arXiv subjects

Hông Vân Lê

Publications and source records attributed to Hông Vân Lê.

At least 19 recordsLinked to original sources

arXiv2026

Azumaya descent and étale-local cyclotomic comparison: a noncommutative filtration

Bhatt-Morrow-Scholze motivic filtration on topological Hochschild homology ($THH$) is extended to Azumaya algebras over quasisyntomic rings. We prove the Azumaya descent theorem: $THH$ of such an algebra is equivalent, as an underlying spectrum, to $THH$ of its center, normalized on $π_0$ by the reduced trace instead of the unit map. This comparison $Φ_A$ is not shown to be étale-locally cyclotomic. But what is cyclotomic étale-locally is a separate Morita trace, agreeing with $Φ_A$ only on $π_0$. The resulting transported filtration is exactly the center BMS filtration, well defined up to the automorphism ambiguity of $Φ_A$. It is insensitive to the Brauer class, reflected instead by $K$-theoretic and module-categorical pairings $THH$ does not see. We establish the analogous statement for derived Azumaya algebras over connective $\mathbb{E}_\infty$-rings, under a connectivity hypothesis.

math.FA↗

arXiv2026

Estimating conditional expectation

In this paper, we consider the problem of estimating conditional expectations as an ill-posed inverse problem. We propose a solution based on a generalization of Vapnik's theorem \cite[Theorem 7.2]{Vapnik1998} for solving stochastic ill-posed problems in Hilbert spaces. As an application, we derive a new upper bound for sample errors of conditional expectation estimation.

math.ST↗

arXiv2026

Batch learning equals online learning in Bayesian supervised learning

In this paper we study Bayesian supervised learning models proposed by Lê in \cite{Le2025}. Using functoriality of probabilistic morphisms, we prove that sequential and batch Bayesian inversions coincide in supervised learning models with conditionally independent (possibly non-i.i.d.) data \cite{Le2025}. This equivalence holds without domination or discreteness assumptions on sampling operators. We derive a recursive formula for posterior predictive distributions, which reduces to the Kalman filter in Gaussian process regression. For Souslin label spaces $\mathcal{Y}$ and arbitrary input sets $\mathcal{X}$, we characterize probability measures on $\mathcal{P}(\mathcal{Y})^{\mathcal{X}}$ via projective systems, generalizing Orbanz \cite{Orbanz2011}. We revisit MacEachern's Dependent Dirichlet Processes (DDP) \cite{MacEachern2000} using copula-based constructions \cite{BJQ2012} and show how to compute posterior predictive distributions in universal Bayesian supervised models with DDP priors.

math.ST↗

arXiv2026

Minimal Unital Cyclic $C_\infty$-Algebras and the Real and Rational Homotopy Type of Closed Manifolds

Using the notion of isotopy modulo $k$, for $k\in\mathbb N^+$, we introduce a stratification on the set of minimal $C_\infty$-algebra enhancements of a finite-dimensional graded commutative algebra $H^*$. We prove that two such enhancements are $C_\infty$-isotopic if and only if they are isotopic modulo $k$ for every $k\in\mathbb N^+$. We define obstruction sets governing the extension of an isotopy modulo $k$ to an isotopy modulo $(k+1)$ and establish their generalized additivity. We prove that if $M$ is a closed $(r-1)$-connected manifold of dimension $n\leq \ell(r-1)+2,\, r\geq2,\, \ell\geq 4$, then its real and rational homotopy types are determined by its cohomology algebra $H^*(M;\mathbb F)$ together with the isotopy class modulo $(\ell-2)$ of the corresponding minimal unital cyclic $C_\infty$-algebra enhancement, for $\mathbb F=\mathbb R$ and $\mathbb F=\mathbb Q$, respectively. Combining this obstruction theory with the Hodge homotopy method introduced in \cite{FKLS2021} and further developed in \cite{FiorenzaLe2025}, we give a new proof of a theorem of Crowley--Nordström \cite{CN}: if $M$ is a closed $(r-1)$-connected manifold of dimension $4r-1$ with $b_r(M)\leq3$, and there exists a class $φ\in H^{2r-1}(M;\mathbb R)$ such that multiplication by $φ$ induces an isomorphism $H^r(M;\mathbb R)\longrightarrow H^{3r-1}(M;\mathbb R),\,x\longmapstoφ\smile x$, then $M$ is intrinsically formal. Finally, we prove a borderline extension of a vanishing theorem of Fiorenza--Lê: if an $(r-1)$-connected Poincaré DGCA over $\mathbb Q$ of degree $n\leq5r-2$ admits a Hodge homotopy and satisfies $b^r\leq2$, then the operations of its transferred minimal unital cyclic $C_\infty$-algebra vanish in every arity $k\geq4$.

