J. C. Breton, C. Houdré
The purpose of these notes is to further complete our understanding of the stable concentration phenomenon, by obtaining the finite range behavior of $P(F-E[F]\geq x)$, with $F=f(X)$ where $f$ is a Lipschitz function and $X$ is a stable random vector or with $F$ a stochastic functional on the Poisson space equipped with a stable Lévy measure.
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Concentration of measure is studied, and obtained, for stable and related random vectors.
Concentration around a stable equilibrium for the non-autonomous $Φ_3^4$ model
针对三维环面上含时奇异随机偏微分方程，采用仿控分布框架，证明非自治 $\Phi_3^4$ 模型解在稳定平衡态附近满足高斯型尾部界。
We derive concentration inequalities for functions of the empirical measure of large random matrices with infinitely divisible entries and, in particular, stable ones. We also give concentration results for some other functionals of these random matrices, such as the largest eigenvalue or the largest singular value.
Concentration of measure for Graphon particle system
本文研究了图论粒子系统中的测度集中性，通过均场型相互作用和图论结构，推导了有限粒子近似在有限时间内的指数集中估计。
In this paper, we consider graphon particle systems with heterogeneous mean-field type interactions and the associated finite particle approximations. Under suitable growth (resp. convexity) assumptions, we obtain uniform-in-time concentration estimates, over finite (resp. infinite) time horizon, for the Wasserstein distance between the empirical measure and its limit, extending the work of Bolley--Guillin--Villani.
We give concentration bounds for martingales that are uniform over finite times and extend classical Hoeffding and Bernstein inequalities. We also demonstrate our concentration bounds to be optimal with a matching anti-concentration inequality, proved using the same method. Together these constitute a finite-time version of the law of the iterated logarithm, and shed light on the relationship between it and the central limit theorem.
We present classes of models in which particles are dropped on an arbitrary fixed finite connected graph, obeying adhesion rules with screening. We prove that there is an invariant distribution for the resulting height profile, and Gaussian concentration for functions depending on the paths of the profiles. As a corollary we obtain a law of large numbers for the maximum height. This describes the asymptotic speed with which the maximal height increases. The results incorporate the case of independent particle droppings but extend to droppings according to a driving Markov chain, and to droppin…
We consider additive functionals of Markov processes in continuous time with general (metric) state spaces. We derive concentration bounds for their exponential moments and moments of finite order. Applications include diffusions, interacting particle systems and random walks. The method is based on coupling estimates and not spectral theory, hence reversibility is not needed. We bound the exponential moments(or the moments of finite order) in terms of a so-called coupled function difference, which in turn is estimated using the generalized coupling time. Along the way we prove a general relat…
Concentration of Equilibria and Relative Instability in Disordered Non-Relaxational Dynamics
研究随机自治 ODEs 系统中平衡点的集中现象，证明当 p>9 时平衡点及稳定平衡点数量集中于其均值，并证实模型经历从相对到绝对不稳定性的转变。
We show an extension of Sanov's theorem on large deviations, controlling the tail probabilities of i.i.d. random variables with matching concentration and anti-concentration bounds. This result has a general scope, applies to samples of any size, and has a short information-theoretic proof using elementary techniques.
A mixture with varying concentrations is a modification of a finite mixture model in which the mixing probabilities (concentrations of mixture components) may be different for different observations. In the paper, we assume that the concentrations are known and the distributions of components are completely unknown. Nonparametric technique is proposed for testing hypotheses on functional moments of components.
We provide a simple, general argument to obtain improvements of concentration-type inequalities starting from improvements of their corresponding isoperimetric-type inequalities. We apply this argument to obtain robust improvements of the Brunn-Minkowski inequality (for Minkowski sums between generic sets and convex sets) and of the Gaussian concentration inequality. The former inequality is then used to obtain a robust improvement of the Riesz rearrangement inequality under certain natural conditions. These conditions are compatible with the applications to a finite-range nonlocal isoperimetr…
A Random-Walk Concentration Principle for Occupancy Processes on Finite Graphs
该研究提出有限图离散时间占位过程的两个随机游走集中定理，可应用于动态随机图等边过程，仅需利普希茨连续性，适用于稠密与稀疏图，能估计子图密度偏差。
We prove an exponential decay concentration inequality to bound the tail probability of the difference between the log-likelihood of discrete random variables on a finite alphabet and the negative entropy. The concentration bound we derive holds uniformly over all parameter values. The new result improves the convergence rate in an earlier result of Zhao (2020), from $(K^2\log K)/n=o(1)$ to $ (\log K)^2/n=o(1)$, where $n$ is the sample size and $K$ is the size of the alphabet. We further prove that the rate $(\log K)^2/n=o(1)$ is optimal. The results are extended to misspecified log-likelihood…
LLY Ricci Reweighting in Stochastic Block Models: Uniform Curvature Concentration and Finite-Horizon Tracking
本文提出利用LLY里奇曲率重新加权方法，在随机块模型中实现社区恢复，通过有限时间范围内的迭代重新加权提升块内连接性，并提供非渐进行扰界限和误聚类保证。
Given a finite group, we study the Gaussian series of the matrices in the image of its left regular representation. We propose such random matrices as a benchmark for improvements to the noncommutative Khintchine inequality, and we highlight an application to the matrix Spencer conjecture.
