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Gaofeng Huang

Publications and source records attributed to Gaofeng Huang.

15 recordsLinked to original sources
arXiv2026

Holomorphic Approximation for Real Diffeomorphism Groups

We show that every diffeomorphism of $\mathbb{R}^n$ for $n \ge 2$ can be approximated by an automorphism of $\mathbb{C}^n$ in the Whitney $C^k$-topology for any positive integer $k$ using the notion of density property. More precisely we find sufficient conditions for this holomorphic approximation of diffeomorphisms to hold and prove that the split real forms of most linear algebraic groups satisfy these conditions. In the same manner we also show holomorphic approximations of volume-preserving diffeomorphisms for the split real forms of linear algebraic groups equipped with the left-invariant volume form.

math.CV↗
arXiv2025

Parametric Factorization of Matrices

In this survey paper we study parametric versions of writing a matrix in $SL_n (\mathbb{C})$ as a product of lower and upper unitriangular matrices in interchanging order as well as generalizations to other classical groups. We give an account of algebraic, continuous and holomorphic factorization results, from the standpoint of Several Complex Variables. Out of the wealth of algebraic results, we only concentrate on those which are related to holomorphic factorization and often formulate them in a specific form, e.g. for the field of complex numbers in place of more general fields or principal ideal domains. The number of unitriangular matrices needed is a difficult problem and is solved in very specific cases only. We give a new lower bound for factorizing matrices in $SL_2 (\mathbb{C})$ continuously parametrized by normal topological spaces of dimension bigger than one.

math.CV↗
arXiv2025

Direct Products for the Hamiltonian Density Property

We show that the direct product of two Stein manifolds with the Hamiltonian density property enjoys the Hamiltonian density property as well. We investigate the relation between the Hamiltonian density property and the symplectic density property. We then establish the Hamiltonian and the symplectic density property for $(\mathbb{C}^\ast)^{2n}$ and for the so-called traceless Calogero--Moser spaces. As an application we obtain a Carleman-type approximation for Hamiltonian diffeomorphisms of a real form of the traceless Calogero--Moser space.

math.CV↗
arXiv2025

Untriangular factorization of holomorpic symplectic matrices

We prove that every holomorphic symplectic matrix can be factorized as a product of holomorphic unitriangular matrices with respect to the symplectic form $ \left[\begin{array}{ccc} 0 & L_n \\ -L_n & 0\end{array}\right]$ where $L$ is the $n \times n$ matrix with $1$ along the skew-diagonal. Also we prove that holomorphic unitriangular matrices with respect to this symplectic form are products of not more than $7$ holomorphic unitriangular matrices with respect to the standard symplectic form $\left[\begin{array}{ccc} 0 & I_n \\ -I_n & 0\end{array}\right]$, thus solving an open problem posed in \cite{HKS}. Combining these two results allows for estimates of the optimal number of factors in the factorization by holomorphic unitriangular matrices with respect to the standard symplectic form. The existence of that factorization was obtained earlier by Ivarsson-Kutzschebauch and Schott, however without any estimates. Another byproduct of our results is a new, much less technical and more elegant proof of this factorization.

math.CV↗
arXiv2025

A Language Vision Model Approach for Automated Tumor Contouring in Radiation Oncology

Background: Lung cancer ranks as the leading cause of cancer-related mortality worldwide. The complexity of tumor delineation, crucial for radiation therapy, requires expertise often unavailable in resource-limited settings. Artificial Intelligence(AI), particularly with advancements in deep learning (DL) and natural language processing (NLP), offers potential solutions yet is challenged by high false positive rates. Purpose: The Oncology Contouring Copilot (OCC) system is developed to leverage oncologist expertise for precise tumor contouring using textual descriptions, aiming to increase the efficiency of oncological workflows by combining the strengths of AI with human oversight. Methods: Our OCC system initially identifies nodule candidates from CT scans. Employing Language Vision Models (LVMs) like GPT-4V, OCC then effectively reduces false positives with clinical descriptive texts, merging textual and visual data to automate tumor delineation, designed to elevate the quality of oncology care by incorporating knowledge from experienced domain experts. Results: Deployments of the OCC system resulted in a significant reduction in the false discovery rate by 35.0%, a 72.4% decrease in false positives per scan, and an F1-score of 0.652 across our dataset for unbiased evaluation. Conclusions: OCC represents a significant advance in oncology care, particularly through the use of the latest LVMs to improve contouring results by (1) streamlining oncology treatment workflows by optimizing tumor delineation, reducing manual processes; (2) offering a scalable and intuitive framework to reduce false positives in radiotherapy planning using LVMs; (3) introducing novel medical language vision prompt techniques to minimize LVMs hallucinations with ablation study, and (4) conducting a comparative analysis of LVMs, highlighting their potential in addressing medical language vision challenges.

eess.IV↗
arXiv2025

A Criterion for the Algebraic Density Property of Affine $SL_2$-Manifolds

Let $B$ be an affine $k$-domain which admits a nontrivial fundamental pair $(D,U)$ of locally nilpotent derivations, i.e., if $E=[D,U]$ then $(D,U,E)$ is an $\mathfrak{sl}_2$-triple. We prove an algebraic criterion, characterizing under which conditions the fundamental pair $(D,U)$ resp. the triple $(D,U,E)$ is compatible in a technical sense that allows us to construct many vector fields on the spectrum of $B$ from the complete ones. This criterion enables us to prove the algebraic density property for the following widely studied classes of $\mathrm{SL}_2$-varieties arising in physics: Classical Calogero--Moser spaces, Calogero--Moser spaces with "inner degrees of freedom'' and a smooth cyclic quiver variety.

