arXiv subjects

Hanwen Liu

Publications and source records attributed to Hanwen Liu.

At least 19 recordsLinked to original sources

arXiv2026

Non-K\"ahler Critical Hermitian Metrics of the Dinew--Popovici Functional

The Dinew--Popovici functional is an energy functional for Hermitian symplectic metrics in a fixed Aeppli cohomology class. Its vanishing characterizes the K\"ahler metrics in that class, providing a variational approach to K\"ahler geometry. Dinew and Popovici proved that every critical point is K\"ahler in complex dimension three. We give a negative answer to Erfan Soheil's question about higher dimensions by constructing non-K\"ahler critical metrics on products of two K\"ahler surfaces $(S_1,\eta_1)$ and $(S_2,\eta_2)$. These metrics are critical under every variation on the product 4-fold. We characterize the critical product metrics and prove that non-K\"ahler critical products exist in the Aeppli class $[\eta_1+\eta_2]_A$ precisely when the canonical bundles of the two surface factors are smoothly trivial. As a by-product, we develop a geometric flow driven by the torsion tensor and converging to the fixed K\"ahler background when this background metric has nonnegative holomorphic bisectional curvature.

math.DG↗

arXiv2026

On the Hessian Conjecture in Lorentzian Signature: Constant Pivots and Hesse Systems

As a close relative of the Jacobian conjecture, the Hessian conjecture in dimension $n$ states that the local Legendre transform of a polynomial solution to the Monge--Amp\`ere equation $\det(\operatorname{Hess}(\phi))=\pm1$ is also a polynomial solution. The general Hessian conjecture is false for $n\geq5$, while in Riemannian signature it follows from the J\"orgens--Calabi--Pogorelov theorem. We study the four-dimensional Hessian conjecture in Lorentzian signature. For a polynomial potential $\phi$ in four real variables whose Hessian matrix has index $1$ and determinant $-1$, we define a constant pivot for $\phi$ to be a nonzero constant vector $\xi$ such that the second directional derivative $D_\xi^2\phi$ is constant. We then prove that the gradient mapping of every potential admitting a pivot is a polynomial automorphism, and that a pivot always exists when $\phi$ decomposes into homogeneous pieces as $\phi=\phi_d+\phi_{d-1}+\phi_2+\phi_1+\phi_0$ with $d\geq4$. More generally, we prove the same conclusion when $$\phi=\phi_d+\cdots+\phi_{d-k}+\phi_2+\phi_1+\phi_0,$$ where $k\geq0$ and $d\geq4k+3$. Then, we associate with each potential a linear system of quadrics, called the Hesse system, and a canonical homomorphism $\mu_\phi$. We prove that the existence of a pivot is equivalent to $\operatorname{rank}(\mu_\phi)\leq55$. As an application, we prove that the gradient mapping is a polynomial automorphism whenever the Hesse system has complex dimension at most 4. We also show that if ${\det(\operatorname{Hess}(\phi-\phi_2))\equiv0}$, then the potential $\phi$ admits a pivot. After that, we then give an analytic degeneracy criterion for $\operatorname{Hess}(\phi-\phi_2)$. Finally, we prove the Hessian conjecture in this setting for every polynomial potential of degree at most five.

math.AG↗

arXiv2026

Scaling Unmodified Multithreaded Applications with Elastic CXL-based Distributed Shared Memory

While CXL presents a promising hardware substrate for Distributed Shared Memory (DSM), seamlessly scaling multithreaded applications across multiple nodes remains a formidable challenge. Existing CXL-based DSMs fall short: they require manual code modifications to share non-heap data, employ rigid data placement policies that fail under diverse and dynamic workloads, and suffer from severe page-fault processing overheads in sub-microsecond ($\mu\mathrm{s}$) environments. We present xDSM, a full-space, elastic DSM system built over CXL that transparently scales unmodified multithreaded applications. To eliminate the burden of manual code rewrites, xDSM employs an OS-runtime co-design that establishes a globally coordinated address space, seamlessly sharing all memory segments. To mask CXL access penalties, xDSM abandons static placement rules in favor of a dynamic, latency-driven policy that actively balances data between local DRAM and CXL memory. Finally, to resolve the fundamental tension between high base-page fault overheads and severe huge-page false sharing, xDSM introduces spatial locality-aware elasticity, dynamically coalescing and splitting pages on the fly to amortize processing costs. Evaluated across diverse workloads using 15 system configurations, xDSM outperforms CXL-only baselines by 1.5$\times$ to 2.2$\times$ and state-of-the-art hybrid DSMs by 1.1$\times$ to 2.2$\times$, while achieving near-linear scalability.

