arXiv subjects

Sayan Banerjee

Publications and source records attributed to Sayan Banerjee.

At least 55 records · Page 3Linked to original sources

arXiv2022

The Inert Drift Atlas Model

Consider a massive (inert) particle impinged from above by N Brownian particles that are instantaneously reflected upon collision with the inert particle. The velocity of the inert particle increases due to the influence of an external Newtonian potential (e.g. gravitation) and decreases in proportion to the total local time of collisions with the Brownian particles. This system models a semi-permeable membrane in a fluid having microscopic impurities (Knight (2001)). We study the long-time behavior of the process $(V,\mathbf{Z})$, where $V$ is the velocity of the inert particle and $\mathbf{Z}$ is the vector of gaps between successive particles ordered by their relative positions. The system is not hypoelliptic, not reversible, and has singular form interactions. Thus the study of stability behavior of the system requires new ideas. We show that this process has a unique stationary distribution that takes an explicit product form which is Gaussian in the velocity component and Exponential in the other components. We also show that convergence in total variation distance to the stationary distribution happens at an exponential rate. We further obtain certain law of large numbers results for the particle locations and intersection local times.

math.PR↗

arXiv2022

Fluctuation Bounds for Continuous Time Branching Processes and Evolution of Growing Trees With a Change Point

We consider dynamic random trees constructed using an attachment function $f : \mathbb{N} \to \mathbb{R}_+$ where, at each step of the evolution, a new vertex attaches to an existing vertex $v$ in the current tree with probability proportional to $f$(degree(v)). We explore the effect of a change point in the system; the dynamics are initially driven by a function f until the tree reaches size $τ(n) \in (0,n)$, at which point the attachment function switches to another function, $g$, until the tree reaches size $n$. Two change point time scales are considered, namely the standard model where $τ(n) = γn$, and the quick big bang model where $τ(n) = n^γ$, for some $0 < γ< 1$. In the former case, we obtain deterministic approximations for the evolution of the empirical degree distribution (EDF) in sup-norm and use these to devise a provably consistent non-parametric estimator for the change point $γ$. In the latter case, we show that the effect of pre-change point dynamics asymptotically vanishes in the EDF, although this effect persists in functionals such as the maximal degree. Our proofs rely on embedding the discrete time tree dynamics in an associated (time) inhomogeneous continuous time branching process (CTBP). In the course of proving the above results, we develop novel mathematical techniques to analyze both homogeneous and inhomogeneous CTBPs and obtain rates of convergence for functionals of such processes, which are of independent interest.

math.PR↗

arXiv2022

Long Time Behavior of Finite and Infinite Dimensional Reflected Brownian Motions

This article presents a review of some old and new results on the long time behavior of reflected diffusions. First, we present a summary of prior results on construction, ergodicity and geometric ergodicity of reflected diffusions in the positive orthant $\mathbb{R}^d_+$, $d \in \mathbb{N}$. The geometric ergodicity results, although very general, usually give implicit convergence rates due to abstract couplings and Lyapunov functions used in obtaining them. This leads us to some recent results on an important subclass of reflected Brownian motions (RBM) (constant drift and diffusion coefficients and oblique reflection at boundaries), known as the Harrison-Reiman class, where explicit rates of convergence are obtained as functions of the system parameters and underlying dimension. In addition, sufficient conditions on system parameters of the RBM are provided under which local convergence to stationarity holds at a `dimension-free' rate, that is, for any fixed $k \in \mathbb{N}$, the rate of convergence of the $k$-marginal to equilibrium does not depend on the dimension of the whole system. Finally, we study the long time behavior of infinite dimensional rank-based diffusions, including the well-studied infinite Atlas model. The gaps between the ordered particles evolve as infinite dimensional RBM and this gap process has uncountably many explicit product form stationary distributions. Sufficient conditions for initial configurations to lie in the weak domain of attraction of the various stationary distributions are provided. Finally, it is shown that, under conditions, all of these explicit stationary distributions are extremal (equivalently, ergodic) and, in some sense, the only product form invariant probability distributions. Proof techniques involve a pathwise analysis of RBM using explicit synchronous and mirror couplings and constructing Lyapunov functions.