In this note, we show that the relative entropy of an empirical distribution of $n$ samples drawn from a set of size $k$ with respect to the true underlying distribution is exponentially concentrated around its expectation, with central moment generating function bounded by that of a gamma distribution with shape $2k$ and rate $n/2$. This improves on recent work of Bhatt and Pensia (arXiv 2021) on the same problem, who showed such a similar bound with an additional polylogarithmic factor of $k$ in the shape, and also confirms a recent conjecture of Mardia et al. (Information and Inference 2020…
Gaussian concentration, integral probability metrics, and coupling functionals for infinite lattice systems
本文提出一种运输-熵框架，用于研究无限积空间上的高斯集中性不等式，揭示了运输成本与熵之间的关系，并在有限体积内证明了积分概率度量与耦合函数的等价性。
Recent research in both the experimental and mathematical communities has focused on biochemical interaction systems that satisfy an "absolute concentration robustness" (ACR) property. The ACR property was first discovered experimentally when, in a number of different systems, the concentrations of key system components at equilibrium were observed to be robust to the total concentration levels of the system. Followup mathematical work focused on deterministic models of biochemical systems and demonstrated how chemical reaction network theory can be utilized to explain this robustness. Later m…
Tight lower bounds for anti-concentration of Rademacher sums and Tomaszewski's counterpart problem
本文证明有限Rademacher和的反集中性概率下界为7/32，证实Hitczenko与Kwapień1994年猜想，改进已有结果，还完全确定相关下确界函数，证实Lowther猜想并部分回答Keller和Klein的问题。
We show that the moment generating function of the Kullback-Leibler divergence (relative entropy) between the empirical distribution of $n$ independent samples from a distribution $P$ over a finite alphabet of size $k$ (i.e. a multinomial distribution) and $P$ itself is no more than that of a gamma distribution with shape $k - 1$ and rate $n$. The resulting exponential concentration inequality becomes meaningful (less than 1) when the divergence $\varepsilon$ is larger than $(k-1)/n$, whereas the standard method of types bound requires $\varepsilon > \frac{1}{n} \cdot \log{\binom{n+k-1}{k-1}} …
Thermodynamic Concentration Inequalities: Controlling Uncertainty in Finite-Time and Small-Sample Thermodynamic Inference
该研究推导了热力学推断中有限时间、小样本下广义电流偏离概率的非渐近上界，揭示了平衡内外涨落差异，得到逆热力学不确定性关系，构造了小数据下的非渐近置信区间并用于检测细致平衡破缺。
Finite sample bounds for barycenter estimation in geodesic spaces
研究测地线空间中给定独立同分布数据时估计分布重心的问题，在特定条件下建立期望和高概率的有限样本误差界，推广相关不等式并为重心计算算法提供统计保证。
Barycenters (aka Fréchet means) were introduced in statistics in the 1940's and popularized in the fields of shape statistics and, later, in optimal transport and matrix analysis. They provide the most natural extension of linear averaging to non-Euclidean geometries, which is perhaps the most basic and widely used tool in data science. In various setups, their asymptotic properties, such as laws of large numbers and central limit theorems, have been established, but their non-asymptotic behaviour is still not well understood. In this work, we prove finite sample concentration inequalities (na…
Finitary coding and Gaussian concentration for random fields
本文研究了通过有限码编码得到的随机场的高斯集中性不等式，探讨了集中性与编码结构的关系。主要结果表明，在有限二阶矩条件下，有限码保持高斯集中性。应用结果到Gibbs测度、Z^d上的马尔可夫随机场及一维过程，得出经典晶格模型中高斯集中性的充分必要条件。
We consider models of Bayesian inference of signals with vectorial components of finite dimensionality. We show that, under a proper perturbation, these models are replica symmetric in the sense that the overlap matrix concentrates. The overlap matrix is the order parameter in these models and is directly related to error metrics such as minimum mean-square errors. Our proof is valid in the optimal Bayesian inference setting. This means that it relies on the assumption that the model and all its hyper-parameters are known so that the posterior distribution can be written exactly. Examples of i…
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