arXiv subjects
Publications and source records attributed to Tianqi Wu.
It is well-known that the Euclidean plane has a standard 6-regular geodesic triangulation , and the unit sphere has a 5-regular geodesic triangulation, which is induced from the regular Dodecahedron, and the hyperbolic plane has an n-regular geodesic triangulation for any n > 6. Here we constructed a 6-regular geodesic triangulation of the hyperbolic plane.
It has been shown that spaces of geodesic triangulations of closed negatively curved surfaces are contractible. Here we prove that these spaces are homeomorphic to Euclidean spaces $\mathbb{R}^n$.
Purpose: Motion compromises the utility of high-resolution 3D MRI, an established tool in quantitative neuroimaging research. Deep learning-based methods have shown promise for mitigating motion-induced artifacts, but their development typically requires simulated motion-corrupted data. Several open-source tools exist for this task, each implementing different algorithms. However, no scheme currently exists for evaluating the accuracy of these simulations, making it difficult for users to choose the most suitable tool. Developing such a scheme is the aim of this study. Methods: The essential ingredient of the desired scheme is a ground-truth reference simulation that does not suffer from sampling-induced error. To meet this requirement, the proposed scheme, APHABAMAS, leverages a digital phantom whose representations in both the image and Fourier domains can be expressed analytically under arbitrary rigid-body transformations. Results: APHABAMAS is used to quantify the sampling-induced errors of three existing simulation algorithms, establishing their first definitive accuracy-based ranking. Conclusions: APHABAMAS provides a rigorous tool for assessing the accuracy of high-resolution 3D MRI motion-artifact simulations. It allows the accuracy-based ranking of existing simulation algorithms to be established, thereby enabling informed selection of the most suitable algorithm for synthesizing motion-corrupted data.
We estimate the probability that the discrete Gaussian free field on a planar domain with Dirichlet boundary conditions stays positive in the bulk. Improving upon the result by Bolthausen, Deuschel and Giacomin from 2001, we derive the order of the subleading term of this probability when a sequence of discretized scale-ups of given domain and compactly included smooth bulk are considered. A main ingredient in the proof is the double exponential decay of the right tail of the centered minimum of the field in the bulk, conditioned on a certain weighted average of its values to be zero.
Sliding-window attention offers a hardware-efficient solution to the memory and throughput challenges of Large Language Models (LLMs) in long-context scenarios. Existing methods typically employ a single window length across all attention heads and input sizes. However, this uniform approach fails to capture the heterogeneous attention patterns inherent in LLMs, ignoring their distinct accuracy-latency trade-offs. To address this challenge, we propose *Mixture of Attention Spans* (MoA), which automatically tailors distinct sliding-window length configurations to different heads and layers. MoA constructs and navigates a search space of various window lengths and their scaling rules relative to input sizes. It profiles the model, evaluates potential configurations, and pinpoints the optimal length configurations for each head. MoA adapts to varying input sizes, revealing that some attention heads expand their focus to accommodate longer inputs, while other heads consistently concentrate on fixed-length local contexts. Experiments show that MoA increases the effective context length by 3.9x with the same average sliding-window length, boosting retrieval accuracy by 1.5-7.1x over the uniform-window baseline across Vicuna-{7B, 13B} and Llama3-{8B, 70B} models. Moreover, MoA narrows the performance gap with full attention, reducing the maximum relative performance drop from 9%-36% to within 5% across three long-context understanding benchmarks. MoA achieves a 1.2-1.4x GPU memory reduction, boosting decode throughput by 6.6-8.2x and 1.7-1.9x over FlashAttention2 and vLLM, with minimal performance impact. Our code is available at: https://github.com/thu-nics/MoA
We propose a discrete approach for approximating solutions to the prescribed Gaussian curvature problem in two-dimensional manifolds, based on the notion of discrete conformality. Our approach provides an efficient numerical method to compute the solution by minimizing a convex functional.
Generative models have achieved remarkable success across various applications, driving the demand for multi-GPU computing. Inter-GPU communication becomes a bottleneck in multi-GPU computing systems, particularly on consumer-grade GPUs. By exploiting concurrent hardware execution, overlapping computation and communication latency becomes an effective technique for mitigating the communication overhead. We identify that an efficient and adaptable overlapping design should satisfy (1) tile-wise overlapping to maximize the overlapping opportunity, (2) interference-free computation to maintain the original computational performance, and (3) communication agnosticism to reduce the development burden against varying communication primitives. Nevertheless, current designs fail to simultaneously optimize for all of those features. To address the issue, we propose FlashOverlap, which utilizes a novel signaling mechanism: when part of the output finishes, the computation kernel sends a signal to trigger the communication of that part, while continuing the computation of the remaining part (interference-free computation). Consequently, the communication of the finished part and the computation of the remaining part can be overlapped. On top of the signaling mechanism, FlashOverlap comprises two key components: (1) the determination of the signaling timing to boost the overlap efficiency (tile-wise overlapping), and (2) a pre-communication reordering to create the contiguous address for finished data, enabling communication by simply calling NCCL APIs (communication agnosticism), and a post-communication reordering to correct the data order. Experiments show that FlashOverlap achieves up to 1.65x speedup through overlap, outperforming existing works in most cases. Code is available at https://github.com/infinigence/FlashOverlap.
We give a probabilistic proof for the emergence of the Stable-$1$ Law for the random fluctuations of the mass of the extremal process of branching Brownian Motion away from its tip. This result was already shown by Mytnik et al. albeit using PDE techniques. As a consequence, we demystify the origin of these fluctuations and the meaning of the deterministic centering function required.
