arXiv subjects

Romain Abraham

Publications and source records attributed to Romain Abraham.

At least 19 recordsLinked to original sources

arXiv2026

Coupling some conditioned L{\'e}vy trees with the Kesten tree

We consider locally compact L{\'e}vy trees conditioned to be large, with respect to different criterion: its height, its maximal ''size'' vertex and its total ''mass''. In the critical case, we provide a coupling with a truncated Kesten tree which then allows to directly prove the local convergence in distribution of the conditioned L{\'e}vy tree to be large towards the Kesten tree. We also consider the sub-critical and super-critical cases. In the former case the results can be partial, due to a possible condensation phenomenon which is outside the mathematical framework used in this paper.

math.PR↗

arXiv2026

Self-Supervised Super-Resolution for Sentinel-5P Hyperspectral Images

Sentinel-5P (S5P) plays a critical role in atmospheric monitoring; however, its spatial resolution limits fine-scale analysis. Existing super-resolution (SR) approaches rely on supervised learning with synthetic low-resolution (LR) data, since true high-resolution (HR) data do not exist, limiting their applicability to real observations. We propose a self-supervised hyperspectral SR framework for S5P that enables training without HR ground truth. The method combines Stein's Unbiased Risk Estimator (SURE) with an equivariant imaging constraint, incorporating the S5P degradation operator and noise statistics derived from signal-to-noise ratio (SNR) metadata. We also introduce depthwise separable convolution U-Net architectures designed for efficiency and spectral fidelity. The framework is evaluated in two settings: (i) LR-HR, where synthetic LR data are used for direct comparison with supervised learning, and (ii) GT-SHR, where super-resolved images surpass the native spatial resolution without HR reference. Results across multiple bands show that self-supervised models achieve performance comparable to supervised methods while maintaining strong consistency. Qualitative analysis shows improved spatial detail over bicubic interpolation, and validation with EMIT data confirms that reconstructed structures are physically meaningful. Code is available at https://github.com/hyamomar/Sentinel-5P-Super-Resolution/tree/main/self_supervised

cs.CV↗

arXiv2025

Local limits of conditioned marked Galton Watson trees

We consider a Galton-Watson tree where each node is marked independently of each others with a probability depending on its outdegree. We give a complete picture of the local convergence of critical or sub-critical marked Galton-Watson trees conditioned on having a large number of marks. In the critical and sub-critical generic case, the limit is a random marked tree with an infinite spine, named marked Kesten's tree. We focus also on the non-generic case, where the local limit is a random marked tree with a node with infinite out-degree. This case corresponds to the so-called marked condensation phenomenon.

math.PR↗

arXiv2025

Site Frequency Spectrum in stationary branching populations

This paper explores the Site Frequency Spectrum (SFS) in stationary branching populations. We derive estimates for the SFS associated with a sample from a continuous-state branching process conditioned to never go extinct, utilizing a quadratic branching mechanism. The genealogy of such processes is represented by a real tree with a semi-infinite branch, and we compute the expectation of the SFS under the infinitely-many-sites assumption as the sample size approaches infinity. Additionally, we present a continuum version of the SFS as a random point measure on the positive real line and compute the density of its expected measure explicitly. Finally, we derive estimates for the size of the clonal subpopulation carrying the same genotype as the most recent common ancestor of the whole population at a given time.

math.PR↗

arXiv2025

Depth Separable architecture for Sentinel-5P Super-Resolution

Sentinel-5P (S5P) satellite provides atmospheric measurements for air quality and climate monitoring. While the S5P satellite offers rich spectral resolution, it inherits physical limitations that restricts its spatial resolution. Super-resolution (SR) techniques can overcome these limitations and enhance the spatial resolution of S5P data. In this work, we introduce a novel SR model specifically designed for S5P data that have eight spectral bands with around 500 channels for each band. Our proposed S5-DSCR model relies on Depth Separable Convolution (DSC) architecture to effectively perform spatial SR by exploiting cross-channel correlations. Quantitative evaluation demonstrates that our model outperforms existing methods for the majority of the spectral bands. This work highlights the potential of leveraging DSC architecture to address the challenges of hyperspectral SR. Our model allows for capturing fine details necessary for precise analysis and paves the way for advancements in air quality monitoring as well as remote sensing applications.

eess.IV↗

arXiv2024

MyData: A Comprehensive Database of Mycetoma Tissue Microscopic Images for Histopathological Analysis