math.AT↗

arXiv2026

Empirical Hodge Laplacians: Spectral Convergence and Harmonic Forms from Point Clouds

Let $M^n\subset\mathbb R^d$ be a closed, connected, orientable $C^4$-smooth Riemannian submanifold of dimension $n\ge3$. We construct, for each degree $0\le k\le n$, a family of deformed Hodge Laplacians $Δ_t^k$, $t>0$, defined in terms of the extrinsic geometry of $M^n$, and prove that $Δ_t^k$ converges uniformly to the classical Hodge Laplacian $Δ^k$ as $t\to0^+$. Given an i.i.d. uniformly distributed point cloud $S_m\subset M^n$, we define empirical Hodge operators $\widehatΔ_{t,S_m}^k$. Under the scaling $t=m^{-1/(2n)}$, we prove uniform consistency in probability and compact Mosco convergence in probability of the associated quadratic forms. Consequently, the empirical spectral cluster near zero contains exactly the $k$-th Betti number $b_k$ of eigenvalues, counted with multiplicity, and converges in the transported discrete $L^2$-sense to the space of harmonic $k$-forms. We also construct consistent empirical estimators of the tangent projection, the second fundamental form, the Riemannian curvature tensor, and the Weitzenböck curvature endomorphisms. As applications, we obtain consistent recovery of the Betti numbers and harmonic representatives of de Rham cohomology, as well as of the Pontryagin forms and their periods. Evaluating these periods on a homology basis recovers the corresponding real Pontryagin classes from sampled data.

math.DG↗

arXiv2025

Probabilistic morphisms and Bayesian supervised learning

In this paper, we develop category theory of Markov kernels to study categorical aspects of Bayesian inversions. As a result, we present a unified model for Bayesian supervised learning, encompassing Bayesian density estimation. We illustrate this model with Gaussian process regressions.

math.ST↗

arXiv2025

Categorical and geometric methods in statistical, manifold, and machine learning

We present and discuss applications of the category of probabilistic morphisms, initially developed in \cite{Le2023}, as well as some geometric methods to several classes of problems in statistical, machine and manifold learning which shall be, along with many other topics, considered in depth in the forthcoming book \cite{LMPT2024}.

stat.ML↗

arXiv2025

Supervised learning with probabilistic morphisms and kernel mean embeddings

In this paper I propose a generative model of supervised learning that unifies two approaches to supervised learning, using a concept of a correct loss function. Addressing two measurability problems, which have been ignored in statistical learning theory, I propose to use convergence in outer probability to characterize the consistency of a learning algorithm. Building upon these results, I extend a result due to Cucker-Smale, which addresses the learnability of a regression model, to the setting of a conditional probability estimation problem. Additionally, I present a variant of Vapnik-Stefanuyk's regularization method for solving stochastic ill-posed problems, and using it to prove the generalizability of overparameterized supervised learning models.

math.ST↗

arXiv2025

Unital $C_\infty$-algebras and the real homotopy type of $(r-1)$-connected compact manifolds of dimension $\le \ell(r-1)+2$

We encode the real homotopy type of an $n$-dimensional $(r-1)$-connected compact manifold $M$, $ r\ge 2$ into a minimal unital $C_\infty$-structure on $H^* (M,\mathbb R)$, obtained via homotopy transfer of the unital DGCA structure of the small quotient algebra associated with a Hodge decomposition of the de Rham algebra $\mathcal A^*(M)$, which has been proposed by Fiorenza-Kawai-Lê-Schwachhöfer in [Ann. Sc. Norm. Super Pisa (5), vol. XXII (2021), 79-107]. We prove that if $n \le \ell (r-1) +2$, with $\ell \geq 4$, the multiplication $μ_k$ on the minimal unital $C_\infty$-algebra $H^*(M,\mathbb R)$ vanishes for all $k \ge \ell-1$. This extends the results from [loc. cit.], extending the bound on the dimension from $5r-3$ to the general bound $\ell(r-1) +2$. We also prove a variant of this result, conjectured by Zhou, stating that if $n \le \ell(r-1)+4$ and $b_r (M) =1$ then the multiplication $μ_k$ for all $k \ge \ell-1$ vanishes. This implies two formality results by Cavalcanti [Math. Proc. Cambridge Philos. Soc. 141 (2006), 101-112]. We show that in any dimension $n$ the Harrison cohomology class $[μ_3]\in \mathrm {HHarr}^{3,-1}(H^* (M, \mathbb R), H^*(M, \mathbb R)) $ is a homotopy invariant of $M$ and the first obstruction to formality, and provide a detailed proof that if $n\leq 4r-1$ this is the only obstruction. Furthermore, we show that in any dimension $n$ the class $[μ_3]$ and the Bianchi-Massey tensor invented by Crowley-Nordström in [J. Topol. 13(2020), 539-575] define each other uniquely.

