arXiv subjects

Tianle Liu

Publications and source records attributed to Tianle Liu.

At least 19 recordsLinked to original sources

arXiv2026

Intrinsic Restriction Traces and Toric Dynamics

We prove Morita invariance of the Campbell--Lind--Malkiewich--Ponto--Zakharevich restriction-system trace after passage to perfect modules. It therefore defines an intrinsic integral restriction trace for an exact endofunctor of a small idempotent-complete stable $\infty$-category. On $\pi_0$, the $m$-th ghost is the laced trace of the $m$-fold iterate, compatibly with Frobenius. For lattice-graded algebras, the ghost targets have twisted cocenters in degree zero; over the open parameter torus, this applies to the cyclic bimodule of Dinkins--Karpov--Krylov. For finite monomial endomorphisms of toric varieties, we construct motivic restriction classes whose ghosts are sums over cones fixed by the iterates. In one example, two classes have the same first ghost, while their second ghosts differ after rational Betti realization.

math.AG↗

arXiv2026

RoLA: Rotary-Positioned Low-Rank Linear Attention for Efficient Diffusion Transformers

Diffusion Transformers (DiTs) achieve strong video generation quality, but their dense spatiotemporal self-attention scales quadratically with sequence length and quickly becomes the dominant inference bottleneck. Sparse low-rank hybrids alleviate this cost by combining a local sparse branch with a global compressed branch. In video DiTs equipped with 3D Rotary Position Embeddings (RoPE), the global branch faces a structural compatibility issue: when RoPE is applied before a nonlinear feature map, the rotation and nonlinearity generally do not commute, making it difficult to keep a query-independent linear summary while preserving relative rotary geometry. Existing work often sidesteps this issue by replacing genuine cross-token global aggregation with coordinate-conditioned surrogates or learnable absolute positional modules. These compromises can be effective, but they approximate relative decay from absolute coordinates and introduce extra positional parameters. We propose \textbf{RoLA}, a rotary-positioned low-rank linear-attention branch that keeps genuine cross-token aggregation while remaining compatible with a reusable linear summary. The design applies RoPE \emph{outside} the nonlinear low-rank feature map and reuses a truncated subset of the pre-trained rotary schedule matched to the low-rank bottleneck. This yields a linear-time low-rank global branch with relative positional behavior by design and no additional positional parameters; the full sparse--low-rank module still includes the fixed-sparsity sparse branch. Experiments on open-source video DiTs show that the resulting method remains competitive in generation quality at 90\% sparsity while achieving 2.63$\times$ end-to-end inference speedup on Wan2.1-14B (720p, 81 frames, measured on an NVIDIA H100 GPU).

cs.CV↗

arXiv2026

Universal Beta Incidence Angles: Cauchy Rigidity and Infinite Arrangements

Let $U$ be Haar-uniform on $\mathbb S^{p-1}$, let $a_1,\ldots,a_k$ be arbitrary nonzero vectors, and let $w_1,\ldots,w_k$ be simplex weights. Define \[ g(U)=\sum_{j=1}^k w_j\frac{a_j}{a_j^\top U}, \qquad N(U)=\frac{g(U)}{\|g(U)\|}. \] We prove the universal incidence law \[ \{U^\top N(U)\}^2\sim\operatorname{Beta}\!\left(\frac12,\frac{p-1}{2}\right), \] independently of the number, arrangement, rank, or overcompleteness of the directions and of the weights. Thus a deterministic, generally non-Haar function of $U$ has the same squared-cosine law as an independent Haar direction. One proof combines a Herglotz--Cauchy boundary principle, a Haar-random two-plane with one common phase, and an exact Beta--Cauchy tangent-projection equivalence. A second proof specializes the positive-semidefinite Pillai--Meng identity. The planar structure leads to converses: plane-conditional Cauchy laws recover positivity, while for signed measures an exact phase-cancellation deficit equals twice the hidden negative mass. This yields local-to-global rigidity under a phase-norming condition strictly weaker than injectivity and an unconditional exclusion of negative atoms. The law extends to probability measures under almost-sure reciprocal integrability. We characterize this condition by an exact Wiener--Dini belt series, prove finite Shannon entropy to be the sharp universal criterion for countable weights, and give an entropy--geometry extension for clustered measures. Every compact carrier of zero one-dimensional Hausdorff measure is admissible, whereas a nonzero rectifiable arc component forces divergence on a set of positive Haar measure. In orthogonal coordinates, the theorem also gives a weight-free scaled $F$ law for Pearson divergence from a fixed simplex vector to a $\operatorname{Dirichlet}(1/2,\ldots,1/2)$ vector.

