model plausible pathological evolutions, the temporal sources are defined as monotonic and independent Gaussian Processes. We also estimate an individual time-shift parameter for each patient to automatically position him/her along the sources time-axis. To encode the spatial continuity of the brain sub-structures, the spatial sources are modeled as Gaussian random fields. We test our algorithm on grey matter maps extracted from brain structural images. The experiments highlight differential temporal progression patterns mapping brain regions key to the AD pathology, and reveal a disease-specific time scale associated with the decline of volumetric biomarkers across clinical stages.

stat.AP↗

arXiv2018

Multi-Channel Stochastic Variational Inference for the Joint Analysis of Heterogeneous Biomedical Data in Alzheimer's Disease

The joint analysis of biomedical data in Alzheimer's Disease (AD) is important for better clinical diagnosis and to understand the relationship between biomarkers. However, jointly accounting for heterogeneous measures poses important challenges related to the modeling of the variability and the interpretability of the results. These issues are here addressed by proposing a novel multi-channel stochastic generative model. We assume that a latent variable generates the data observed through different channels (e.g., clinical scores, imaging, ...) and describe an efficient way to estimate jointly the distribution of both latent variable and data generative process. Experiments on synthetic data show that the multi-channel formulation allows superior data reconstruction as opposed to the single channel one. Moreover, the derived lower bound of the model evidence represents a promising model selection criterion. Experiments on AD data show that the model parameters can be used for unsupervised patient stratification and for the joint interpretation of the heterogeneous observations. Because of its general and flexible formulation, we believe that the proposed method can find important applications as a general data fusion technique.

stat.ME↗

arXiv2017

A Stochastic Analysis of a Network with Two Levels of Service

In this paper a stochastic model of a call center with a two-level architecture is analyzed. A first-level pool of operators answers calls, identifies, and handles non-urgent calls. A call classified as urgent has to be transferred to specialized operators at the second level. When the operators of the second level are all busy, the operator of first level handling the urgent call is blocked until an operator at the second level is available. Under a scaling assumption, the evolution of the number of urgent calls blocked at level~$1$ is investigated. It is shown that if the ratio of the number of operators at level $2$ and~$1$ is greater than some threshold, then, essentially, the system operates without congestion, with probability close to $1$, no urgent call is blocked after some finite time. Otherwise, we prove that a positive fraction of the operators of the first level are blocked due to the congestion of the second level. Stochastic calculus with Poisson processes, coupling arguments and formulations in terms of Skorokhod problems are the main mathematical tools to establish these convergence results.

cs.NI↗

arXiv2017

A Large Scale Analysis of Unreliable Stochastic Networks

The problem of reliability of a large distributed system is analyzed via a new mathematical model. A typical framework is a system where a set of files are duplicated on several data servers. When one of these servers breaks down, all copies of files stored on it are lost. In this way, repeated failures may lead to losses of files. The efficiency of such a network is directly related to the performances of the mechanism used to duplicate files on servers. In this paper we study the evolution of the network using a natural duplication policy giving priority to the files with the least number of copies. We investigate the asymptotic behavior of the network when the number $N$ of servers is large. The analysis is complicated by the large dimension of the state space of the empirical distribution of the state of the network. A stochastic model of the evolution of the network which has values in state space whose dimension does not depend on $N$ is introduced. Despite this description does not have the Markov property, it turns out that it is converging in distribution, when the number of nodes goes to infinity, to a nonlinear Markov process. The rate of decay of the network, which is the key characteristic of interest of these systems, can be expressed in terms of this asymptotic process. The corresponding mean-field convergence results are established. A lower bound on the exponential decay, with respect to time, of the fraction of the number of initial files with at least one copy is obtained.

math.PR↗

arXiv2017

Analysis of a Stochastic Model of Replication in Large Distributed Storage Systems: A Mean-Field Approach

Distributed storage systems such as Hadoop File System or Google File System (GFS) ensure data availability and durability using replication. This paper is focused on the analysis of the efficiency of replication mechanism that determines the location of the copies of a given file at some server. The variability of the loads of the nodes of the network is investigated for several policies. Three replication mechanisms are tested against simulations in the context of a real implementation of a such a system: Random, Least Loaded and Power of Choice. The simulations show that some of these policies may lead to quite unbalanced situations: if $β$ is the average number of copies per node it turns out that, at equilibrium, the load of the nodes may exhibit a high variability. It is shown in this paper that a simple variant of a power of choice type algorithm has a striking effect on the loads of the nodes: at equilibrium, the distribution of the load of a node has a bounded support, most of nodes have a load less than $2β$ which is an interesting property for the design of the storage space of these systems. Mathematical models are introduced and investigated to explain this interesting phenomenon. The analysis of these systems turns out to be quite complicated mainly because of the large dimensionality of the state spaces involved. Our study relies on probabilistic methods, mean-field analysis, to analyze the asymptotic behavior of an arbitrary node of the network when the total number of nodes gets large. An additional ingredient is the use of stochastic calculus with marked Poisson point processes to establish some of our results.