cs.OS↗

arXiv2026

A Self-Dual Frame Formalism of the SO(3) Yang-Mills Theory

For a closed oriented Riemannian $4$-manifold $(M,g)$, we consider $\operatorname{SO}(3)$ connections on the bundle $\Lambda^+$ of self-dual $2$-forms. On the open locus where the self-dual curvature is an orientation-preserving frame, pointwise polar decomposition removes the gauge freedom and replaces the connection by a positive self-adjoint endomorphism field $h$ of $\Lambda^+$. Using the classical reconstruction of the compatible connection $A(h)$, we obtain a global formulation of the Yang--Mills equation as the determined second order system $$ \Phi_g(h):=F_{A(h)}^+h^{-1}-\operatorname{Id}=0. $$ Here, the tensor $F_{A(h)}^+$ is the self-dual curvature of $A(h)$, regarded as an endomorphism of $\Lambda^+$, and $\operatorname{Id}$ is the identity of $\Lambda^+$. We establish a variational formulation, automatic irreducibility, elliptic regularity, and a Fredholm index theorem for the linearized operator $D_h\Phi_g$. For fields of the form $h=e^{2\omega}\operatorname{Id}$, where $\omega$ is a smooth real-valued function on $M$, the equation $\Phi_g(h)=0$ is equivalent to anti-self-duality and constant scalar curvature $6\sqrt{2}$ of the conformal metric $\hat g=e^{2\omega}g$. Consequently, every anti-self-dual conformal class of positive Yamabe constant gives a global solution of $\Phi_g(h)=0$.

math.DG↗

arXiv2026

A Note on the Rainich Problem for SU(2) Gauge

We provide a resolution to the non-Abelian Rainich problem. By canonically identifying traceless symmetric $(0,2)$-tensors with Hermitian forms on the vector bundle of chiral 2-forms, we define the internal square roots of a stress-energy tensor. We then prove that the existence of a local $\operatorname{SU}(2)$ Yang-Mills field with prescribed stress-energy tensor $T$ is equivalent to a single differential condition on internal square roots of $T$.

math.DG↗

arXiv2026

On Conservative Statistical Riemann Surfaces

We establish a correspondence between information geometry and gauge theory. First, we define an important class of statistical manifolds, that is normalized and satisfies a conservation field equation. Second, we prove that for a conservative statistical structure on an orientable surface, the Chebyshev 1-form is constrained to be harmonic, and the traceless part of the Amari--Chentsov tensor descends to a holomorphic cubic differential. Then, we demonstrate that normalized conservative statistical structures are geometrically generated by solutions to the scalar Tzitz\'eica equation on Higgs bundles with general linear holonomy, generalizing the Labourie-Loftin correspondence. Finally, we prove that the moduli space of normalized conservative statistical structures on a closed orientable surface of genus at least 2 is completely parameterized by a holomorphic vector bundle over the Teichm\"uller space, consisting of Abelian differentials and cubic differentials.

math-ph↗

arXiv2026

On the Rigidity of Analytic Mappings in Complex Analysis and Geometry

We establish rigidity results for holomorphic mappings and plurisubharmonic functions in complex geometry. First, under mild conditions, we show that the gradient of a $\operatorname{U}(1)$-invariant strictly plurisubharmonic function in $\mathbb{C}^2$ possesses finite fibers and induces a analytic mapping of topological degree $1$ on the symplectic quotient. Second, we prove that continuous fiber-wise holomorphic maps on proper fibrations elevate to global holomorphic maps when anchored by mutually disjoint sections, yielding rigidity for homomorphisms between elliptic fibrations and Abelian schemes. Third, we demonstrate that a fiber-wise holomorphic map of mapping degree $1$ from a fibered compact Kobayashi hyperbolic manifold to a projective variety is a biholomorphism, provided it is injective on a very ample hypersurface. Finally, we prove that a holomorphic Lie group action with sufficiently large orbits confines the critical locus of a proper invariant strictly plurisubharmonic function to the fixed-point set, guaranteeing a unique global minimum and yielding a sharp differential topological obstruction on the orbit dimensions of compact Lie group actions.