math.PR↗

arXiv2026

Kac's Walk on Rotation Matrices Mixes in $\boldsymbol{\Theta(n^2)}$ Steps: A Proof Discovered with AI

Let $N=\binom n2=\dim\mathrm{SO}(n)$. We prove that the coordinate-plane Kac walk on $\mathrm{SO}(n)$ has total-variation mixing time of order $N$: for every fixed $0<\varepsilon<1$, \[ t_{\mathrm{mix}}^{(n)}(\varepsilon)=\Theta_\varepsilon(n^2). \] The lower bound is the dimensional singularity obstruction before $N$ steps. The upper bound removes the final logarithm from the previously known $O(n^2\log n)$ estimate. The proof combines the discrete Malliavin coupling and low-degree pseudo-mixing inputs with a new log-free analysis of the derivative shells. Its static core is a circuit-anchored, arbitrary-spectrum root/pass identity for the physical five-box prime. Keeping one normalization base per original circuit permits simultaneous scalar regluing without paying for artificial cuts. Its temporal core is an exact chronological calculus: passive singleton runs acquire a coboundary/Riesz gain, while root-interrupted components are allocated by vertex-labelled packets before absolute values are taken. The curvature split into pure-Weyl and Ricci parts is kept at its physical tensor type. All-Weyl packets retain a full $N^{-1}$ resource; mixed packets contain a typed $O(n^{-1/2})$ Ricci debit; and the final packetless Ricci cell is closed by a joint invariant-column estimate on its two root-hit circuits and an exact causal restoration of the marked root time. These estimates yield an $O(n)$ squared first-derivative shell and a summable all-order marked-shell expansion through logarithmic degree. The resulting score energy is $O(n/c^2)$ after $cN$ steps. A weighted submersion integration-by-parts argument and the Haar log-Sobolev inequality then give the uniform total-variation upper bound. No cutoff profile or cutoff window is asserted.

math.PR↗

arXiv2026

Stein Kernels and Normal Approximation for Log-Concave Bilinear Forms

Jiang, Lee, and Vempala conjectured that if $X,Y\in\mathbb{R}^n$ are independent isotropic log-concave random vectors, then $W_2(L(\langle X,Y\rangle),N(0,n))$ is bounded by a universal constant. Subject to Theorems 1.2 and 2.5 of arXiv:2607.24164v1, we prove this conjecture and a rectangular bilinear-form extension. For independent isotropic log-concave $X\in\mathbb{R}^m$, $Y\in\mathbb{R}^n$, and nonzero $B\in\mathbb{R}^{m\times n}$, put \[ r_4(B)=\frac{(\operatorname{Tr}(B^\top B))^2} {\operatorname{Tr}((B^\top B)^2)}. \] We construct a nonnegative scalar Stein kernel for $X^\top B Y/\|B\|_F$ whose squared $L^2$ discrepancy is at most $20/r_4(B)$, and consequently obtain the same bound for squared $2$-Wasserstein distance to $N(0,1)$. The proof develops an exact covariance identity and deficit decomposition for trace observables of moment-map Stein kernels, together with a stability theorem for positive Stein kernels under log-concave approximation. Taking $B=I_n$ yields \[ W_2^2\left(L\left(\frac{\langle X,Y\rangle}{\sqrt{n}}\right),N(0,1)\right)\leq\frac{20}{n}, \] which is the Jiang--Lee--Vempala conjecture.

math.PR↗

arXiv2026

A Forward-Inverse Dynamic Game Framework for Enhanced Multi-Agent Trajectory Planning