cs.DC↗

arXiv2017

Allocation Schemes of Ressources with Downgrading

We consider a server with large capacity delivering video files encoded in various resolutions. We assume that the system is under saturation in the sense that the total demand exceeds the server capacity $C$. In such a case, requests may be rejected. For the policies considered in this paper, instead of rejecting a video request, it is downgraded. When the occupancy of the server is above some value $C_0{<}C$, the server delivers the video at a minimal bit rate. The quantity $C_0$ is the bit rate adaptation threshold. For these policies, request blocking is thus replaced by bit rate adaptation. Under the assumptions of Poisson request arrivals and exponential service times, we show that, by rescaling the system, a process associated to the occupancy of the server converges to some limiting process whose invariant distribution is computed by a Wiener-Hopf technique. This allows us to derive an asymptotic expression of the key performance measure of such a policy, the equilibrium probability that a request is transmitted without downgrading. Numerical applications of these results are presented.

cs.NI↗

arXiv2016

A Scaling Analysis of a Star Network with Logarithmic Weights

The paper investigates the properties of a class of resource allocation algorithms for communication networks: if a node of this network has $x$ requests to transmit, then it receives a fraction of the capacity proportional to $\log(1{+}L)$, the logarithm of its current load $L$. A stochastic model of such an algorithm is investigated in the case of the star network, in which $J$ nodes can transmit simultaneously, but interfere with a central node $0$ in such a way that node $0$ cannot transmit while one of the other nodes does. One studies the impact of the log policy on these $J+1$ interacting communication nodes. A fluid scaling analysis of the network is derived with the scaling parameter $N$ being the norm of the initial state. It is shown that the asymptotic fluid behaviour of the system is a consequence of the evolution of the state of the network on a specific time scale $(N^t,\, t{\in}(0,1))$. The main result is that, on this time scale and under appropriate conditions, the state of a node with index $j\geq 1$ is of the order of $N^{a_j(t)}$, with $0{\leq}a_j(t){<}1$, where $t\mapsto a_j(t)$ is a piecewise linear function. Convergence results on the fluid time scale and a stability property are derived as a consequence of this study.

math.PR↗

arXiv2016

A Stochastic Analysis of Autoregulation of Gene Expression

This paper analyzes, in the context of a prokaryotic cell, the stochastic variability of the number of proteins when there is a control of gene expression by an autoregulation scheme. The goal of this work is to estimate the efficiency of the regulation to limit the fluctuations of the number of copies of a given protein. The autoregulation considered in this paper relies mainly on a negative feedback: the proteins are repressors of their own gene expression. The efficiency of a production process without feedback control is compared to a production process with an autoregulation of the gene expression assuming that both of them produce the same average number of proteins. The main characteristic used for the comparison is the standard deviation of the number of proteins at equilibrium. With a Markovian representation and a simple model of repression, we prove that, under a scaling regime, the repression mechanism follows a Hill repression scheme with an hyperbolic control. An explicit asymptotic expression of the variance of the number of proteins under this regulation mechanism is obtained. Simulations are used to study other aspects of autoregulation such as the rate of convergence to equilibrium of the production process and the case where the control of the production process of proteins is achieved via the inhibition of mRNAs.

q-bio.MN↗

arXiv2016

Analysis of an offloading scheme for data centers in the framework of fog computing

In the context of fog computing, we consider a simple case when data centers are installed at the edge of the network and assume that if a request arrives at an overloaded data center, then it is forwarded to a neighboring data center with some probability. Data centers are assumed to have a large number of servers and that traffic at some of them is causing saturation. In this case the other data centers may help to cope with this saturation regime by accepting some of the rejected requests. Our aim is to qualitatively estimate the gain achieved via cooperation between neighboring data centers. After proving some convergence results, related to the scaling limits of loss systems, for the process describing the number of free servers at both data centers, we show that the performance of the system can be expressed in terms of the invariant distribution of a random walk in the quarter plane. By using and developing existing results in the technical literature, explicit formulas for the blocking rates of such a system are derived.

cs.NI↗

arXiv2016

Asymptotics of Stochastic Protein Assembly Models

Self-assembly of proteins is a biological phenomenon which gives rise to spontaneous formation of amyloid fibrils or polymers. The starting point of this phase, called nucleation exhibits an important variability among replicated experiments.To analyse the stochastic nature of this phenomenon, one of the simplest models considers two populations of chemical components: monomers and polymerised monomers. Initially there are only monomers. There are two reactions for the polymerization of a monomer: either two monomers collide to combine into two polymerised monomers or a monomer is polymerised after the encounter of a polymerised monomer. It turns out that this simple model does not explain completely the variability observed in the experiments. This paper investigates extensions of this model to take into account other mechanisms of the polymerization process that may have impact an impact on fluctuations.The first variant consists in introducing a preliminary conformation step to take into account the biological fact that, before being polymerised, a monomer has two states, regular or misfolded. Only misfolded monomers can be polymerised so that the fluctuations of the number of misfolded monomers can be also a source of variability of the number of polymerised monomers. The second variant represents the reaction rate $α$ of spontaneous formation of a polymer as of the order of $N^{-ν}$, with $ν$ some positive constant. First and second order results for the starting instant of nucleation are derived from these limit theorems. The proofs of the results rely on a study of a stochastic averaging principle for a model related to an Ehrenfest urn model, and also on a scaling analysis of a population model.

