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Songzi Li

Publications and source records attributed to Songzi Li.

At least 19 recordsLinked to original sources
arXiv2026

The invariance principle for inhomogeneous Diophantine approximations

We establish the central limit theorem and the invariance principle for the inhomogeneous Diophantine approximations. The proof employs the cumulant method, which was developed by Bj\"orklund and Gorodnik to prove the central limit theorem in the homogeneous setting. Our approach also relies on the effective mixing of expanding translates for high-order correlations on the affine lattice space, extending the previous result by Kim.

math.PR↗
arXiv2024

Langevin deformation for R\'enyi entropy on Wasserstein space over Riemannian manifolds

We introduce the Langevin deformation for the R\'enyi entropy on the $L^2$-Wasserstein space over $\mathbb{R}^n$ or a Riemannian manifold, which interpolates between the porous medium equation and the Benamou-Brenier geodesic flow on the $L^2$-Wasserstein space and can be regarded as the compressible Euler equations for isentropic gas with damping. We prove the $W$-entropy-information formulae and the the rigidity theorems for the Langevin deformation for the R\'enyi entropy on the Wasserstein space over complete Riemannian manifolds with non-negative Ricci curvature or CD$(0, m)$-condition. Moreover, we prove the monotonicity of the Hamiltonian and the convexity of the Lagrangian along the Langevin deformation of flows. Finally, we prove the convergence of the Langevin deformation for the R\'enyi entropy as $c\rightarrow 0$ and $c\rightarrow \infty$ respectively. Our results are new even in the case of Euclidean spaces and compact or complete Riemannian manifolds with non-negative Ricci curvature.

math.PR↗
arXiv2024

The periodic Lorentz gas in the Boltzmann-Grad limit: the free path length and the K-S entropy

We study the periodic Lorentz gas in the Boltzmann-Grad limit, whose convergence was rigorously established in the seminal work of Marklof-Str\"ombergsson. Extending the two dimensional results of Boca-Zaharescu to higher dimensions, we present a more detailed description of this convergence. More precisely, we derive the asymptotic formula of the distribution function of the free path length; and we explicitly compute the constant in the asymptotic formula for the Kolmogorov-Sinai (K-S) entropy of the billiard map.

math.PR↗
arXiv2023

AI in Pharma for Personalized Sequential Decision-Making: Methods, Applications and Opportunities

In the pharmaceutical industry, the use of artificial intelligence (AI) has seen consistent growth over the past decade. This rise is attributed to major advancements in statistical machine learning methodologies, computational capabilities and the increased availability of large datasets. AI techniques are applied throughout different stages of drug development, ranging from drug discovery to post-marketing benefit-risk assessment. Kolluri et al. provided a review of several case studies that span these stages, featuring key applications such as protein structure prediction, success probability estimation, subgroup identification, and AI-assisted clinical trial monitoring. From a regulatory standpoint, there was a notable uptick in submissions incorporating AI components in 2021. The most prevalent therapeutic areas leveraging AI were oncology (27%), psychiatry (15%), gastroenterology (12%), and neurology (11%). The paradigm of personalized or precision medicine has gained significant traction in recent research, partly due to advancements in AI techniques \cite{hamburg2010path}. This shift has had a transformative impact on the pharmaceutical industry. Departing from the traditional "one-size-fits-all" model, personalized medicine incorporates various individual factors, such as environmental conditions, lifestyle choices, and health histories, to formulate customized treatment plans. By utilizing sophisticated machine learning algorithms, clinicians and researchers are better equipped to make informed decisions in areas such as disease prevention, diagnosis, and treatment selection, thereby optimizing health outcomes for each individual.

stat.ME↗
arXiv2023

Liouville theorem for $V$-harmonic maps under non-negative $(m, V)$-Ricci curvature for non-positive $m$