This paper studies feedback Nash equilibrium (FBNE) seeking for multi-agent trajectory planning in nonlinear dynamical systems with unknown agents' objectives and state-dependent inter-agent coupling. While dynamic game theory provides a principled framework for such problems, existing approaches typically assume fully rational agents with known objectives or rely on fixed regularization, limiting their ability to capture bounded rationality and spatially varying interaction intensity in safety-critical settings. To this end, we propose a KL-regularized dynamic game with a state-dependent weight that adaptively balances optimality and behavioral priors. To infer unknown cost parameters from demonstrated behaviors, we develop a context-aware inverse game module based on maximum-entropy inverse reinforcement learning with physics-informed regularization, ensuring structural consistency with the forward game. We establish per-iteration well-posedness of the regularized local game and show that the adaptive weighting function remains Lipschitz continuous under bounded nominal-trajectory updates. Numerical simulations and multi-robot experiments on cooperative navigation and merging scenarios validate the effectiveness of the proposed framework.

cs.RO↗

arXiv2026

Torus-enriched Motivic Bruhat Complexes and Maximal Compact Groups

Bruhat decompositions give cellular models for split algebraic groups, flag varieties, and maximal compact groups, but motivic boundaries retain orientation and torus-translation data lost in the flag quotient. Over a perfect field of characteristic zero, let the group be connected, split, semisimple, and simply connected. Fixing a Borel subgroup with split maximal torus and unipotent radical, we construct a torus-enriched motivic cellular complex for the basic affine space and compute its boundary in every degree. Each cover in Bruhat order contributes a two-face operator determined by a transported coroot, a tail determinant weight, and an explicit Milnor--Witt frame degree. Bott--Samelson purity proves the formula, while the unipotent torsor identifies the complex with that of the group. Over the real numbers, realization identifies it at chain level with the extended-Weyl complex of a maximal compact subgroup, while torus augmentation gives the flag complex. A single motivic complex therefore interpolates between the two incidence theories. A finite torus-support filtration makes this explicit; after inversion of two it splits by the characters of the component group of the real split torus, and the support spectral sequence degenerates. Calculations in the rank-three special linear and exceptional rank-two cases exhibit the first higher differentials beyond the previously known range.

math.AG↗

arXiv2026

Cellular $\mathbb{A}^1$-Homology from Bruhat Boundary Matrices of Split Semisimple Flag Varieties

Let $k$ be a perfect field of characteristic different from 2, we compute the cellular $\mathbb{A}^1$-homology of the flag varieties $G/P_\Theta$ attached to split semisimple simply connected groups over $k$ and describe the differentials in the cellular $\mathbb{A}^1$-chain complex concretely. The construction applies uniformly to the type $A$ coefficient formula, to the type $B_n,C_n,D_n$ for $n\leq 7$, and to the exceptional types for $F_4,E_6,E_7$. Under real realization over $k=\mathbb{R}$, this computation recovers the corresponding results of real flag manifolds. We also provide a detailed computation for $SL_3/B$ and an application to the full split flag variety of type $F_4$.

math.AG↗

arXiv2026

Autonomous End-to-End SOH Prediction Services for Battery Systems via Temporal-Contrastive Representation Learning

Accurate state of health (SOH) estimation is a critical diagnostic service for lithium-ion battery management. However, reliance on labor-intensive manual feature engineering and opaque black-box models hinders scalable industrial deployment. To address this, we introduce TC-SOH: a modular, plug-and-play service architecture for autonomous, end-to-end SOH prediction. TC-SOH employs a temporal-contrastive mechanism and a cross-window prediction pretext task to extract degradation-relevant representations directly from raw operational data. To improve transparency, we connect model efficacy with representation diagnostics: visualization, sensitivity analysis, redundancy analysis, bidirectional probing, future-SOH probing, and temporal shuffling show that learned features overlap with selected expert descriptors while retaining additional SOH-relevant variation, and that ordered temporal context improves subsequent-SOH prediction. Across four public datasets, TC-SOH outperforms the considered physics-informed and data-driven baselines, reducing MAPE by 1.91 times and RMSE by 2.13 times.

cs.LG↗

arXiv2026

Discrete-WAM: Unified Discrete Vision-Action Token Editing for World-Policy Learning