math.PR↗

arXiv2022

Anomalous Fano factor as a signature of Bogoliubov Fermi surfaces

Noise spectroscopy is a key technique to investigate the nature and dynamics of charge carriers in superconductors. The recently discovered superconducting hybrids with Bogoliubov Fermi surfaces exhibit a particularly intriguing and rich charge dynamics, as their charge carriers consist of both Cooper pairs and an extensive number of Bogoliubov quasiparticles. Motivated by this, we compute the noise spectra of Bogoliubov Fermi surfaces and identify their key signatures in the differential conductance and the Fano factor. Specifically, we consider a semiconductor/superconductor hybrid device with an in-plane magnetic field, which exhibits several Bogoliubov Fermi surfaces. The number and orientation of the Bogoliubov Fermi surfaces in this device can be readily controlled by the applied magnetic field, which in turn alters the noise signal. In particular, we find that the Fano factor exhibits a reduced value, substantially lower than two, whenever the charge dynamics is governed by a large number of Bogoliubov quasiparticles. Using experimentally relevant parameters, we make a number of specific predictions for the noise spectra, that can be used as direct evidence of Bogoliubov Fermi surfaces. In particular, we find that the Fano factor as a function of magnetic field and spin-orbit coupling exhibits characteristic discontinuities at the transition lines that separate phases with different number of Bogoliubov Fermi surfaces.

cond-mat.supr-con↗

arXiv2022

Dimension-free local convergence and perturbations for reflected Brownian motions

We describe and analyze a class of positive recurrent reflected Brownian motions (RBMs) in $\mathbb{R}^d_+$ for which local statistics converge to equilibrium at a rate independent of the dimension $d$. Under suitable assumptions on the reflection matrix, drift and diffusivity coefficients, dimension-independent stretched exponential convergence rates are obtained by estimating contractions in an underlying weighted distance between synchronously coupled RBMs. We also study the Symmetric Atlas model as a first step in obtaining dimension-independent convergence rates for RBMs not satisfying the above assumptions. By analyzing a pathwise derivative process and connecting it to a random walk in a random environment, we obtain polynomial convergence rates for the gap process of the Symmetric Atlas model started from appropriate perturbations of stationarity.

math.PR↗

arXiv2022

Domains of attraction of invariant distributions of the infinite Atlas model

The infinite Atlas model describes a countable system of competing Brownian particles where the lowest particle gets a unit upward drift and the rest evolve as standard Brownian motions. The stochastic process of gaps between the particles in the infinite Atlas model does not have a unique stationary distribution and in fact for every $a \ge 0$, $π_a := \bigotimes_{i=1}^{\infty} \operatorname{Exp}(2 + ia)$ is a stationary measure for the gap process. We say that an initial distribution of gaps is in the weak domain of attraction of the stationary measure $π_a$ if the time averaged laws of the stochastic process of the gaps, when initialized using that distribution, converge to $π_a$ weakly in the large time limit. We provide general sufficient conditions on the initial gap distribution of the Atlas particles for it to lie in the weak domain of attraction of $π_a$ for each $a\ge 0$. The cases $a=0$ and $a>0$ are qualitatively different as is seen from the analysis and the sufficient conditions that we provide. Proofs are based on the analysis of synchronous couplings, namely, couplings of the ranked particle systems started from different initial configurations, but driven using the same set of Brownian motions.

math.PR↗

arXiv2021

PageRank Asymptotics on Directed Preferential Attachment Networks

We characterize the tail behavior of the distribution of the PageRank of a uniformly chosen vertex in a directed preferential attachment graph and show that it decays as a power law with an explicit exponent that is described in terms of the model parameters. Interestingly, this power law is heavier than the tail of the limiting in-degree distribution, which goes against the commonly accepted {\em power law hypothesis}. This deviation from the power law hypothesis points at the structural differences between the inbound neighborhoods of typical vertices in a preferential attachment graph versus those in static random graph models where the power law hypothesis has been proven to hold (e.g., directed configuration models and inhomogeneous random digraphs). In addition to characterizing the PageRank distribution of a typical vertex, we also characterize the explicit growth rate of the PageRank of the oldest vertex as the network size grows.