Let $V$ be a $C^1$-vector field on an $n$-dimensional complete Riemannian manifold $(M, g)$. We prove a Liouville theorem for $V$-harmonic maps satisfying various growth conditions from complete Riemannian manifolds with non-negative $(m, V)$-Ricci curvature for $m\in\,[\,-\infty,\,0\,]\,\cup\,[\,n,\,+\infty\,]$ into Cartan-Hadam\-ard manifolds, which extends Cheng's Liouville theorem proved S.~Y.~Cheng for sublinear growth harmonic maps from complete Riemannian manifolds with non-negative Ricci curvature into Cartan-Hadamard manifolds. We also prove a Liouville theorem for $V$-harmonic maps from complete Riemannian manifolds with non-negative $(m, V)$-Ricci curvature for $m\in\,[\,-\infty,\,0\,]\,\cup\,[\,n,\,+\infty\,]$ into regular geodesic balls of Riemannian manifolds with positive upper sectional curvature bound, which extends the results of Hildebrandt-Jost-Wideman and Choi. Our probabilistic proof of Liouville theorem for several growth $V$-harmonic maps into Hadamard manifolds enhances an incomplete argument by Stafford. Our results extend the results due to Chen-Jost-Qiu\cite{ChenJostQiu} and Qiu\cite{Qiu} in the case of $m=+\infty$ on the Liouville theorem for bounded $V$-harmonic maps from complete Riemannian manifolds with non-negative $(\infty, V)$-Ricci curvature into regular geodesic balls of Riemannian manifolds with positive sectional curvature upper bound. Finally, we establish a connection between the Liouville property of $V$-harmonic maps and the recurrence property of $\Delta_V$-diffusion processes on manifolds. Our results are new even in the case $V=\nabla f$ for $f\in C^2(M)$.

math.DG↗
arXiv2021

$W$-entropy formulas and Langevin deformation of flows on Wasserstein space over Riemannian manifolds

We introduce Perelman's $W$-entropy and prove the $W$-entropy formula along the geodesic flow on the $L^2$-Wasserstein space over compact Riemannian manifolds equipped with Otto's Riemannian metric, which allows us to recapture a previous result due to Lott and Villani on the displacement convexity of $s{\rm Ent}+ns\log s$ on $P^\infty_2(M)$ over Riemannian manifolds with non-negative Ricci curvature. To better understand the similarity between the $W$-entropy formula for the geodesic flow on the Wasserstein space and the $W$-entropy formula for the heat equation of the Witten Laplacian on the underlying manifolds, we introduce the Langevin deformation of flows on the Wasserstein space over Riemannian manifold, which interpolates the gradient flow and the geodesic flow on the Wasserstein space over Riemannian manifolds, and can be regarded as the potential flow of the compressible Euler equation with damping on manifolds. We prove the existence, uniqueness and regularity of the Langevin deformation on the Wasserstein space over the Euclidean space and compact Riemannian manifolds, and prove the convergence of the Langevin deformation for $c\rightarrow 0$ and $c\rightarrow \infty$ respectively. We prove an analogue of the Perelman type $W$-entropy formula along the Langevin deformation on the Wasserstein space on Riemannian manifolds. A rigidity theorem is proved for the $W$-entropy for the geodesic flow, and a rigidity model is also provided for the Langevin deformation on the Wasserstein space over complete Riemannian manifolds with the $CD(0, m)$-condition.

math.PR↗
arXiv2021

The convergence rates for the superdiffusion in the Boltzmann-Grad limit of the periodic Lorentz gas

In this article, we obtain the rates of convergence for superdiffusion in the Boltzmann-Grad limit of the periodic Lorentz gas, which is one of the fundamental models to study diffusions in deterministic systems. In their seminal work, Marklof and Str\"ombergsson proved the Boltzmann-Grad limit of the periodic Lorentz gas~\cite{M-Sannals2}, and then Marklof and T\'oth established a superdiffusive central limit theorem in large time for the Boltzmann-Grad limit~\cite{M-T16}. Based on their work, we apply Stein's method to derive the convergence rates for the superdiffusion in the Boltzmann-Grad limit of the periodic Lorentz gas. For the discrete time displacement the rate of convergence in Wasserstein distance is obtained, while in the context of the continuous time displacement the result is presented for the Berry-Essen type bound.

math.PR↗
arXiv2020

On the Renyi entropy power and the Gagliardo-Nirenberg-Sobolev inequality on Riemannian manifolds

In this paper, we prove the concavity of the Renyi entropy power for nonlinear diffusion equation (NLDE) associated with the Laplacian and the Witten Laplacian on compact Riemannian manifolds with non-negative Ricci curvature or $CD(0,m)$-condition and on compact manifolds equipped with time dependent metrics and potentials. Our results can be regarded as natural extensions of a result due to Savaré and Toscani \cite{ST} on the concavity of the Renyi entropy for NLDE on Euclidean spaces. Moreover, we prove that the rigidity models for the Renyi entropy power are the Einstein or quasi-Einstein manifolds and a special $(K,m)$-Ricci flow with Hessian solitons. Inspired by Lu-Ni-Vazquez-Villani \cite{LNVV}, we prove the Aronson-Benilan estimates for NLDE on compact Riemannian manifolds with $CD(0,m)$-condition. We also prove the NIW formula which indicates an intrinsic relationship between the second order derivative of the Renyi entropy power $N_p$, the $p$-th Fisher information $I_p$ and the time derivative of the $W$-entropy associated with NLDE. Finally, we prove the entropy isoperimetric inequality for the Renyi entropy power and the Gagliardo-Nirenberg-Sobolev inequality on complete Riemannian manifolds with non-negative Ricci curvature or $CD(0, m)$-condition and maximal volume growth condition.

