arXiv subjects

Xiaojun Tan

Publications and source records attributed to Xiaojun Tan.

9 recordsLinked to original sources

arXiv2026

Residual bounds for Schur-stable polynomials

Let $r_n$ be the infimum of \[ \frac{\lVert P'-P'(0)P\rVert_{H^2}}{\lVert P\rVert_{H^2}} \] over all degree-$n$ polynomials $P$ satisfying $P(0)=1$ whose zeros lie in the closed unit disk. We prove the quantitative residual bound \[ r_n\geq \exp\!\bigl(-(1+o(1))\sqrt n\log n\bigr) \qquad(n\to\infty). \] As an application, we answer Erd\H{o}s Problem 973 on exterior power sums in the negative, in a form quantitatively stronger than the answer first obtained by Luo, Yang, and Zhu.

math.CV↗

arXiv2026

Ordinary Least Squares is a Special Case of Transformer

The statistical essence of the Transformer architecture has long remained elusive: Is it a universal approximator, or a neural network version of known computational algorithms? Through rigorous algebraic proof, we show that the latter better describes Transformer's basic nature: Ordinary Least Squares (OLS) is a special case of the single-layer Linear Transformer. Using the spectral decomposition of the empirical covariance matrix, we construct a specific parameter setting where the attention mechanism's forward pass becomes mathematically equivalent to the OLS closed-form projection. This means attention can solve the problem in one forward pass, not by iterating. Building upon this prototypical case, we further uncover a decoupled slow and fast memory mechanism within Transformers. Finally, the evolution from our established linear prototype to standard Transformers is discussed. This progression facilitates the transition of the Hopfield energy function from linear to exponential memory capacity, thereby establishing a clear continuity between modern deep architectures and classical statistical inference.

cs.LG↗

arXiv2024

InScope: A New Real-world 3D Infrastructure-side Collaborative Perception Dataset for Open Traffic Scenarios

Perception systems of autonomous vehicles are susceptible to occlusion, especially when examined from a vehicle-centric perspective. Such occlusion can lead to overlooked object detections, e.g., larger vehicles such as trucks or buses may create blind spots where cyclists or pedestrians could be obscured, accentuating the safety concerns associated with such perception system limitations. To mitigate these challenges, the vehicle-to-everything (V2X) paradigm suggests employing an infrastructure-side perception system (IPS) to complement autonomous vehicles with a broader perceptual scope. Nevertheless, the scarcity of real-world 3D infrastructure-side datasets constrains the advancement of V2X technologies. To bridge these gaps, this paper introduces a new 3D infrastructure-side collaborative perception dataset, abbreviated as inscope. Notably, InScope is the first dataset dedicated to addressing occlusion challenges by strategically deploying multiple-position Light Detection and Ranging (LiDAR) systems on the infrastructure side. Specifically, InScope encapsulates a 20-day capture duration with 303 tracking trajectories and 187,787 3D bounding boxes annotated by experts. Through analysis of benchmarks, four different benchmarks are presented for open traffic scenarios, including collaborative 3D object detection, multisource data fusion, data domain transfer, and 3D multiobject tracking tasks. Additionally, a new metric is designed to quantify the impact of occlusion, facilitating the evaluation of detection degradation ratios among various algorithms. The Experimental findings showcase the enhanced performance of leveraging InScope to assist in detecting and tracking 3D multiobjects in real-world scenarios, particularly in tracking obscured, small, and distant objects. The dataset and benchmarks are available at https://github.com/xf-zh/InScope.

cs.CV↗

arXiv2023

Graphical Representations and Worm Algorithms for the O($N$) Spin Model

We present a family of graphical representations for the O($N$) spin model, where $N \ge 1$ represents the spin dimension, and $N=1,2,3$ corresponds to the Ising, XY and Heisenberg models, respectively. With an integer parameter $0 \le \ell \le N/2$, each configuration is the coupling of $\ell$ copies of subgraphs consisting of directed flows and $N -2\ell$ copies of subgraphs constructed by undirected loops, which we call the XY and Ising subgraphs, respectively. On each lattice site, the XY subgraphs satisfy the Kirchhoff flow-conservation law and the Ising subgraphs obey the Eulerian bond condition. Then, we formulate worm-type algorithms and simulate the O($N$) model on the simple-cubic lattice for $N$ from 2 to 6 at all possible $\ell$. It is observed that the worm algorithm has much higher efficiency than the Metropolis method, and, for a given $N$, the efficiency is an increasing function of $\ell$. Beside Monte Carlo simulations, we expect that these graphical representations would provide a convenient basis for the study of the O($N$) spin model by other state-of-the-art methods like the tensor network renormalization.

cond-mat.stat-mech↗

arXiv2022

LMBAO: A Landmark Map for Bundle Adjustment Odometry in LiDAR SLAM

LiDAR odometry is one of the essential parts of LiDAR simultaneous localization and mapping (SLAM). However, existing LiDAR odometry tends to match a new scan simply iteratively with previous fixed-pose scans, gradually accumulating errors. Furthermore, as an effective joint optimization mechanism, bundle adjustment (BA) cannot be directly introduced into real-time odometry due to the intensive computation of large-scale global landmarks. Therefore, this letter designs a new strategy named a landmark map for bundle adjustment odometry (LMBAO) in LiDAR SLAM to solve these problems. First, BA-based odometry is further developed with an active landmark maintenance strategy for a more accurate local registration and avoiding cumulative errors. Specifically, this paper keeps entire stable landmarks on the map instead of just their feature points in the sliding window and deletes the landmarks according to their active grade. Next, the sliding window length is reduced, and marginalization is performed to retain the scans outside the window but corresponding to active landmarks on the map, greatly simplifying the computation and improving the real-time properties. In addition, experiments on three challenging datasets show that our algorithm achieves real-time performance in outdoor driving and outperforms state-of-the-art LiDAR SLAM algorithms, including Lego-LOAM and VLOM.