math.CV↗

arXiv2026

Hausdorff Dimension of Union of Lines Covering a Curve: Applications to Mathematical Physics

We prove that for any nonlinear $f \in C^{1,\alpha}([0,1])$, the union of lines covering its graph has a Hausdorff dimension of at least $1+\alpha$, and this dimension bound is sharp. We then apply these geometric results to mathematical physics, proving that spacetime observability sets for conservation laws with $\alpha$-H\"older initial wave speeds possess a dimension of at least $\alpha$. Finally, we prove that if an absolutely integrable vector field $v$ on the boundary of a polyhedron exhibits a strictly positive total flux, then the union of the line field spanned by $v$ possesses a Hausdorff dimension of 3.

math.AP↗

arXiv2026

CTC-TTS: LLM-based dual-streaming text-to-speech with CTC alignment

Large-language-model (LLM)-based text-to-speech (TTS) systems can generate natural speech, but most are not designed for low-latency dual-streaming synthesis. High-quality dual-streaming TTS depends on accurate text--speech alignment and well-designed training sequences that balance synthesis quality and latency. Prior work often relies on GMM-HMM based forced-alignment toolkits (e.g., MFA), which are pipeline-heavy and less flexible than neural aligners; fixed-ratio interleaving of text and speech tokens struggles to capture text--speech alignment regularities. We propose CTC-TTS, which replaces MFA with a CTC based aligner and introduces a bi-word based interleaving strategy. Two variants are designed: CTC-TTS-L (token concatenation along the sequence length) for higher quality and CTC-TTS-F (embedding stacking along the feature dimension) for lower latency. Experiments show that CTC-TTS outperforms fixed-ratio interleaving and MFA-based baselines on streaming synthesis and zero-shot tasks.

eess.AS↗

arXiv2026

Towards a Hybrid Quantum-Classical Computing Framework for Database Optimization Problems in Real Time Setup

Quantum computing has shown promise for solving complex optimization problems in databases, such as join ordering and index selection. Prior work often submits formulated problems directly to black-box quantum or quantum-inspired solvers with the expectation of directly obtaining a good final solution. Due to the black-box nature of these solvers, users cannot perform fine-grained control over the solving procedure to balance the accuracy and efficiency, which in turn limits flexibility in real-time settings where most database problems arise. Moreover, it leads to limited potential for handling large-scale database optimization problems. In this paper, we propose a vision for the first real-time quantum-augmented database system, enabling transparent solutions for database optimization problems. We develop two complementary scalability strategies to address large-scale challenges, overcomplexity, and oversizing that exceed hardware limits. We integrate our approach with a database query optimizer as a preliminary prototype, evaluating on real-world workload, achieving up to 14x improvement over the classical query optimizer. We also achieve both better efficiency and solution quality than a black-box quantum solver.

cs.DB↗

arXiv2026

Is Quantum Computing Ready for Real-Time Database Optimization?