math.PR↗

arXiv2021

Heavy Traffic Scaling Limits for shortest remaining processing time queues with heavy tailed processing time distributions

We study a single server queue operating under the shortest remaining processing time (SRPT) scheduling policy; that is, the server preemptively serves the job with the shortest remaining processing time first. In this work we are interested in studying the asymptotic behavior of suitably scaled measure-valued state descriptors that describe the evolution of a sequence of SRPT queuing systems. Gromoll, Kruk, and Puha (2011) have studied this problem under diffusive scaling. In the setting where the processing time distributions have unbounded support, under suitable conditions, they show that the diffusion scaled measures converge in distribution to the process that is identically zero. In Puha (2015) for the setting where the processing time distributions have unbounded support and light tails, a non-standard scaling of the queue length process is shown to give rise to a form of state space collapse that results in a nonzero limit. In the current work we consider the case where processing time distributions have finite second moments and regularly varying tails. We show that the measure valued process, under a non-standard scaling, converges in distribution in the space of paths of measures. In sharp contrast with previous results, there is no state space collapse. Nevertheless, the description of the limit is simple and given explicitly in terms of a certain $\mathbb{R}_+$ valued random field which is determined from a single Brownian motion. Along the way we establish convergence of suitably scaled workload and queue length processes. We also show that as the tail of the distribution of job processing times becomes lighter in an appropriate fashion, the difference between the limiting queue length process and the limiting workload process converges to zero, thereby approaching the behavior of state space collapse.

math.PR↗

arXiv2021

Root finding algorithms and persistence of Jordan centrality in growing random trees

We consider models of growing random trees $\{\mathcal{T}_f(n):n\geq 1\}$ with model dynamics driven by an attachment function $f:\mathbb{Z}_+\to \mathbb{R}_+$. At each stage a new vertex enters the system and connects to a vertex $v$ in the current tree with probability proportional to $f(\text{degree}(v))$. The main goal of this study is to understand the performance of root finding algorithms. A large body of work (e.g. the work of Bubeck, Devroye and Lugosi or Jog and Loh) has emerged in the last few years in using techniques based on the Jordan centrality measure and its variants to develop root finding algorithms. Given an unlabelled unrooted tree, one computes the Jordan centrality for each vertex in the tree and for a fixed budget $K$ outputs the optimal $K$ vertices (as measured by Jordan centrality). Under general conditions on the attachment function $f$, we derive necessary and sufficient bounds on the budget $K(ε)$ in order to recover the root with probability at least $1-ε$. For canonical examples such as linear preferential attachment and uniform attachment, these general results give matching upper and lower bounds for the budget. We also prove persistence of the optimal $K$ Jordan centers for any $K$, i.e. the existence of an almost surely finite random time $n^*$ such that for $n \geq n^*$ the identity of the $K$-optimal Jordan centers in $\{\mathcal{T}_f(n):n\geq n^*\}$ does not change, thus describing robustness properties of this measure. Key technical ingredients in the proofs of independent interest include sufficient conditions for the existence of exponential moments for limits of (appropriately normalized) continuous time branching processes within which the models $\{\mathcal{T}_f(n):n\geq n^*\}$ can be embedded, as well as rates of convergence results to these limits.