Mycetoma is a chronic and neglected inflammatory disease prevalent in tropical and subtropical regions. It can lead to severe disability and social stigma. The disease is classified into two types based on the causative microorganisms: eumycetoma (fungal) and actinomycetoma (bacterial). Effective treatment strategies depend on accurately identifying the causative agents. Current identification methods include molecular, cytological, and histopathological techniques, as well as grain culturing. Among these, histopathological techniques are considered optimal for use in endemic areas, but they require expert pathologists for accurate identification, which can be challenging in rural areas lacking such expertise. The advent of digital pathology and automated image analysis algorithms offers a potential solution. This report introduces a novel dataset designed for the automated detection and classification of mycetoma using histopathological images. It includes the first database of microscopic images of mycetoma tissue, detailing the entire pipeline from species distribution and patient sampling to acquisition protocols through histological procedures. The dataset consists of images from 142 patients, totalling 864 images, each annotated with binary masks indicating the presence of grains, facilitating both detection and segmentation tasks.

eess.IV↗

arXiv2024

Penalization of Galton Watson trees with marked vertices

We consider a Galton-Watson tree where each node is marked independently of each others with a probability depending on itsout-degree. Using a penalization method, we exhibit new martingales where the number of marks up to level n -- 1 appears. Then, we use these martingales to define new probability measures via a Girsanov transformation and describe the distribution of the random trees under these new probabilities.

math.PR↗

arXiv2023

Probability-graphons: Limits of large dense weighted graphs

We introduce probability-graphons which are probability kernels that generalize graphons to the case of weighted graphs. Probability-graphons appear as the limit objects to study sequences of large weighted graphs whose distribution of subgraph sampling converge. The edge-weights are taken from a general Polish space, which also covers the case of decorated graphs. Here, graphs can be either directed or undirected. Starting from a distance $d_m$ inducing the weak topology on measures, we define a cut distance on probability-graphons, making it a Polish space, and study the properties of this cut distance. In particular, we exhibit a tightness criterion for probability-graphons related to relative compactness in the cut distance. We also prove that under some conditions on the distance $d_m$, which are satisfied for some well-know distances like the Prohorov distance, and the Fortet-Mourier and Kantorovitch-Rubinstein norms, the topology induced by the cut distance on the spaceof probability-graphons is independent from the choice of $d_m$. Eventually, we prove that this topology coincides with the topology induced by the convergence in distribution of the sampled subgraphs.

cs.DM↗

arXiv2023

Conditioning Bienaym{é}-Galton-Watson trees to have large sub-populations

We study the local limit in distribution of Bienaym{é}-Galton-Watson trees conditioned on having large sub-populations. Assuming a generic and aperiodic condition on the offspring distribution, we prove the existence of a limit given by a Kesten's tree associated with a certain critical offspring distribution.

math.PR↗

arXiv2022

Conditioning (sub)critical L{é}vy trees by their maximal degree: Decomposition and local limit

We study the maximal degree of (sub)critical L{é}vy trees which arise as the scaling limits of Bienaym{é}-Galton-Watson trees. We determine the genealogical structure of large nodes and establish a Poissonian decomposition of the tree along those nodes. Furthermore, we make sense of the distribution of the L{é}vy tree conditioned to have a fixed maximal degree. In the case where the L{é}vy measure is diffuse, we show that the maximal degree is realized by a unique node whose height is exponentially distributed and we also prove that the conditioned L{é}vy tree can be obtained by grafting a L{é}vy forest on an independent size-biased L{é}vy tree with a degree constraint at a uniformly chosen leaf. Finally, we show that the L{é}vy tree conditioned on having large maximal degree converges locally to an immortal tree (which is the continuous analogue of the Kesten tree) in the critical case and to a condensation tree in the subcritical case. Our results are formulated in terms of the exploration process which allows to drop the Grey condition.

math.PR↗

arXiv2022

Brownian continuum random tree conditioned to be large

We consider a Feller diffusion (Zs, s $\ge$ 0) (with diffusion coefficient $\sqrt$ 2$\beta$ and drift $\theta$ $\in$ R) that we condition on {Zt = at}, where at is a deterministic function, and we study the limit in distribution of the conditioned process and of its genealogical tree as t $\rightarrow$ +$\infty$. When at does not increase too rapidly, we recover the standard size-biased process (and the associated genealogical tree given by the Kesten's tree). When at behaves as $\alpha$$\beta$ 2 t 2 when $\theta$ = 0 or as $\alpha$ e 2$\beta$|$\theta$|t when $\theta$ = 0, we obtain a new process whose distribution is described by a Girsanov transformation and equivalently by a SDE with a Poissonian immigration. Its associated genealogical tree is described by an infinite discrete skeleton (which does not satisfy the branching property) decorated with Brownian continuum random trees given by a Poisson point measure. As a by-product of this study, we introduce several sets of trees endowed with a Gromovtype distance which are of independent interest and which allow here to define in a formal and measurable way the decoration of a backbone with a family of continuum random trees.

math.PR↗

arXiv2020

Global Regime for General Additive Functionals of Conditioned Bienaym{é}-Galton-Watson Trees