Autonomous driving requires reasoning about how ego actions shape future world evolution, rather than merely mapping observations to actions. However, most end-to-end methods rely on direct state-to-action imitation, while existing world models often remain weakly aligned with downstream policy generation. We introduce Discrete-WAM, a unified discrete vision-action world-policy framework that represents visual observations, future states, high-level decisions, and ego actions within a shared token space. Built on this discrete alignment, Discrete-WAM jointly trains world modeling, world-policy modeling, and policy modeling through multi-task and multi-stage pretraining, allowing action-conditioned future prediction to directly support policy generation. For downstream planning, Discrete-WAM further decomposes policy generation into hierarchical decision prediction and parallel action-token editing, where the decision token provides a high-level planning skeleton and confidence-based scheduling refines dense future actions efficiently. Experiments on large-scale autonomous-driving benchmarks show that Discrete-WAM achieves strong planning performance while supporting controllable future generation, counterfactual evaluation, surprise-based world-model analysis, and efficient parallel policy decoding. These results suggest that discrete representation alignment, unified world-policy training, and hierarchical token editing provide a promising design paradigm for physical AI.

cs.RO↗

arXiv2026

GS-Playground: A High-Throughput Photorealistic Simulator for Vision-Informed Robot Learning

Embodied AI research is undergoing a shift toward vision-centric perceptual paradigms. While massively parallel simulators have catalyzed breakthroughs in proprioception-based locomotion, their potential remains largely untapped for vision-informed tasks due to the prohibitive computational overhead of large-scale photorealistic rendering. Furthermore, the creation of simulation-ready 3D assets heavily relies on labor-intensive manual modeling, while the significant sim-to-real physical gap hinders the transfer of contact-rich manipulation policies. To address these bottlenecks, we propose GS-Playground, a multi-modal simulation framework designed to accelerate end-to-end perceptual learning. We develop a novel high-performance parallel physics engine, specifically designed to integrate with a batch 3D Gaussian Splatting (3DGS) rendering pipeline to ensure high-fidelity synchronization. Our system achieves a breakthrough throughput of 10^4 FPS at 640x480 resolution, significantly lowering the barrier for large-scale visual RL. Additionally, we introduce an automated Real2Sim workflow that reconstructs photorealistic, physically consistent, and memory-efficient environments, streamlining the generation of complex simulation-ready scenes. Extensive experiments on locomotion, navigation, and manipulation demonstrate that GS-Playground effectively bridges the perceptual and physical gaps across diverse embodied tasks. Project homepage: https://gsplayground.github.io.

cs.RO↗

arXiv2026

Difficulty-Estimated Policy Optimization

Recent advancements in Large Reasoning Models (LRMs), exemplified by DeepSeek-R1, have underscored the potential of scaling inference-time compute through Group Relative Policy Optimization (GRPO). However, GRPO frequently suffers from gradient signal attenuation when encountering problems that are either too trivial or overly complex. In these scenarios, the disappearance of inter-group advantages makes the gradient signal susceptible to noise, thereby jeopardizing convergence stability. While variants like DAPO attempt to rectify gradient vanishing, they do not alleviate the substantial computational overhead incurred by exhaustive rollouts on low-utility samples. In this paper, we propose Difficulty-Estimated Policy Optimization (DEPO), a novel framework designed to optimize the efficiency and robustness of reasoning alignment. DEPO integrates an online Difficulty Estimator that dynamically assesses and filters training data before the rollout phase. This mechanism ensures that computational resources are prioritized for samples with high learning potential. Empirical results demonstrate that DEPO achieves up to a 2x reduction in rollout costs without compromising model performance. Our approach significantly lowers the computational barrier for training high-performance reasoning models, offering a more sustainable path for reasoning scaling. Code and data will be released upon acceptance.

cs.AI↗

arXiv2026

Fast Converging 3D Gaussian Splatting for 1-Minute Reconstruction

We present a fast 3DGS reconstruction pipeline designed to converge within one minute, developed for the SIGGRAPH Asia 3DGS Fast Reconstruction Challenge. The challenge consists of an initial round using SLAM-generated camera poses (with noisy trajectories) and a final round using COLMAP poses (highly accurate). To robustly handle these heterogeneous settings, we develop a two-stage solution. In the first round, we use reverse per-Gaussian parallel optimization and compact forward splatting based on Taming-GS and Speedy-splat, load-balanced tiling, an anchor-based Neural-Gaussian representation enabling rapid convergence with fewer learnable parameters, initialization from monocular depth and partially from feed-forward 3DGS models, and a global pose refinement module for noisy SLAM trajectories. In the final round, the accurate COLMAP poses change the optimization landscape; we disable pose refinement, revert from Neural-Gaussians back to standard 3DGS to eliminate MLP inference overhead, introduce multi-view consistency-guided Gaussian splitting inspired by Fast-GS, and introduce a depth estimator to supervise the rendered depth. Together, these techniques enable high-fidelity reconstruction under a strict one-minute budget. Our method achieved the top performance with a PSNR of 28.43 and ranked first in the competition.