math.PR↗

arXiv2021

Persistence of hubs in growing random networks

We consider models of evolving networks $\{\mathcal{G}_n:n\geq 0\}$ modulated by two parameters: an attachment function $f:\mathbb{N}_0\to\mathbb{R}_+$ and a (possibly random) attachment sequence $\{m_i:i\geq 1\}$. Starting with a single vertex, at each discrete step $i\geq 1$ a new vertex $v_i$ enters the system with $m_i\geq 1$ edges which it sequentially connects to a pre-existing vertex $v\in \mathcal{G}_{i-1}$ with probability proportional to $f(\operatorname{degree}(v))$. We consider the problem of emergence of persistent hubs: existence of a finite (a.s.) time $n^*$ such that for all $n\geq n^*$ the identity of the maximal degree vertex (or in general the $K$ largest degree vertices for $K\geq 1$) does not change. We obtain general conditions on $f$ and $\{m_i:i\geq 1\}$ under which a persistent hub emerges, and also those under which a persistent hub fails to emerge. In the case of lack of persistence, for the specific case of trees ($m_i\equiv 1$ for all $i$), we derive asymptotics for the maximal degree and the index of the maximal degree vertex (time at which the vertex with current maximal degree entered the system) to understand the movement of the maximal degree vertex as the network evolves. A key role in the analysis is played by an inverse rate weighted martingale constructed from a continuous time embedding of the discrete time model. Asymptotics for this martingale, including concentration inequalities and moderate deviations, play a major role in the analysis of the model.

math.PR↗

arXiv2020

Directed Variational Cross-encoder Network for Few-shot Multi-image Co-segmentation

In this paper, we propose a novel framework for multi-image co-segmentation using class agnostic meta-learning strategy by generalizing to new classes given only a small number of training samples for each new class. We have developed a novel encoder-decoder network termed as DVICE (Directed Variational Inference Cross Encoder), which learns a continuous embedding space to ensure better similarity learning. We employ a combination of the proposed DVICE network and a novel few-shot learning approach to tackle the small sample size problem encountered in co-segmentation with small datasets like iCoseg and MSRC. Furthermore, the proposed framework does not use any semantic class labels and is entirely class agnostic. Through exhaustive experimentation over multiple datasets using only a small volume of training data, we have demonstrated that our approach outperforms all existing state-of-the-art techniques.

cs.CV↗

arXiv2020

Rates of convergence to equilibrium for Potlatch and Smoothing processes

We analyze the local and global smoothing rates of the smoothing process and obtain convergence rates to stationarity for the dual process known as the potlatch process. For general finite graphs, we connect the smoothing and convergence rates to the spectral gap of the associated Markov chain. We perform a more detailed analysis of these processes on the torus. Polynomial corrections to the smoothing rates are obtained. They show that local smoothing happens faster than global smoothing. These polynomial rates translate to rates of convergence to stationarity in $L^2$-Wasserstein distance for the potlatch process on $\mathbb{Z}^d$.

math.PR↗

arXiv2020

Ergodicity and steady state analysis for Interference Queueing Networks

We analyze an interacting queueing network on $\mathbb{Z}^d$ that was introduced in Sankararaman-Baccelli-Foss (2019) as a model for wireless networks. We show that the marginals of the minimal stationary distribution have exponential tails. This is used to furnish asymptotics for the maximum steady state queue length in growing boxes around the origin. We also establish a decay of correlations which shows that the minimal stationary distribution is strongly mixing, and hence, ergodic with respect to translations on $\mathbb{Z}^d$.

math.PR↗

arXiv2020

NENET: An Edge Learnable Network for Link Prediction in Scene Text

Text detection in scenes based on deep neural networks have shown promising results. Instead of using word bounding box regression, recent state-of-the-art methods have started focusing on character bounding box and pixel-level prediction. This necessitates the need to link adjacent characters, which we propose in this paper using a novel Graph Neural Network (GNN) architecture that allows us to learn both node and edge features as opposed to only the node features under the typical GNN. The main advantage of using GNN for link prediction lies in its ability to connect characters which are spatially separated and have an arbitrary orientation. We show our concept on the well known SynthText dataset, achieving top results as compared to state-of-the-art methods.

cs.LG↗

arXiv2020

DeFraudNet:End2End Fingerprint Spoof Detection using Patch Level Attention

In recent years, fingerprint recognition systems have made remarkable advancements in the field of biometric security as it plays an important role in personal, national and global security. In spite of all these notable advancements, the fingerprint recognition technology is still susceptible to spoof attacks which can significantly jeopardize the user security. The cross sensor and cross material spoof detection still pose a challenge with a myriad of spoof materials emerging every day, compromising sensor interoperability and robustness. This paper proposes a novel method for fingerprint spoof detection using both global and local fingerprint feature descriptors. These descriptors are extracted using DenseNet which significantly improves cross-sensor, cross-material and cross-dataset performance. A novel patch attention network is used for finding the most discriminative patches and also for network fusion. We evaluate our method on four publicly available datasets:LivDet 2011, 2013, 2015 and 2017. A set of comprehensive experiments are carried out to evaluate cross-sensor, cross-material and cross-dataset performance over these datasets. The proposed approach achieves an average accuracy of 99.52%, 99.16% and 99.72% on LivDet 2017,2015 and 2011 respectively outperforming the current state-of-the-art results by 3% and 4% for LivDet 2015 and 2011 respectively.