We give an invariance principle for very general additive functionals of conditioned Bienaym{é}-Galton-Watson trees in the global regime when the offspring distribution lies in the domain of attraction of a stable distribution, the limit being an additive functional of a stable L{é}vy tree. This includes the case when the offspring distribution has finite variance (the L{é}vy tree being then the Brownian tree). We also describe, using an integral test, a phase transition for toll functions depending on the size and height.

math.PR↗

arXiv2020

Some properties of stationary continuous state branching processes

We consider the genealogical tree of a stationary continuous state branching process with immigration. For a sub-critical stable branching mechanism, we consider the genealogical tree of the extant population at some fixed time and prove that, up to a deterministic time-change, it is distributed as a continuous-time Galton-Watson process with immigration. We obtain similar results for a critical stable branching mechanism when only looking at immigrants arriving in some fixed time-interval. For a general sub-critical branching mechanism, we consider the number of individuals that give descendants in the extant population. The associated processes (forward or backward in time) are pure-death or pure-birth Markov processes, for which we compute the transition rates.

math.PR↗

arXiv2020

Exact simulation of the genealogical tree for a stationary branching population and application to the asymptotics of its total length

We consider a model of stationary population with random size given by a continuous state branching process with immigration with a quadratic branching mechanism. We give an exact elementary simulation procedure of the genealogical tree of $n$ individuals randomly chosen among the extant population at a given time. Then, we prove the convergence of the renormalized total length of this genealogical tree as $n$ goes to infinity, see also Pfaffelhuber, Wakolbinger and Weisshaupt (2011) in the context of a constant size population. The limit appears already in Bi and Delmas (2016) but with a different approximation of the full genealogical tree. The proof is based on the ancestral process of the extant population at a fixed time which was defined by Aldous and Popovic (2005) in the critical case.

math.PR↗

arXiv2018

Penalization of Galton-Watson processes

We apply the penalization technique introduced by Roynette, Vallois, Yor for Brownian motion to Galton-Watson processes with a penalizing function of the form $P (x)s^x$ where P is a polynomial of degree p and s $\in$ [0, 1]. We prove that the limiting martingales obtained by this method are most of the time classical ones, except in the super-critical case for s = 1 (or s $\rightarrow$ 1) where we obtain new martingales. If we make a change of probability measure with this martingale, we obtain a multi-type Galton-Watson tree with p distinguished infinite spines.

math.PR↗

arXiv2017

Asymptotic properties of expansive Galton-Watson trees

We consider a super-critical Galton-Watson tree whose non-degenerate offspring distribution has finite mean. We consider the random trees $τ$n distributed as $τ$ conditioned on the n-th generation, Zn, to be of size an $\in$ N. We identify the possible local limits of $τ$n as n goes to infinity according to the growth rate of an. In the low regime, the local limit $τ$ 0 is the Kesten tree, in the moderate regime the family of local limits, $τ$ $θ$ for $θ$ $\in$ (0, +$\infty$), is distributed as $τ$ conditionally on {W = $θ$}, where W is the (non-trivial) limit of the renormalization of Zn. In the high regime, we prove the local convergence towards $τ$ $\infty$ in the Harris case (finite support of the offspring distribution) and we give a conjecture for the possible limit when the offspring distribution has some exponential moments. When the offspring distribution has a fat tail, the problem is open. The proof relies on the strong ratio theorem for Galton-Watson processes. Those latter results are new in the low regime and high regime, and they can be used to complete the description of the (space-time) Martin boundary of Galton-Watson processes. Eventually, we consider the continuity in distribution of the local limits ($τ$ $θ$ , $θ$ $\in$ [0, $\infty$]).

math.PR↗

arXiv2017

Reversal property of the Brownian tree

We consider the Brownian tree introduced by Aldous and the associated Q-process which consists in an infinite spine on which are grafted independent Brownian trees. We present a reversal procedure on these trees that consists in looking at the tree downward from its top: the branching points becoming leaves and leaves becoming branching points. We prove that the distribution of the tree is invariant under this reversal procedure, which provides a better understanding of previous results from Bi and Delmas (2016).

math.PR↗

arXiv2017

Very fat geometric galton-watson trees

Let $τ$n be a random tree distributed as a Galton-Watson tree with geometric offspring distribution conditioned on {Zn = an} where Zn is the size of the n-th generation and (an, n $\in$ N *) is a deterministic positive sequence. We study the local limit of these trees $τ$n as n $\rightarrow$ $\infty$ and observe three distinct regimes: if (an, n $\in$ N *) grows slowly, the limit consists in an infinite spine decorated with finite trees (which corresponds to the size-biased tree for critical or subcritical offspring distributions), in an intermediate regime, the limiting tree is composed of an infinite skeleton (that does not satisfy the branching property) still decorated with finite trees and, if the sequence (an, n $\in$ N *) increases rapidly, a condensation phenomenon appears and the root of the limiting tree has an infinite number of offspring.

math.PR↗