math.AT↗

arXiv2024

Semisimple elements and the little Weyl group of real semisimple $Z_m$-graded Lie algebras

We consider the semisimple orbits of a Vinberg $θ$-representation. First we take the complex numbers as base field. By a case by case analysis we show a technical result stating the equality of two sets of hyperplanes, one corresponding to the restricted roots of a Cartan subspace, the other corresponding to the complex reflections in the (little) Weyl group. The semisimple orbits have representatives in a finite number of sets that correspond to reflection subgroups of the (little) Weyl group. One of the consequences of our technical result is that the elements in a fixed such set all have the same stabilizer in the acting group. Secondly we study what happens when the base field is the real numbers. We look at Cartan subspaces and show that the real Cartan subspaces can be classified by the first Galois cohomology set of the normalizer of a fixed real Cartan subspace. In the real case the orbits can be classified using Galois cohomology. However, in order for that to work we need to know which orbits have a real representative. We show a theorem that characterizes the orbits of homogeneous semisimple elements that do have such a real representative. This closely follows and generalizes a theorem from \cite{bgl}.

math.RT↗

arXiv2023

Almost formality of manifolds of low dimension

In this paper we introduce the notion of Poincaré DGCAs of Hodge type, which is a subclass of Poincaré DGCAs encompassing the de Rham algebras of closed orientable manifolds. Then we introduce the notion of the small algebra and the small quotient algebra of a Poincaré DGCA of Hodge type. Using these concepts, we investigate the equivalence class of $(r-1)$ connected $(r>1)$ Poincaré DGCAs of Hodge type. In particular, we show that a $(r-1)$ connected Poincaré DGCA of Hodge type ${\mathcal A}^\ast$ of dimension $n \le 5r-3$ is $A_\infty$-quasi-isomorphic to an $A_3$-algebra and prove that the only obstruction to the formality of ${\mathcal A}^\ast$ is a distinguished Harrison cohomology class $[μ_3] \in {\mathsf{Harr}}^{3,-1} (H^*({\mathcal A}^\ast), H^*({\mathcal A}^\ast))$. Moreover, the cohomology class $[μ_3]$ and the DGCA isomorphism class of $H^*({\mathcal A}^\ast)$ determine the $A_\infty$-quasi-isomorphism class of ${\mathcal A}^\ast$. This can be seen as a Harrison cohomology version of the Crowley-Nordström results [D. Crowley, J. Nordström, The rational homotopy type of $(n-1)$-connected manifolds of dimension up to $5n-3$, arXiv:1505.04184v2] on rational homotopy type of $(r-1)$-connected $(r>1)$ closed manifolds of dimension up to $5r-3$. We also derive the almost formality of closed $G_2$-manifolds, which have been discovered recently by Chan-Karigiannis-Tsang in [K.F. Chan, S. Karigiannis and C.C. Tsang, The ${\mathcal L}_B$-cohomology on compact torsion-free ${\rm G}_2$ manifolds and an application to `almost' formality, arXiv:1801.06410, to appear in Ann. Global Anal. Geom.], from our results and the Cheeger-Gromoll splitting theorem.

math.DG↗

arXiv2022

Natural differentiable structures on statistical models and the Fisher metric

In this paper I discuss the relation between the concept of the Fisher metric and the concept of differentiability of a family of probability measures. I compare the concepts of smooth statistical manifolds, differentiable families of measures, $k$-integrable parameterized measure models, diffeological statistical models, differentiable measures, which arise in Information Geometry, mathematical statistics and measure theory, and discuss some related problems.

math.DG↗

arXiv2022

CR-twistor spaces over manifolds with $G_2$- and $Spin(7)$-structures

In 1984 LeBrun constructed a CR-twistor space over an arbitrary conformal Riemannian 3-manifold and proved that the CR-structure is formally integrable. This twistor construction has been generalized by Rossi in 1985 for $m$-dimensional Riemannian manifolds endowed with a $(m-1)$-fold vector cross product (VCP). In 2011 Verbitsky generalized LeBrun's construction of twistor-spaces to $7$-manifolds endowed with a $G_2$-structure. In this paper we unify and generalize LeBrun's, Rossi's and Verbitsky's construction of a CR-twistor space to the case where a Riemannian manifold $(M, g)$ has a VCP structure. We show that the formal integrability of the CR-structure is expressed in terms of a torsion tensor on the twistor space, which is a Grassmanian bundle over $(M, g)$. If the VCP structure on $(M,g)$ is generated by a $G_2$- or $Spin(7)$-structure, then the vertical component of the torsion tensor vanishes if and only if $(M, g)$ has constant curvature, and the horizontal component vanishes if and only if $(M,g)$ is a torsion-free $G_2$ or $Spin(7)$-manifold. Finally we discuss some open problems.