math.DG↗
arXiv2018

$W$-entropy, super Perelman Ricci flows and $(K, m)$-Ricci solitons

In this paper, we prove the characterization of the $(K, \infty)$-super Perelman Ricci flows by various functional inequalities and gradient estimate for the heat semigroup generated by the Witten Laplacian on manifolds equipped with time dependent metrics and potentials. As a byproduct, we derive the Hamilton type dimension free Harnack inequality on manifolds with $(K, \infty)$-super Perelman Ricci flows. Based on a new second order differential inequality on the Boltzmann-Shannon entropy for the heat equation of the Witten Laplacian, we introduce a new $W$-entropy quantity and prove its monotonicity for the heat equation of the Witten Laplacian on complete Riemannian manifolds with the $CD(K, \infty)$-condition and on compact manifolds with $(K, \infty)$-super Perelman Ricci flows. Our results characterize the $(K, \infty)$-Ricci solitons and the $(K, \infty)$-Perelman Ricci flows. We also prove a second order differential entropy inequality on $(K, m)$-super Ricci flows, which can be used to characterize the $(K, m)$-Ricci solitons and the $(K, m)$-Ricci flows. Finally, we give a probabilistic interpretation of the $W$-entropy for the heat equation of the Witten Laplacian on manifolds with the $CD(K, m)$-condition.

math.DG↗
arXiv2017

K-groups: A Generalization of K-means Clustering

We propose a new class of distribution-based clustering algorithms, called k-groups, based on energy distance between samples. The energy distance clustering criterion assigns observations to clusters according to a multi-sample energy statistic that measures the distance between distributions. The energy distance determines a consistent test for equality of distributions, and it is based on a population distance that characterizes equality of distributions. The k-groups procedure therefore generalizes the k-means method, which separates clusters that have different means. We propose two k-groups algorithms: k-groups by first variation; and k-groups by second variation. The implementation of k-groups is partly based on Hartigan and Wong's algorithm for k-means. The algorithm is generalized from moving one point on each iteration (first variation) to moving $m$ $(m > 1)$ points. For univariate data, we prove that Hartigan and Wong's k-means algorithm is a special case of k-groups by first variation. The simulation results from univariate and multivariate cases show that our k-groups algorithms perform as well as Hartigan and Wong's k-means algorithm when clusters are well-separated and normally distributed. Moreover, both k-groups algorithms perform better than k-means when data does not have a finite first moment or data has strong skewness and heavy tails. For non--spherical clusters, both k-groups algorithms performed better than k-means in high dimension, and k-groups by first variation is consistent as dimension increases. In a case study on dermatology data with 34 features, both k-groups algorithms performed better than k-means.

stat.ME↗
arXiv2017

$W$-entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds

In this survey paper, we give an overview of our recent works on the study of the $W$-entropy for the heat equation associated with the Witten Laplacian on super-Ricci flows and the Langevin deformation on Wasserstein space over Riemannian manifolds. Inspired by Perelman's seminal work on the entropy formula for the Ricci flow, we prove the $W$-entropy formula for the heat equation associated with the Witten Laplacian on $n$-dimensional complete Riemannian manifolds with the $CD(K, m)$-condition, and the $W$-entropy formula for the heat equation associated with the time dependent Witten Laplacian on $n$-dimensional compact manifolds equipped with a $(K, m)$-super Ricci flow, where $K\in \mathbb{R}$ and $m\in [n, \infty]$. Furthermore, we prove an analogue of the $W$-entropy formula for the geodesic flow on the Wasserstein space over Riemannian manifolds. Our result recaptures an important result due to Lott and Villani on the displacement convexity of the Boltzmann-Shannon entropy on Riemannian manifolds with non-negative Ricci curvature. To better understand the similarity between above two $W$-entropy formulas, we introduce the Langevin deformation of geometric flows on the cotangent bundle over the Wasserstein space and prove an extension of the $W$-entropy formula for the Langevin deformation. Finally, we make a discussion on the $W$-entropy for the Ricci flow from the point of view of statistical mechanics and probability theory.

math.DG↗
arXiv2017

Hamilton differential Harnack inequality and $W$-entropy for Witten Laplacian on Riemannian manifolds

