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Huitao Feng

Publications and source records attributed to Huitao Feng.

15 recordsLinked to original sources
arXiv2026

Demystifying Oversmoothing in Sheaf Neural Networks: An Index-Theoretic Criterion

To combat oversmoothing in Graph Convolutional Networks, Sheaf Neural Networks (SNNs) were proposed as a generalization by equipping the graph with a sheaf structure and replacing the graph Laplacian with a sheaf Laplacian $\mathcal{L}$. Existing analyses connect sheaf diffusion to oversmoothing via the harmonic space ($\ker\mathcal{L}$), taking its absolute dimension as an indicator of anti-oversmoothing capacity. However, absolute dimension alone is not a reliable measure: certain sheaf configurations inflate $\dim \ker \mathcal{L}$ while their harmonic sections remain entirely constant, without enriching discriminative capacity. We instead introduce the first relative, geometric approach, yielding a precise characterisation of anti-oversmoothing capacity. Under natural conditions on stalk transportation and global sheaf structure, we establish an index-theoretic comparison criterion showing that one sheaf's harmonic space genuinely contains another's beyond trivial inflation. We illustrate this with a concrete instance and further introduce \textit{GyroSheaf}, a sheaf with curved gyrovector-space stalks, extending the criterion to the non-linear setting via local tangent-space linearization. Experiments across ten models confirm the theoretical criterion: sheaf models violating the criterion collapse despite possessing index jumps, while compliant models maintain depth-stable representations.

cs.LG↗
arXiv2026

Sheaf Neural Networks on SPD Manifolds: Second-Order Geometric Representation Learning

Graph neural networks face two fundamental challenges rooted in the linear structure of Euclidean vector spaces: (1) Current architectures represent geometry through vectors (directions, gradients), yet many tasks require matrix-valued representations that capture relationships between directions-such as how atomic orientations covary in a molecule. These second-order representations are naturally captured by points on the symmetric positive definite matrices (SPD) manifold; (2) Standard message passing applies shared transformations across edges. Sheaf neural networks address this via edge-specific transformations, but existing formulations remain confined to vector spaces and therefore cannot propagate matrix-valued features. We address both challenges by developing the first sheaf neural network operates natively on the SPD manifold. Our key insight is that the SPD manifold admits a Lie group structure, enabling well-posed analogs of sheaf operators without projecting to Euclidean space. Theoretically, we prove that SPD-valued sheaves are strictly more expressive than Euclidean sheaves: they admit consistent configurations (global sections) that vector-valued sheaves cannot represent, directly translating to richer learned representations. Empirically, our sheaf convolution transforms effectively rank-1 directional inputs into full-rank matrices encoding local geometric structure. Our dual-stream architecture achieves SOTA on 6/7 MoleculeNet benchmarks, with the sheaf framework providing consistent depth robustness.

cs.LG↗
arXiv2020

An equivalence theorem of a class of Minkowski norms and its applications

In this paper, the Cartan tensors of the $(α,β)$-norms are investigated in details. Then an equivalence theorem of $(α,β)$-norms is proved. As a consequence in Finsler geometry, general $(α,β)$-metrics on smooth manifolds of dimension $n\geq4$ with vanishing Landsberg curvatures must be Berwald manifolds.

math.DG↗
arXiv2019

Complex Finsler vector bundles with positive Kobayashi curvature

In this short note, we prove that a complex Finsler vector bundle with positive Kobayashi curvature must be ample, which partially solves a problem of S. Kobayashi posed in 1975. As applications, a strongly pseudoconvex complex Finsler manifold with positive Kobayashi curvature must be biholomorphic to the complex projective space; we also show that all Schur polynomials are numerically positive for complex Finsler vector bundles with positive Kobayashi curvature.

math.DG↗
arXiv2018

Geodesic-Einstein metrics and nonlinear stabilities

In this paper, we introduce notions of nonlinear stabilities for a relative ample line bundle over a holomorphic fibration and define the notion of a geodesic-Einstein metric on this line bundle, which generalize the classical stabilities and Hermitian-Einstein metrics of holomorphic vector bundles. We introduce a Donaldson type functional and show that this functional attains its absolute minimum at geodesic-Einstein metrics, and we also discuss the relations between the existence of geodesic-Einstein metrics and the nonlinear stabilities of the line bundle. As an application, we will prove that a holomorphic vector bundle admits a Finsler-Einstein metric if and only if it admits a Hermitian-Einstein metric, which answers a problem posed by S. Kobayashi.

