arXiv subjects

Lizhen Ji

Publications and source records attributed to Lizhen Ji.

At least 19 recordsLinked to original sources

arXiv2024

Polyhedral Horofunction Compactification as a Polyhedral Ball

In this paper we answer positively a question raised by Kapovich and Leeb in a paper titled "Finsler bordifications of symmetric and certain locally symmetric spaces". Specifically, we show that for a finite-dimensional vector space with a polyhedral norm, its horofunction compactification is homeomorphic to the dual unit ball of the norm by an explicit map. To prove this, we establish a criterion for converging sequences in the horofunction compactification and generalize the basic notion of the moment map in the theory of toric varieties.

math.GT↗

arXiv2023

Information Theoretical Importance Sampling Clustering

A current assumption of most clustering methods is that the training data and future data are taken from the same distribution. However, this assumption may not hold in most real-world scenarios. In this paper, we propose an information theoretical importance sampling based approach for clustering problems (ITISC) which minimizes the worst case of expected distortions under the constraint of distribution deviation. The distribution deviation constraint can be converted to the constraint over a set of weight distributions centered on the uniform distribution derived from importance sampling. The objective of the proposed approach is to minimize the loss under maximum degradation hence the resulting problem is a constrained minimax optimization problem which can be reformulated to an unconstrained problem using the Lagrange method. The optimization problem can be solved by both an alternative optimization algorithm or a general optimization routine by commercially available software. Experiment results on synthetic datasets and a real-world load forecasting problem validate the effectiveness of the proposed model. Furthermore, we show that fuzzy c-means is a special case of ITISC with the logarithmic distortion, and this observation provides an interesting physical interpretation for fuzzy exponent $m$.

stat.ML↗

arXiv2023

An information-theoretic learning model based on importance sampling

A crucial assumption underlying the most current theory of machine learning is that the training distribution is identical to the test distribution. However, this assumption may not hold in some real-world applications. In this paper, we develop a learning model based on principles of information theory by minimizing the worst-case loss at prescribed levels of uncertainty. We reformulate the empirical estimation of the risk functional and the distribution deviation constraint based on the importance sampling method. The objective of the proposed approach is to minimize the loss under maximum degradation and hence the resulting problem is a minimax problem which can be converted to an unconstrained minimum problem using the Lagrange method with the Lagrange multiplier $T$. We reveal that the minimization of the objective function under logarithmic transformation is equivalent to the minimization of the p-norm loss with $p=\frac{1}{T}$. We applied the proposed model to the face verification task on Racial Faces in the Wild datasets and showed that the proposed model performs better under large distribution deviations.

stat.ML↗

arXiv2020

Towards Reducing Severe Defocus Spread Effects for Multi-Focus Image Fusion via an Optimization Based Strategy

Multi-focus image fusion (MFF) is a popular technique to generate an all-in-focus image, where all objects in the scene are sharp. However, existing methods pay little attention to defocus spread effects of the real-world multi-focus images. Consequently, most of the methods perform badly in the areas near focus map boundaries. According to the idea that each local region in the fused image should be similar to the sharpest one among source images, this paper presents an optimization-based approach to reduce defocus spread effects. Firstly, a new MFF assessmentmetric is presented by combining the principle of structure similarity and detected focus maps. Then, MFF problem is cast into maximizing this metric. The optimization is solved by gradient ascent. Experiments conducted on the real-world dataset verify superiority of the proposed model. The codes are available at https://github.com/xsxjtu/MFF-SSIM.

cs.CV↗

arXiv2019

Metrics and compactifications of Teichmüller spaces of flat tori

Using the identification of the symmetric space $\mathrm{SL}(n,\mathbb{R})/\mathrm{SO}(n)$ with the Teichmüller space of flat $n$-tori of unit volume, we explore several metrics and compactifications of these spaces, drawing inspiration both from Teichmüller theory and symmetric spaces. We define and study analogs of the Thurston, Teichmüller, and Weil-Petersson metrics. We show the Teichmüller metric is a symmetrization of the Thurston metric, which is a polyhedral Finsler metric, and the Weil-Petersson metric is the Riemannian metric of $\mathrm{SL}(n,\mathbb{R})/\mathrm{SO}(n)$ as a symmetric space. We also construct a Thurston-type compactification using measured foliations on $n$-tori, and show that the horofunction compactification with respect to the Thurston metric is isomorphic to it, as well as to a minimal Satake compactification.