We prove that the Koebe circle domain conjecture is equivalent to the Weyl type problem that every complete hyperbolic surface of genus zero is isometric to the boundary of the hyperbolic convex hull of the complement of a circle domain. It provides a new way to approach the Koebe's conjecture using convex geometry. Combining our result with the work of He-Schramm on the Koebe conjecture, one establishes that every simply connected non-compact polyhedral surface is discrete conformal to the complex plane or the open unit disk. The main tool we use is Schramm's transboundary extremal lengths.
We proved a rigidity result for Delaunay triangulations of the plane under Luo's discrete conformal change, extending previous results on hexagonal triangulations. Our result is a discrete analogue of the conformal rigidity of the plane. We followed Zhengxu He's analytical approach in his work on the rigidity of disk patterns, and developed a discrete Schwarz lemma and a discrete Liouville theorem. The main tools include conformal modulus, discrete extremal length, and maximum principles in discrete conformal geometry.
We study the limiting extremal and cluster point processes of branching Brownian motion. The former records the heights of all extreme values of the process, while the latter records the relative heights of extreme values in a genealogical neighborhood of order unity around a local maximum thereof. For the extremal point process, we show that the mass of upper level sets $[-v, \infty)$ grows as $C_\star Z v e^{\sqrt{2} v}(1+o(1))$ as $v \to \infty$, almost surely, where $Z$ is the limit of the associated derivative martingale and $C_\star \in (0, \infty)$ is a universal constant. For the cluster point process, we show that the logarithm of the mass of $[-v, \infty)$ grow as $\sqrt{2}v$ minus random fluctuations of order $v^{2/3}$, which are governed by an explicit law in the limit. The first result improves upon the works of Cortines et al. (arXiv:1703.06529) and Mytnik et al. (arXiv:2009.02042) in which asymptotics are shown in probability, while the second makes rigorous the derivation in the physics literature by Mueller et al. (arXiv:1910.06382) and Le et al. (arXiv:2207.07672) and resolves a conjecture thereof.
We prove that for a convex polytope decomposition of a domain in $\mathbb R^n$, an integral curve of a piecewise analytic vector field is chopped by the decomposition into finitely many pieces. As a consequence, we prove the finiteness of the number of the edge flips in a discrete Yamabe flow.
Given a triangulated surface, a polyhedral metric could be constructed by gluing Euclidean triangles edge-to-edge. We carefully describe the construction and prove that such a polyhedral metric is the only intrinsic metric on the glued surface that preserves the lengths of the curves on the Euclidean triangles. We also discuss the edge length coordinates of the Teichmüller space of the polyhedral metrics on a marked surface.
Let $S$ be the 2-sphere and $V \subset S$ be a finite set of at least three points. We show that for each function $κ: V \rightarrow (0, 2π)$ satisfying elementary necessary conditions, in each discrete conformal class of spherical cone-metrics there exists a unique metric realizing $κ$ as its discrete curvature. This can be seen as a discrete version of a result of Luo and Tian.
Many real-world networks exhibit the so-called small-world phenomenon: their typical distances are much smaller than their sizes. One mathematical model for this phenomenon is a long-range percolation graph on a $d$-dimensional box $\{0, 1, \cdots, N\}^d$, in which edges are independently added between far-away sites with probability falling off as a power of the Euclidean distance. A natural question is how the resulting diameter of the box of size $N$, measured in graph-theoretical distance, scales with $N$. This question has been intensely studied in the past and the answer depends on the exponent $s$ in the connection probabilities. In this work we focus on the critical regime $s = d$ studied earlier in a work by Coppersmith, Gamarnik, and Sviridenko and improve the bounds obtained there to a sharp leading-order asymptotic, by exploiting the high degree of concentration due to the large amount of independence in the model.
In this paper, we propose a meshless method of computing eigenvalues and eigenfunctions of a given surface embedded in $\mathbb R^3$. We use point cloud data as input and generate the lattice approximation for some neighborhood of the surface. We compute the eigenvalues and eigenvectors of the cubic lattice graph as an approximation of the eigenvalues and eigenfunctions of the Laplace-Beltrami operator on the surface. We perform extensive numerical experiments on surfaces with various topology and compare our computed eigenvalues from point cloud surface with exact solutions and standard finite element methods using triangle mesh.
We show that a uniformly acute triangulation of the plane is rigid under Luo's discrete conformal change, extending previous results on hexagonal triangulations. Our result is a discrete analogue of the conformal rigidity of the plane. We followed He's analytical approach in his work on the rigidity of disk patterns. The main tools include maximum principles, a discrete Liouville theorem, smooth and discrete extremal lengths on networks. The key step is relating the Euclidean discrete conformality to the hyperbolic discrete conformality, to obtain an L-infinity bound on the discrete conformal factor.
Protein complexes are macromolecules essential to the functioning and well-being of all living organisms. As the structure of a protein complex, in particular its region of interaction between multiple protein subunits (i.e., chains), has a notable influence on the biological function of the complex, computational methods that can quickly and effectively be used to refine and assess the quality of a protein complex's 3D structure can directly be used within a drug discovery pipeline to accelerate the development of new therapeutics and improve the efficacy of future vaccines. In this work, we introduce the Equivariant Graph Refiner (EGR), a novel E(3)-equivariant graph neural network (GNN) for multi-task structure refinement and assessment of protein complexes. Our experiments on new, diverse protein complex datasets, all of which we make publicly available in this work, demonstrate the state-of-the-art effectiveness of EGR for atomistic refinement and assessment of protein complexes and outline directions for future work in the field. In doing so, we establish a baseline for future studies in macromolecular refinement and structure analysis.