math.DG↗
arXiv2018

Chern forms of holomorphic Finsler vector bundles and some applications

In this paper, we present two kinds of total Chern forms $c(E,G)$ and $\mathcal{C}(E,G)$ as well as a total Segre form $s(E,G)$ of a holomorphic Finsler vector bundle $π:(E,G)\to M$ expressed by the Finsler metric $G$, which answers a question of J. Faran (\cite{Faran}) to some extent. As some applications, we show that the signed Segre forms $(-1)^ks_k(E,G)$ are positive $(k,k)$-forms on $M$ when $G$ is of positive Kobayashi curvature; we prove, under an extra assumption, that a Finsler-Einstein vector bundle in the sense of Kobayashi is semi-stable; we introduce a new definition of a flat Finsler metric, which is weaker than Aikou's one (\cite{Aikou}) and prove that a holomorphic vector bundle is Finsler flat in our sense if and only if it is Hermitian flat.

math.DG↗
arXiv2018

A Donaldson type functional on a holomorphic Finsler vector bundle

In this paper, we solve a problem of Kobayashi posed in \cite{Ko4} by introducing a Donaldson type functional on the space $F^+(E)$ of strongly pseudo-convex complex Finsler metrics on $E$ -- a holomorphic vector bundle over a closed Kähler manifold $M$. This Donaldson type functional is a generalization in the complex Finsler geometry setting of the original Donaldson functional and has Finsler-Einstein metrics on $E$ as its only critical points, at which this functional attains the absolute minimum.

math.DG↗
arXiv2017

Flat vector bundles and open coverings

We establish a generic counting formula for the Euler number of a flat vector bundle of rank $2n$ over a $2n$ dimensional closed manifold, in terms of vertices of transversal open coverings of the underlying manifold. We use the Mathai-Quillen formalism to prove our result.

math.DG↗
arXiv2014

Rational curves on Hermitian manifolds

By using analytic method, we prove that there exist rational curves on compact Hermitian manifolds with positive holomorphic bisectional curvature. It confirms a question of S.-T. Yau. It is well-known that Mori proved in \cite{Mori79} that every compact complex manifold $N$ with $c_1(N)>0$ contains at least one rational curve. However, as a borderline example, we show that the standard Hopf surface $S^1\times S^3$ has a Hermitian metric with non-negative holomorphic bisectional curvature (in particular, $c_1(S^1\times S^3)\geq 0$), but it contains no rational curve.

math.DG↗
arXiv2012

Adiabatic limit and connections in Finsler Geometry

In this paper, we identify the Bott connection on the natural foliation of the projective sphere bundle of a Finsler manifold to the Chern connection of this manifold. As a consequence, the symmetrization of the Bott connection turns out to be the Cartan connection of the Finsler manifold. Following Liu-Zhang \cite{LiuZ}, the Cartan connection can also be obtained through an adiabatic limit process. Furthermore, a Chern-Simons type form is defined and its conformal properties are discussed.

math.DG↗
arXiv2010

A Poincaré-Hopf type formula for Chern character numbers

For two complex vector bundles admitting a homomorphism with isolated singularities between them, we establish a Poincaré-Hopf type formula for the difference of the Chern character numbers of these two vector bundles. As a consequence, we extend the original Poincaré-Hopf index formula to the case of complex vector fields (to appear in Mathematische Zeitschrift)

math.GT↗
arXiv2007

Real embeddings, eta invariant and Chern-Simons current

We present an alternate proof of the Bismut-Zhang localization formula for $η$-invariants without using the analytic techniques developed by Bismut-Lebeau. A Riemann-Roch property for Chern-Simons currents, which is of independent interest, is established in due course.

math.DG↗
arXiv2003

Holomorphic Equivariant Cohomology via a Transversal Holomorphic Vector Field

In this paper an analytic proof of a generalization of a theorem of Bismut ([Bis1, Theorem 5.1]) is given, which says that, when $v$ is a transversal holomorphic vector field on a compact complex manifold $X$ with a zero point set $Y$, the embedding $j:Y\to X$ induces a natural isomorphism between the holomorphic equivariant cohomology of $X$ via $v$ with coefficients in $ξ$ and the Dolbeault cohomology of $Y$ with coefficients in $ξ|_Y$, where $ξ\to X$ is a holomorphic vector bundle over $X$.

math.DG↗