math.DG↗

arXiv2017

A new modular characterization of the hyperbolic plane

We develop a natural and geometric way to realize the hyperbolic plane as the moduli space of marked genus 1 Riemann surfaces. To do so, a metric is defined on the Teichmüller space of the torus, inspired by Thurston's Lipschitz metric for the case of hyperbolic surfaces. Based on extremal Lipschitz maps, the Teichmüller space of the torus with this new metric is shown to be isometric to the hyperbolic plane under the usual identification. This also gives a new way to recover the complex-analytic Teichmüller metric via metric geometry on the underlying surfaces. Along the way, we prove a few results about this metric analogous to Thurston's Lipschitz metric in the case of hyperbolic surfaces, and analogous to the Teichmüller metric.

math.GT↗

arXiv2017

Toric Varieties vs. Horofunction Compactifications of Polyhedral Norms

We establish a natural and geometric 1-1 correspondence between projective toric varieties of dimension $n$ and horofunction compactifications of $\mathbb{R}^n$ with respect to rational polyhedral norms. For this purpose, we explain a topological model of toric varieties. Consequently, toric varieties in algebraic geometry, normed spaces in convex analysis, and horofunction compactifications in metric geometry are directly and explicitly related.

math.MG↗

arXiv2016

Universal moduli spaces of Riemann surfaces

We construct a moduli space for Riemann surfaces that is universal in the sense that it represents compact Riemann surfaces of any finite genus. This moduli space is stratifed according to genus, and it carries a metric and a measure that induce a Riemannian metric and a finite volume measure on each stratum. Applications to the Plateau-Douglas problem for minimal surfaces of varying genus and to the partition function of Bosonic string theory are outlined. The construction starts with a universal moduli space of Abelian varieties. This space carries a structure of an infinite dimensional locally symmetric space which is of interest in its own right. The key to our construction of the universal moduli space then is the Torelli map that assigns to every Riemann surface its Jacobian and its extension to the Satake-Baily-Borel compactifications.

math.AG↗

arXiv2016

Actions of the Absolute Galois Group

We review some ideas of Grothendieck and others on actions of the absolute Galois group Γ Q of Q (the automorphism group of the tower of finite extensions of Q), related to the geometry and topology of surfaces (mapping class groups, Teichm{ü}ller spaces and moduli spaces of Riemann surfaces). Grothendieck's motivation came in part from his desire to understand the absolute Galois group. But he was also interested in Thurston's work on surfaces, and he expressed this in his Esquisse d'un programme, his R{é}coltes et semailles and on other occasions. He introduced the notions of dessin d'enfant, Teichm{ü}ller tower, and other related objects, he considered the actions of Γ Q on them or on their etale fundamental groups, and he made conjectures on some natural homomorphisms between the absolute Galois group and the automor-phism groups (or outer automorphism groups) of these objects. We mention several ramifications of these ideas, due to various authors. We also report on the works of Sullivan and others on nonlinear actions of Γ Q , in particular in homotopy theory. The final version of this paper will appear as a chapter in Volume VI of the Handbook of Teichm{ü}ller theory. This volume is dedicated to the memory of Alexander Grothendieck.

math.GT↗

arXiv2016

On Grothendieck's tame topology

Grothendieck's Esquisse d'un programme is often referred to for the ideas it contains on dessins d'enfants, the Teichm{ü}ller tower, and the actions of the absolute Galois group on these objects or their etale fundamental groups. But this program contains several other important ideas. In particular, motivated by surface topology and moduli spaces of Riemann surfaces, Grothendieck calls there for a recasting of topology, in order to make it fit to the objects of semialgebraic and semianalytic geometry, and in particular to the study of the Mumford-Deligne compactifications of moduli spaces. A new conception of manifold, of submanifold and of maps between them is outlined. We review these ideas in the present chapter, because of their relation to the theory of moduli and Te-ichm{ü}ller spaces. We also mention briefly the relations between Grothendieck's ideas and earlier theories developed by Whitney, Lojasiewicz and Hironaka and especially Thom, and with the more recent theory of o-minimal structures. The final version of this paper will appear as a chapter in Volume VI of the Handbook of Teichm{ü}ller theory. This volume is dedicated to the memory of Alexander Grothendieck.