Database systems encompass several performance-critical optimization tasks, such as join ordering and index tuning. As data volumes grow and workloads become more complex, these problems have become exponentially harder to solve efficiently. Quantum computing, especially quantum annealing, is a promising paradigm that can efficiently explore very large search spaces through quantum tunneling. It can escape local optima by tunneling through energy barriers rather than climbing over them. Earlier works mainly focused on providing an abstract representation (e.g., Quadratic Unconstrained Binary Optimization (QUBO)) for the database optimization problems (e.g., join order) and overlooked the real integration within database systems due to the high overhead of quantum computing services (e.g., a minimum 5s runtime for D-Wave's CQM-Solver). Recently, quantum annealing providers have offered more low-latency solutions, e.g., NL-Solver, which paves the road to actually realizing quantum solutions within DBMSs. However, this raises new systems research challenges in balancing efficiency and solution quality. In this talk, we show that this balance is possible to achieve. As a proof of concept, we present Q2O, the first real Quantum-augmented Query Optimizer. We show the end-to-end workflow: we encode the join order problem as a nonlinear model, a format solvable by the NL-Solver, using actual database statistics; the solution is translated into a plan hint that guides PostgreSQL's optimizer to produce a complete plan. Q2O is capable of handling actual queries in real time.

cs.DB↗

arXiv2025

On Two Dimensional Flat Hessian Potentials

A Riemannian metric is termed a Hessian metric if in some coordinate system it can be locally represented as the Hessian quadratic form of some locally defined smooth potential function. Under very mild extra technical conditions, we first theoretically describe the potentials of flat Hessian metrics on surfaces, and then construct these potentials explicitly using methods from integrable systems.

math.DG↗

arXiv2025

How Particle-System Random Batch Methods Enhance Graph Transformer: Memory Efficiency and Parallel Computing Strategy

Attention mechanism is a significant part of Transformer models. It helps extract features from embedded vectors by adding global information and its expressivity has been proved to be powerful. Nevertheless, the quadratic complexity restricts its practicability. Although several researches have provided attention mechanism in sparse form, they are lack of theoretical analysis about the expressivity of their mechanism while reducing complexity. In this paper, we put forward Random Batch Attention (RBA), a linear self-attention mechanism, which has theoretical support of the ability to maintain its expressivity. Random Batch Attention has several significant strengths as follows: (1) Random Batch Attention has linear time complexity. Other than this, it can be implemented in parallel on a new dimension, which contributes to much memory saving. (2) Random Batch Attention mechanism can improve most of the existing models by replacing their attention mechanisms, even many previously improved attention mechanisms. (3) Random Batch Attention mechanism has theoretical explanation in convergence, as it comes from Random Batch Methods on computation mathematics. Experiments on large graphs have proved advantages mentioned above. Also, the theoretical modeling of self-attention mechanism is a new tool for future research on attention-mechanism analysis.

cs.LG↗

arXiv2025

Unveiling the Spatial-temporal Effective Receptive Fields of Spiking Neural Networks

Spiking Neural Networks (SNNs) demonstrate significant potential for energy-efficient neuromorphic computing through an event-driven paradigm. While training methods and computational models have greatly advanced, SNNs struggle to achieve competitive performance in visual long-sequence modeling tasks. In artificial neural networks, the effective receptive field (ERF) serves as a valuable tool for analyzing feature extraction capabilities in visual long-sequence modeling. Inspired by this, we introduce the Spatio-Temporal Effective Receptive Field (ST-ERF) to analyze the ERF distributions across various Transformer-based SNNs. Based on the proposed ST-ERF, we reveal that these models suffer from establishing a robust global ST-ERF, thereby limiting their visual feature modeling capabilities. To overcome this issue, we propose two novel channel-mixer architectures: \underline{m}ulti-\underline{l}ayer-\underline{p}erceptron-based m\underline{ixer} (MLPixer) and \underline{s}plash-and-\underline{r}econstruct \underline{b}lock (SRB). These architectures enhance global spatial ERF through all timesteps in early network stages of Transformer-based SNNs, improving performance on challenging visual long-sequence modeling tasks. Extensive experiments conducted on the Meta-SDT variants and across object detection and semantic segmentation tasks further validate the effectiveness of our proposed method. Beyond these specific applications, we believe the proposed ST-ERF framework can provide valuable insights for designing and optimizing SNN architectures across a broader range of tasks. The code is available at \href{https://github.com/EricZhang1412/Spatial-temporal-ERF}{\faGithub~EricZhang1412/Spatial-temporal-ERF}.