cs.CV↗

arXiv2021

ASPCNet: A Deep Adaptive Spatial Pattern Capsule Network for Hyperspectral Image Classification

Previous studies have shown the great potential of capsule networks for the spatial contextual feature extraction from {hyperspectral images (HSIs)}. However, the sampling locations of the convolutional kernels of capsules are fixed and cannot be adaptively changed according to the inconsistent semantic information of HSIs. Based on this observation, this paper proposes an adaptive spatial pattern capsule network (ASPCNet) architecture by developing an adaptive spatial pattern (ASP) unit, that can rotate the sampling location of convolutional kernels on the basis of an enlarged receptive field. Note that this unit can learn more discriminative representations of HSIs with fewer parameters. Specifically, two cascaded ASP-based convolution operations (ASPConvs) are applied to input images to learn relatively high-level semantic features, transmitting hierarchical structures among capsules more accurately than the use of the most fundamental features. Furthermore, the semantic features are fed into ASP-based conv-capsule operations (ASPCaps) to explore the shapes of objects among the capsules in an adaptive manner, further exploring the potential of capsule networks. Finally, the class labels of image patches centered on test samples can be determined according to the fully connected capsule layer. Experiments on three public datasets demonstrate that ASPCNet can yield competitive performance with higher accuracies than state-of-the-art methods.

cs.CV↗

arXiv2020

$N$-cluster correlations in four- and five-dimensional percolation

We study $N$-cluster correlation functions in four- and five-dimensional (4D, 5D) bond percolation by extensive Monte Carlo simulation. We reformulate the transfer Monte Carlo algorithm for percolation [Phys. Rev. E {\bf 72}, 016126 (2005)] using the disjoint-set data structure, and simulate a cylindrical geometry $L^{d-1}\times \infty$, with the linear size up to $L=512$ for 4D and $128$ for 5D. We determine with a high precision all possible $N$-cluster exponents, for $N \! =\!2$ and $3$, and the universal amplitude for a logarithmic correlation function. From the symmetric correlator with $N \! = \!2$, we obtain the correlation-length critical exponent as $1/ν\! =\! 1.4610(12)$ for 4D and $1/ν\! =\! 1.737 (2)$ for 5D, significantly improving over the existing results. Estimates for the other exponents and the universal logarithmic amplitude have not been reported before to our knowledge. Our work demonstrates the validity of logarithmic conformal field theory and adds to the growing knowledge for high-dimensional percolation.

cond-mat.stat-mech↗

arXiv2019

Clock Monte Carlo methods

We propose the clock Monte Carlo technique for sampling each successive chain step in constant time. It is built on a recently proposed factorized transition filter and its core features include its O(1) computational complexity and its generality. We elaborate how it leads to the clock factorized Metropolis (clock FMet) method, and discuss its application in other update schemes. By grouping interaction terms into boxes of tunable sizes, we further formulate a variant of the clock FMet algorithm, with the limiting case of a single box reducing to the standard Metropolis method. A theoretical analysis shows that an overall acceleration of ${\rm O}(N^κ)$ ($0 \! \leq \! κ\! \leq \! 1$) can be achieved compared to the Metropolis method, where $N$ is the system size and the $κ$ value depends on the nature of the energy extensivity. As a systematic test, we simulate long-range O$(n)$ spin models in a wide parameter regime: for $n \! = \! 1,2,3$, with disordered algebraically decaying or oscillatory Ruderman-Kittel-Kasuya-Yoshida-type interactions and with and without external fields, and in spatial dimensions from $d \! = \! 1, 2, 3$ to mean-field. The O(1) computational complexity is demonstrated, and the expected acceleration is confirmed. Its flexibility and its independence from the interaction range guarantee that the clock method would find decisive applications in systems with many interaction terms.

cond-mat.stat-mech↗

arXiv2018

Observation of non-scalar and logarithmic correlations in 2D and 3D percolation

Percolation, a paradigmatic geometric system in various branches of physical sciences, is known to possess logarithmic factors in its correlators. Starting from its definition, as the $Q\rightarrow1$ limit of the $Q$-state Potts model with $S_Q$ symmetry, in terms of geometrical clusters, its operator content as $N$-cluster observables has been classified. We extensively simulate critical bond percolation in two and three dimensions and determine with high precision the $N$-cluster exponents and non-scalar features up to $N \! =\! 4$ (2D) and $N \! =\! 3$ (3D). The results are in excellent agreement with the predicted exact values in 2D, while such families of critical exponents have not been reported in 3D, to our knowledge. Finally, we demonstrate the validity of predictions about the logarithmic structure between the energy and two-cluster operators in 3D.

cond-mat.stat-mech↗