math.GT↗

arXiv2016

On Grothendieck's construction of Teichmüller space

In his 1944 paper Veränderliche Riemannsche Flächen , Teichmüller defined a structure of complex manifold on the set of isomorphism classes of marked closed Riemann surfaces of genus g. The complex manifold he obtained is the space called today Teichmüller space. In the same paper, Teichmüller introduced the so-called universal Teichmüller curve -- a space over Teichmüller space where the fiber above each point is a Riemann surface representing that point. In fact, Teichmüller proved the existence of the Teichmüller curve as a space of Riemann surfaces parametrized by an analytic space, with an existence and uniqueness theorem establishing this analytic structure. This result was later reformulated and proved by Grothendieck in a series of ten lectures he gave at Cartan's seminar in 1960-1961. In his approach , Grothendieck replaced Teichmüller's explicit parameters by a general construction of fiber bundles whose base is an arbitrary analytic space. This work on Teichmüller space led him to recast the bases of analytic geometry using the language of categories and functors. In Grothendieck's words, the Teichmüller curve becomes a space representing a functor from the category of analytic spaces into the category of sets. In this survey, we comment on Grothendieck's series of lectures. The survey is primarily addressed to low-dimensional topologists and geometers. In presenting Grothendieck's results, we tried to explain or rephrase in more simple terms some notions that are usually expressed in the language of algebraic geometry. However, it is not possible to short-circuit the language of categories and functors. The survey is also addressed to those algebraic geometers who wish to know how the notion of moduli space evolved in connection with Teichmüller theory. Explaining the origins of mathematical ideas contributes in dispensing justice to their authors and it usually renders the theory that is surveyed more attractive. The final version of this paper will appear as a chapter in Volume VI of the Handbook of Teichmüller theory. This volume is dedicated to the memory of Alexander Grothendieck.

math.GT↗

arXiv2016

On the early history of moduli and Teichm{ü}ller spaces

We survey some major contributions to Riemann's moduli space and Teichm{ü}ller space. Our report has a historical character, but the stress is on the chain of mathematical ideas. We start with the introduction of Riemann surfaces, and we end with the discovery of some of the basic structures of Riemann's moduli space and Teichm{ü}ller space. We point out several facts which seem to be unknown to many algebraic geometers and analysts working in the theory. The period we are interested in starts with Riemann, in 1851, and ends in the early 1960s, when Ahlfors and Bers confirmed that Teichm{ü}ller's results were correct.This paper was written for the book "Lipman Bers, a life in Mathematics," edited by Linda Keen , Irwin Kra and Rubi Rodriguez (Amercian Mathematical Society, 2015). It is dedicated to the memory of Lipman Bers who was above all a complex analyst and spent a large part of his life and energy working on the analytic structure of Teichm{ü}ller space. His work on analysis is nevertheless inseparable from geometry and topology. In this survey, we highlight the relations and the logical dependence between this work and the works of Riemann, Poincar{é}, Klein, Brouwer, Siegel, Teichm{ü}ller, Weil, Grothendieck and others. We explain the motivation behind the ideas. In doing so, we point out several facts which seem to be unknown to many Teichm{ü}ller theorists.

math.HO↗

arXiv2015

The fundamental group of reductive Borel-Serre and Satake compactifications

Let $G$ be an almost simple, simply connected algebraic group defined over a number field $k$, and let $S$ be a finite set of places of $k$ including all infinite places. Let $X$ be the product over $v\in S$ of the symmetric spaces associated to $G(k_v)$, when $v$ is an infinite place, and the Bruhat-Tits buildings associated to $G(k_v)$, when $v$ is a finite place. The main result of this paper is an explicit computation of the fundamental group of the reductive Borel-Serre compactification of $Γ\backslash X$, where $Γ$ is an $S$-arithmetic subgroup of $G$. In the case that $Γ$ is neat, we show that this fundamental group is isomorphic to $Γ/EΓ$, where $EΓ$ is the subgroup generated by the elements of $Γ$ belonging to unipotent radicals of $k$-parabolic subgroups. Analogous computations of the fundamental group of the Satake compactifications are made. It is noteworthy that calculations of the congruence subgroup kernel $C(S,G)$ yield similar results.