cs.CV↗

arXiv2020

Inert drift system in a viscous fluid: Steady state asymptotics and exponential ergodicity

We analyze a system of stochastic differential equations describing the joint motion of a massive (inert) particle in a viscous fluid in the presence of a gravitational field and a Brownian particle impinging on it from below, which transfers momentum proportional to the local time of collisions. We study the long-time fluctuations of the velocity of the inert particle and the gap between the two particles, and we show convergence in total variation to the stationary distribution is exponentially fast. We also produce matching upper and lower bounds on the tails of the stationary distribution and show how these bounds depend on the system parameters. A renewal structure for the process is established, which is the key technical tool in proving the mentioned results.

math.PR↗

arXiv2019

Parameter and dimension dependence of convergence rates to stationarity for Reflecting Brownian Motions

We obtain rates of convergence to stationarity in L^1-Wasserstein distance for a d-dimensional reflected Brownian motion (RBM) in the nonnegative orthant that are explicit in the dimension and the system parameters. The results are then applied to a class of RBMs considered in Blanchet-Chen (2016) and to rank-based diffusions including the Atlas model. In both cases, we obtain explicit rates and bounds on relaxation times. In the first case we improve the relaxation time estimates of O(d^4(log d)^2) obtained in Blanchet-Chen (2016) to O((log d)^2). In the latter case, we give the first results on explicit parameter and dimension dependent rates under the Wasserstein distance. The proofs do not require an explicit form for the stationary measure or reversibility of the process with respect to this measure, and cover settings where these properties are not available. In the special case of the standard Atlas model, we obtain a bound on the relaxation time of O(d^6(log d)^2).

math.PR↗

arXiv2019

Join-the-Shortest Queue Diffusion Limit in Halfin-Whitt Regime: Sensitivity on the Heavy-traffic Parameter

Consider a system of $N$ parallel single-server queues with unit-exponential service time distribution and a single dispatcher where tasks arrive as a Poisson process of rate $λ(N)$. When a task arrives, the dispatcher assigns it to one of the servers according to the Join-the-Shortest Queue (JSQ) policy. Eschenfeldt and Gamarnik (Math. Oper. Res., 43(3):867-886, 2018) identified a novel limiting diffusion process that arises as the weak-limit of the appropriately scaled occupancy measure of the system under the JSQ policy in the Halfin-Whitt regime, where $(N - λ(N)) / \sqrt{N} \to β> 0$ as $N \to \infty$. The analysis of this diffusion goes beyond the state of the art techniques, and even proving its ergodicity is non-trivial, and was left as an open question. Recently, exploiting a generator expansion framework via the Stein's method, Braverman (arXiv:1801.05121, 2018) established its exponential ergodicity, and adapting a regenerative approach, Banerjee and Mukherjee (Ann. Appl. Probab., 29(2):1262-1309, 2018) analyzed the tail properties of the stationary distribution and path fluctuations of the diffusion. However, the analysis of the bulk behavior of the stationary distribution, viz., the moments, remained intractable until this work. In this paper, we perform a thorough analysis of the bulk behavior of the stationary distribution of the diffusion process, and discover that it exhibits different qualitative behavior, depending on the value of the heavy-traffic parameter $β$. Moreover, we obtain precise asymptotic laws of the centered and scaled steady state distribution, as $β$ tends to 0 and $\infty$. Of particular interest, we also establish a certain intermittency phenomena in the $β\to \infty$ regime and a surprising distributional convergence result in the $β\to 0$ regime.

math.PR↗