math.DG↗

arXiv2021

Classification of real trivectors in dimension nine

We classify real trivectors in dimension 9. The corresponding classification over the field C of complex numbers was obtained by Vinberg and Elashvili in 1978. One of the main tools used for their classification was the construction of the representation of SL(9,C) on the space of complex trivectors of C^9 as a theta-representation corresponding to a Z/3Z-grading of the simple complex Lie algebra of type E_8. This divides the trivectors into three groups: nilpotent, semisimple, and mixed trivectors. Our classification follows the same pattern. We use Galois cohomology, first and second, to obtain the classification over R.

math.RT↗

arXiv2021

Unified modelling of epidemics by coupled dynamics via Monte-Carlo Markov Chain algorithms

To forecast the time dynamics of an epidemic, we propose a discrete stochastic model that unifies and generalizes previous approaches to the subject. Viewing a given population of individuals or groups of individuals with given health state attributes as living in and moving between the nodes of a graph, we use Monte-Carlo Markov Chain techniques to simulate the movements and health state changes of the individuals according to given probabilities of stay that have been preassigned to each of the nodes. We utilize this model to either capture and predict the future geographic evolution of an epidemic in time, or the evolution of an epidemic inside a heterogeneous population which is divided into homogeneous sub-populations, or, more generally, its evolution in a combination or superposition of the previous two contexts. We also prove that when the size of the population increases and a natural hypothesis is satisfied, the stochastic process associated to our model converges to a deterministic process. Indeed, when the length of the time step used in the discrete model converges to zero, in the limit this deterministic process is driven by a differential equation yielding the evolution of the expectation value of the number of infected as a function of time. In the second part of the paper, we apply our model to study the evolution of the Covid-19 epidemic. We deduce a decomposition of the function yielding the number of infectious individuals into "wavelets", which allows to trace in time the expectation value for the number of infections inside each sub-population. Within this framework, we also discuss possible causes for the occurrence of multiple epidemiological waves.

math.PR↗

arXiv2021

Real graded Lie algebras, Galois cohomology, and classification of trivectors in R^9

In this paper we classify real trivectors in dimension 9. The corresponding classification over the field C of complex numbers was done by Vinberg and Elashvili in 1978. One of the main tools used for their classification was the construction of the representation of SL(9,C) on the space of complex trivectors of C^9 as a theta-representation corresponding to a Z/3Z-grading of the simple complex Lie algebra of type E8. This divides the trivectors into three groups: nilpotent, semisimple, and mixed trivectors. Our classification follows the same pattern. We use Galois cohomology to obtain the classification over R. For the nilpotent orbits this is in principle rather straightforward and the main problem is to determine the first Galois cohomology sets of a long list of centralizers (we compute the centralizers using computer). For the semisimple and mixed orbits we develop new methods based on Galois cohomology, first and second. We also consider real theta-representations in general, and derive a number of results that are useful for the classification of their orbits.

math.RT↗

arXiv2021

Nonparametric estimations and the diffeological Fisher metric

In this paper, first, we survey the concept of diffeological Fisher metric and its naturality, using functorial language of probability morphisms, and slightly extending Lê's theory in \cite{Le2020} to include weakly $C^k$-diffeological statistical models. Then we introduce the resulting notions of the diffeological Fisher distance, the diffeological Hausdorff--Jeffrey measure and explain their role in classical and Bayesian nonparametric estimation problems in statistics.

math.ST↗

arXiv2021

Probabilistic morphisms and Bayesian nonparametrics

In this paper we develop a functorial language of probabilistic morphisms and apply it to some basic problems in Bayesian nonparametrics. First we extend and unify the Kleisli category of probabilistic morphisms proposed by Lawvere and Giry with the category of statistical models proposed by Chentsov and Morse-Sacksteder. Then we introduce the notion of a Bayesian statistical model that formalizes the notion of a parameter space with a given prior distribution in Bayesian statistics. {We revisit the existence of a posterior distribution, using probabilistic morphisms}. In particular, we give an explicit formula for posterior distributions of the Bayesian statistical model, assuming that the underlying parameter space is a Souslin space and the sample space is a subset in a complete connected finite dimensional Riemannian manifold. Then we give a new proof of the existence of Dirichlet measures over any measurable space using a functorial property of the Dirichlet map constructed by Sethuraman.

math.ST↗