math.GT↗

arXiv2014

Well-rounded equivariant deformation retracts of Teichmüller spaces

In this paper, we construct spines, i.e., $\Mod_g$-equivariant deformation retracts, of the Teichmüller space $\T_g$ of compact Riemann surfaces of genus $g$. Specifically, we define a $\Mod_g$-stable subspace $S$ of positive codimension and construct an intrinsic $\Mod_g$-equivariant deformation retraction from $\T_g$ to $S$. As an essential part of the proof, we construct a canonical $\Mod_g$-deformation retraction of the Teichmüller space $\T_g$ to its thick part $\T_g(\varepsilon)$ when $\varepsilon$ is sufficiently small. These equivariant deformation retracts of $\T_g$ give cocompact models of the universal space $\underline{E}\Mod_g$ for proper actions of the mapping class group $\Mod_g$. These deformation retractions of $\T_g$ are motivated by the well-rounded deformation retraction of the space of lattices in $\R^n$. We also include a summary of results and difficulties of an unpublished paper of Thurston on a potential spine of the Teichmüller space.

math.GT↗

arXiv2013

$L^p$ spectrum and heat dynamics of locally symmetric spaces of higher rank

The aim of this paper is to study the spectrum of the $L^p$ Laplacian and the dynamics of the $L^p$ heat semigroup on non-compact locally symmetric spaces of higher rank. Our work here generalizes previously obtained results in the setting of locally symmetric spaces of rank one to higher rank spaces. Similarly as in the rank one case, it turns out that the $L^p$ heat semigroup on $M$ has a certain chaotic behavior if $p\in(1,2)$ whereas for $p\geq 2$ such a chaotic behavior never occurs.

math.DG↗

arXiv2012

Complete invariant geodesic metrics on outer spaces and Jacobian varieties of tropical curves

Let $\mathrm{Out}(F_n)$ be the outer automorphism group of the free group $F_n$. It acts properly on the outer space $X_n$ of marked metric graphs, which is a finite-dimensional infinite simplicial complex with some simplicial faces missing. In this paper, we construct complete geodesic metrics and complete piecewise smooth Riemannian metrics on $X_n$ which are invariant under $\mathrm{Out}(F_n)$. One key ingredient is the identification of metric graphs with tropical curves and the use of the tropical Jacobian map from the moduli space of tropical curves to the moduli space of principally polarized tropical abelian varieties.

math.GT↗

arXiv2012

A commentary on Teichmüller's paper "Veränderliche Riemannsche Flächen" (Variable Riemann Surfaces)

This is a commentary on Teichmüllers' paper "Veränderliche Riemannsche Flächen" (Variable Riemann Surfaces), published in 1944. This paper is the last one that Teichmüller wrote on the problem of moduli. At most places the paper contains ideas and no technical details. The author presents a completely new approach to Teichmüller space, compared to the approach he took in his first seminal paper "Extremale quasikonforme Abbildungen und quadratische Differentiale" and its sequel "Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Flächen" in which he completed some of the the results stated in the former. In the paper "Extremale quasikonforme ...", Teichmüller led the foundations of what we call today Teichmüller theory (but without the complex structure), defining its metric and introducing in that theory the techniques of quasiconformal mappings and of quadratic differentials as essential tools. In the present paper, the approach is more abstract, through complex analytic geometry. Teichmüller space, equipped with its complex-analytic structure, is characterized here by a certain universal property. Among the other ideas and results contained in the paper, we mention the following: (1) The existence and uniqueness of the universal Teichmüller curve, rediscovered later on by Ahlfors and by Bers. At the same time, this introduced the first fibre bundle over Teichmüller space. (2) The proof of the fact that the automorphisms group of the univeral Teichmüller curve is the extended mapping class group. (3) The idea of a fine moduli space. (4) The idea of using the period map to define a complex structure on Teichmüller space.

math.GT↗

arXiv2012

Spectral theory for the Weil-Petersson Laplacian on the Riemann moduli space

We study the spectral geometric properties of the scalar Laplace-Beltrami operator associated to the Weil-Petersson metric $g_{\mathrm{WP}}$ on $\mathcal M_γ$, the Riemann moduli space of surfaces of genus $γ> 1$. This space has a singular compactification with respect to $g_{\mathrm{WP}}$, and this metric has crossing cusp-edge singularities along a finite collection of simple normal crossing divisors. We prove first that the scalar Laplacian is essentially self-adjoint, which then implies that its spectrum is discrete. The second theorem is a Weyl asymptotic formula for the counting function for this spectrum.

math.DG↗