cs.NE↗

arXiv2025

S$^2$NN: Sub-bit Spiking Neural Networks

Spiking Neural Networks (SNNs) offer an energy-efficient paradigm for machine intelligence, but their continued scaling poses challenges for resource-limited deployment. Despite recent advances in binary SNNs, the storage and computational demands remain substantial for large-scale networks. To further explore the compression and acceleration potential of SNNs, we propose Sub-bit Spiking Neural Networks (S$^2$NNs) that represent weights with less than one bit. Specifically, we first establish an S$^2$NN baseline by leveraging the clustering patterns of kernels in well-trained binary SNNs. This baseline is highly efficient but suffers from \textit{outlier-induced codeword selection bias} during training. To mitigate this issue, we propose an \textit{outlier-aware sub-bit weight quantization} (OS-Quant) method, which optimizes codeword selection by identifying and adaptively scaling outliers. Furthermore, we propose a \textit{membrane potential-based feature distillation} (MPFD) method, improving the performance of highly compressed S$^2$NN via more precise guidance from a teacher model. Extensive results on vision tasks reveal that S$^2$NN outperforms existing quantized SNNs in both performance and efficiency, making it promising for edge computing applications.

cs.CV↗

arXiv2025

On Topology of Compact Hessian Manifolds

We investigate the global topological constraints and structural properties of compact Hessian manifolds. By establishing novel fibration and splitting theorems, we confirm Chern's conjecture on the vanishing of the Euler characteristic for this class of affine manifolds. Applying these techniques to low dimensions, we provide a topological classification of complete Hessian surfaces. Furthermore, utilizing the theory of Hitchin systems and the Cheng-Yau solution to the real Monge-Amp\`ere equation, we establish a geometric classification of closed orientable Hessian $3$-manifolds.

math.DG↗

arXiv2025

SEFRQO: A Self-Evolving Fine-Tuned RAG-Based Query Optimizer

Query optimization is a crucial problem in database systems that has been studied for decades. Learned query optimizers (LQOs) can improve performance over time by incorporating feedback; however, they suffer from cold-start issues and often require retraining when workloads shift or schemas change. Recent LLM-based query optimizers leverage pre-trained and fine-tuned LLMs to mitigate these challenges. Nevertheless, they neglect LLMs' in-context learning and execution records as feedback for continuous evolution. In this paper, we present SEFRQO, a Self-Evolving Fine-tuned RAG-based Query Optimizer. SEFRQO mitigates the cold-start problem of LQOs by continuously learning from execution feedback via a Retrieval-Augmented Generation (RAG) framework. We employ both supervised fine-tuning and reinforcement fine-tuning to prepare the LLM to produce syntactically correct and performance-efficient query hints. Moreover, SEFRQO leverages the LLM's in-context learning capabilities by dynamically constructing prompts with references to similar queries and the historical execution record of the same query. This self-evolving paradigm iteratively optimizes the prompt to minimize query execution latency. Evaluations show that SEFRQO outperforms state-of-the-art LQOs, achieving up to 65.05% and 93.57% reductions in query latency on the CEB and Stack workloads, respectively, compared to PostgreSQL.

cs.DB↗

arXiv2025

Training-Free ANN-to-SNN Conversion for High-Performance Spiking Transformer

Leveraging the event-driven paradigm, Spiking Neural Networks (SNNs) offer a promising approach for energy-efficient Transformer architectures.While ANN-to-SNN conversion avoids the high training cost of directly trained Spiking Transformers, existing approaches still struggle to handle the nonlinear operations within Transformer blocks, and often require additional fine-tuning of pretrained ANNs.To address these limitations, we propose a training-free and high-performance ANN-to-SNN conversion framework tailored for Transformer architectures. Specifically, we introduce a Multi-basis Exponential Decay (MBE) neuron that combines exponential decay with a multi-basis encoding strategy to effectively approximate nonlinear operations, eliminating the need for weight modifications in pretrained ANNs.Extensive experiments across diverse tasks (CV, NLU, NLG) and mainstream Transformer architectures (ViT, RoBERTa, GPT-2) demonstrate that our method achieves near-lossless conversion accuracy with significantly lower latency. This provides a promising pathway for the efficient and scalable deployment of Spiking Transformers in real-world applications.

cs.LG↗