math.AC↗
arXiv2024

Parametric Symplectic Jet Interpolation

We prove a parametric jet interpolation theorem for symplectic holomorphic automorphisms of $\mathbb{C}^{2n}$ with parameters in a Stein space. Moreover, we provide an example of an unavoidable set for symplectic holomorphic maps.

math.CV↗
arXiv2024

Holomorphic Approximation of Symplectic Diffeomorphisms for Calogero--Moser Spaces

The real Calogero--Moser space $\mathcal{C}_n^\mathbb{R}$ is a noncompact, totally real submanifold of the complex Calogero--Moser space $\mathcal{C}_n$. We prove that every symplectic diffeomorphism of $\mathcal{C}_n^\mathbb{R}$ smoothly isotopic to the identity can be approximated in the fine Whitney topology -- the strongest in this context -- by holomorphic symplectic automorphisms of $\mathcal{C}_n$ that preserve $\mathcal{C}_n^\mathbb{R}$. A key ingredient in our proof is a refined version of the symplectic density property of $\mathcal{C}_n$.

math.CV↗
arXiv2024

EXACT-Net:EHR-guided lung tumor auto-segmentation for non-small cell lung cancer radiotherapy

Lung cancer is a devastating disease with the highest mortality rate among cancer types. Over 60% of non-small cell lung cancer (NSCLC) patients, which accounts for 87% of diagnoses, require radiation therapy. Rapid treatment initiation significantly increases the patient's survival rate and reduces the mortality rate. Accurate tumor segmentation is a critical step in the diagnosis and treatment of NSCLC. Manual segmentation is time and labor-consuming and causes delays in treatment initiation. Although many lung nodule detection methods, including deep learning-based models, have been proposed, there is still a long-standing problem of high false positives (FPs) with most of these methods. Here, we developed an electronic health record (EHR) guided lung tumor auto-segmentation called EXACT-Net (EHR-enhanced eXACtitude in Tumor segmentation), where the extracted information from EHRs using a pre-trained large language model (LLM), was used to remove the FPs and keep the TP nodules only. The auto-segmentation model was trained on NSCLC patients' computed tomography (CT), and the pre-trained LLM was used with the zero-shot learning approach. Our approach resulted in a 250% boost in successful nodule detection using the data from ten NSCLC patients treated in our institution.

eess.IV↗
arXiv2023

The Density Property for Generalized Calogero--Moser Spaces with Inner Degrees of Freedom

We prove the density property for generalized Calogero--Moser spaces with inner degrees of freedom. This allows us to describe the holomorphic automorphism group of these complex affine manifolds. These generalized Calogero--Moser spaces can also be understood as quiver varieties corresponding to moduli spaces of $\mathrm{SU}(2)$ instantons on a non-commutative $\mathbb{R}^4$.

math.CV↗
arXiv2023

A Criterion for the Density Property of Stein Manifolds

We generalize a criterion for the density property of Stein manifolds. As an application, we give a new, simple proof of the fact that the Danielewski surfaces have the algebraic density property. Furthermore, we have found new examples of Stein manifolds with the density property.

math.CV↗
arXiv2023

The symplectic holomorphic density property for Calogero-Moser spaces

We introduce the symplectic holomorphic density property and the Hamiltonian holomorphic density property together with the corresponding version of Anders\'en-Lempert theory. We establish these properties for the Calogero-Moser space $\mathcal{C}_n$ of $n$ particles and describe its group of holomorphic symplectic automorphisms.

math.CV↗
arXiv2022

Factorization of Holomorphic Matrices and Kazhdan's property (T)

In this article we deduce some algebraic properties for the group $\mathrm{Sp}_{2n} (\mathcal{O}(X))$ of holomorphic symplectic matrices on a Stein space $X$: holomorphic factorization, exponential factorization, and Kazhdan's property (T). In holomorphic factorization we combine a recent result of the third author and K-theory tools to give explicit bounds for the case when $X$ is one-dimensional or two-dimensional. Next we use them to find bounds for exponential factorization. As a further application, we show that the elementary symplectic group $\mathrm{Ep}_{2n}(\mathcal{O}(X))$ admits Kazhdan's property (T).

math.CV↗
arXiv2020

Enhanced Balancing GAN: Minority-class Image Generation

Generative adversarial networks (GANs) are one of the most powerful generative models, but always require a large and balanced dataset to train. Traditional GANs are not applicable to generate minority-class images in a highly imbalanced dataset. Balancing GAN (BAGAN) is proposed to mitigate this problem, but it is unstable when images in different classes look similar, e.g. flowers and cells. In this work, we propose a supervised autoencoder with an intermediate embedding model to disperse the labeled latent vectors. With the improved autoencoder initialization, we also build an architecture of BAGAN with gradient penalty (BAGAN-GP). Our proposed model overcomes the unstable issue in original BAGAN and converges faster to high quality generations. Our model achieves high performance on the imbalanced scale-down version of MNIST Fashion, CIFAR-10, and one small-scale medical image dataset.

cs.CV↗