q-bio.BM↗

arXiv2015

Analysis of Large Unreliable Stochastic Networks

In this paper a stochastic model of a large distributed system where users' files are duplicated on unreliable data servers is investigated. Due to a server breakdown, a copy of a file can be lost, it can be retrieved if another copy of the same file is stored on other servers. In the case where no other copy of a given file is present in the network, it is definitively lost. In order to have multiple copies of a given file, it is assumed that each server can devote a fraction of its processing capacity to duplicate files on other servers to enhance the durability of the system. A simplified stochastic model of this network is analyzed. It is assumed that a copy of a given file is lost at some fixed rate and that the initial state is optimal: each file has the maximum number $d$ of copies located on the servers of the network. Due to random losses, the state of the network is transient and all files will be eventually lost. As a consequence, a transient $d$-dimensional Markov process $(X(t))$ with a unique absorbing state describes the evolution this network. By taking a scaling parameter $N$ related to the number of nodes of the network. a scaling analysis of this process is developed. The asymptotic behavior of $(X(t))$ is analyzed on time scales of the type $t\mapsto N^p t$ for $0\leq p\leq d{-}1$. The paper derives asymptotic results on the decay of the network: Under a stability assumption, the main results state that the critical time scale for the decay of the system is given by $t\mapsto N^{d-1}t$. When the stability condition is not satisfied, it is shown that the state of the network converges to an interesting local equilibrium which is investigated. As a consequence it sheds some light on the role of the key parameters $λ$, the duplication rate and $d$, the maximal number of copies, in the design of these systems.

math.PR↗

arXiv2015

Insights into the variability of nucleated amyloid polymerization by a minimalistic model of stochastic protein assembly

Self-assembly of proteins into amyloid aggregates is an important biological phenomenon associated with human diseases such as Alzheimer's disease. Amyloid fibrils also have potential applications in nano-engineering of biomaterials. The kinetics of amyloid assembly show an exponential growth phase preceded by a lag phase, variable in duration as seen in bulk experiments and experiments that mimic the small volumes of cells. Here, to investigate the origins and the properties of the observed variability in the lag phase of amyloid assembly currently not accounted for by deterministic nucleation dependent mechanisms, we formulate a new stochastic minimal model that is capable of describing the characteristics of amyloid growth curves despite its simplicity. We then solve the stochastic differential equations of our model and give mathematical proof of a central limit theorem for the sample growth trajectories of the nucleated aggregation process. These results give an asymptotic description for our simple model, from which closed form analytical results capable of describing and predicting the variability of nucleated amyloid assembly were derived. We also demonstrate the application of our results to inform experiments in a conceptually friendly and clear fashion. Our model offers a new perspective and paves the way for a new and efficient approach on extracting vital information regarding the key initial events of amyloid formation.

q-bio.CB↗

arXiv2015

A stochastic analysis of resource sharing with logarithmic weights

The paper investigates the properties of a class of resource allocation algorithms for communication networks: if a node of this network has $x$ requests to transmit, then it receives a fraction of the capacity proportional to $\log(1+x)$, the logarithm of its current load. A detailed fluid scaling analysis of such a network with two nodes is presented. It is shown that the interaction of several time scales plays an important role in the evolution of such a system, in particular its coordinates may live on very different time and space scales. As a consequence, the associated stochastic processes turn out to have unusual scaling behaviors. A heavy traffic limit theorem for the invariant distribution is also proved. Finally, we present a generalization to the resource sharing algorithm for which the $\log$ function is replaced by an increasing function. Possible generalizations of these results with $J>2$ nodes or with the function $\log$ replaced by another slowly increasing function are discussed.

math.PR↗

arXiv2015

On the dynamics of random neuronal networks

We study the mean-field limit and stationary distributions of a pulse-coupled network modeling the dynamics of a large neuronal assemblies. Our model takes into account explicitly the intrinsic randomness of firing times, contrasting with the classical integrate-and-fire model. The ergodicity properties of the Markov process associated to finite networks are investigated. We derive the limit in distribution of the sample path of the state of a neuron of the network when its size gets large. The invariant distributions of this limiting stochastic process are analyzed as well as their stability properties. We show that the system undergoes transitions as a function of the averaged connectivity parameter, and can support trivial states (where the network activity dies out, which is also the unique stationary state of finite networks in some cases) and self-sustained activity when connectivity level is sufficiently large, both being possibly stable.

math.PR↗

arXiv2014

Impatience in mobile networks and its application to data pricing

We consider in this paper an import Quality of Experience (QoE) indicator in mobile networks that is reneging of users due to impatience. We specifically consider a cell under heavy load conditions and compute the reneging probability by using a fluid limit analysis. By solving the fixed point equation, we obtain a new QoE perturbation metric quantifying the impact of reneging on the performance of the system. This metric is then used to devise a new pricing scheme accounting of reneging. We specifically propose several flavors of this pricing around the idea of having a flat rate for accessing the network and an elastic price related to the level of QoE perturbation induced by communications.

cs.NI↗