In this paper, we prove the Hamilton differential Harnack inequality for positive solutions to the heat equation of the Witten Laplacian on complete Riemannian manifolds with the $CD(-K, m)$-condition, where $m\in [n, \infty)$ and $K\geq 0$ are two constants. Moreover, we introduce the $W$-entropy and prove the $W$-entropy formula for the fundamental solution of the Witten Laplacian on complete Riemannian manifolds with the $CD(-K, m)$-condition and on compact manifolds equipped with $(-K, m)$-super Ricci flows.

math.DG↗
arXiv2017

Matrix Dirichlet processes

Matrix Dirichlet processes, in reference to their reversible measure, appear in a natural way in many different models in probability. Applying the language of diffusion operators and the method of boundary equations, we describe Dirichlet processes on the simplex and provide two models of matrix Dirichlet processes, which can be realized by various projections, through the Brownian motion on the special unitary group, the polar decomposition of complex matrices and also through Wishart processes.

math.PR↗
arXiv2016

Harnack inequalities and $W$-entropy formula for Witten Laplacian on Riemannian manifolds with $K$-super Perelman Ricci flow

In this paper, we prove logarithmic Sobolev inequalities and derive the Hamilton Harnack inequality for the heat semigroup of the Witten Laplacian on complete Riemannian manifolds equipped with $K$-super Perelman Ricci flow. We establish the $W$-entropy formula for the heat equation of the Witten Laplacian and prove a rigidity theorem on complete Riemannian manifolds satisfying the $CD(K, m)$ condition, and extend the $W$-entropy formula to time dependent Witten Laplacian on compact Riemannian manifolds with $(K, m)$-super Perelman Ricci flow, where $K\in \mathbb{R}$ and $m\in [n, \infty]$ are two constants. Finally, we prove the Li-Yau and the Li-Yau-Hamilton Harnack inequalities for positive solutions to the heat equation $\partial_t u=Lu$ associated to the time dependent Witten Laplacian on compact or complete manifolds equipped with variants of the $(K, m)$-super Ricci flow.

math.DG↗
arXiv2015

Dyson processes on the octonion algebra

We consider Brownian motion on symmetric matrices of octonions, and study the law of the spectrum. Due to the fact that the octonion algebra is nonassociative, the dimension of the matrices plays a special role. We provide two specific models on octonions, which give some indication of the relation between the multiplicity of eigenvalues and the exponent in the law of the spectrum.

math.PR↗
arXiv2015

On the Law of Large Numbers for the empirical measure process of Generalized Dyson Brownian motion

We study the generalized Dyson Brownian motion (GDBM) of an interacting $N$-particle system with logarithmic Coulomb interaction and general potential $V$. Under reasonable condition on $V$, we prove the existence and uniqueness of strong solution to SDE for GDBM. We then prove that the family of the empirical measures of GDBM is tight on $\mathcal {C}([0,T],\mathscr{P}(\mathbb{R}))$ and all the large $N$ limits satisfy a nonlinear McKean-Vlasov equation. Inspired by previous works due to Biane and Speicher, Carrillo, McCann and Villani, we prove that the McKean-Vlasov equation is indeed the gradient flow of the Voiculescu free entropy on the Wasserstein space of probability measures over $\mathbb{R}$. Using the optimal transportation theory, we prove that if $V"\geq K$ for some constant $K\in \mathbb{R}$, the McKean-Vlasov equation has a unique weak solution. This proves the Law of Large Numbers and the propagation of chaos for the empirical measures of GDBM. Finally, we prove the longtime convergence of the McKean-Vlasov equation for $C^2$-convex potentials $V$.

math.PR↗
arXiv2014

$W$-entropy formula for the Witten Laplacian on manifolds with time dependent metrics and potentials

In this paper, we develop a new approach to prove the $W$-entropy formula for the Witten Laplacian via warped product on Riemannian manifolds and give a natural geometric interpretation of a quantity appeared in the $W$-entropy formula. Then we prove the $W$-entropy formula for the Witten Laplacian on compact Riemannian manifolds with time dependent metrics and potentials, and derive the $W$-entropy formula for the backward heat equation associated with the Witten Laplacian on compact Riemannian manifolds equipped with Lott's modified Ricci flow. We also extend our results to complete Riemannian manifolds with negative $m$-dimensional Bakry-Emery Ricci curvature, and to compact Riemannian manifolds with $K$-super $m$-dimensional Bakry-Emery Ricci flow. As application, we prove that the optimal logarithmic Sobolev constant on compact manifolds equipped with the $K$-super $m$-dimensional Bakry-Emery Ricci flow is decreasing in time.

math.DG↗