cs.CV↗

arXiv2025

CTTA-T: Continual Test-Time Adaptation for Text Understanding via Teacher-Student with a Domain-aware and Generalized Teacher

Text understanding often suffers from domain shifts. To handle testing domains, domain adaptation (DA) is trained to adapt to a fixed and observed testing domain; a more challenging paradigm, test-time adaptation (TTA), cannot access the testing domain during training and online adapts to the testing samples during testing, where the samples are from a fixed domain. We aim to explore a more practical and underexplored scenario, continual test-time adaptation (CTTA) for text understanding, which involves a sequence of testing (unobserved) domains in testing. Current CTTA methods struggle in reducing error accumulation over domains and enhancing generalization to handle unobserved domains: 1) Noise-filtering reduces accumulated errors but discards useful information, and 2) accumulating historical domains enhances generalization, but it is hard to achieve adaptive accumulation. In this paper, we propose a CTTA-T (continual test-time adaptation for text understanding) framework adaptable to evolving target domains: it adopts a teacher-student framework, where the teacher is domain-aware and generalized for evolving domains. To improve teacher predictions, we propose a refine-then-filter based on dropout-driven consistency, which calibrates predictions and removes unreliable guidance. For the adaptation-generalization trade-off, we construct a domain-aware teacher by dynamically accumulating cross-domain semantics via incremental PCA, which continuously tracks domain shifts. Experiments show CTTA-T excels baselines.

cs.CL↗

arXiv2025

LADY: Linear Attention for Autonomous Driving Efficiency without Transformers

End-to-end autonomous driving has emerged as a promising paradigm. However, state-of-the-art methods rely heavily on Transformer architectures. The inherent quadratic complexity of Transformers restricts their ability to model long-range spatial and temporal dependencies, particularly on resource-constrained edge platforms. Given the inherent demand for efficient temporal modeling in autonomous driving, this computational bottleneck severely constrains real-time deployment. While linear attention mechanisms offer a computationally efficient alternative, existing architectures are predominantly limited to self-attention, lacking the cross-modal capabilities essential for autonomous driving. In this work, we propose LADY, the first fully linear attention-based generative model for end-to-end autonomous driving. LADY incorporates a novel, lightweight linear cross-attention (LICA) mechanism to enable effective cross-modal interaction while preserving linearity. A key advantage of our framework is its ability to fuse long-range temporal contexts during inference with constant computational and memory costs ($O(1)$), regardless of the historical sequence length. Experiments on the NAVSIM and Bench2Drive benchmarks demonstrate that LADY achieves performance comparable to state-of-the-art methods, delivering competitive planning accuracy with significantly reduced latency. Furthermore, efficiency benchmarking on edge devices validates the model's feasibility for resource-limited scenarios.

cs.AI↗

arXiv2025

A Looming of phantoms

Following Krah's method, we construct new examples of phantom categories as semiorthogonal components of the derived categories of two types of rational surfaces: the blowup of the plane at 11 points in general position, and the blowup of the second Hirzebruch surface at 9 points in general position. We also pose conjectures about the existence of phantom subcategories in the derived categories of other rational surfaces, obtained as the blowups of the other Hirzebruch surfaces.

math.AG↗

arXiv2025

Dexbotic: Open-Source Vision-Language-Action Toolbox

In this paper, we present Dexbotic, an open-source Vision-Language-Action (VLA) model toolbox based on PyTorch. It aims to provide a one-stop VLA research service for professionals in the field of embodied intelligence. It offers a codebase that supports multiple mainstream VLA policies simultaneously, allowing users to reproduce various VLA methods with just a single environment setup. The toolbox is experiment-centric, where the users can quickly develop new VLA experiments by simply modifying the Exp script. Moreover, we provide much stronger pretrained models to achieve great performance improvements for state-of-the-art VLA policies. Dexbotic will continuously update to include more of the latest pre-trained foundation models and cutting-edge VLA models in the industry.

cs.RO↗

arXiv2025

Generative AI for subgrid turbulence in large-eddy simulations

Turbulence governs the transport of momentum, energy, and scalars in many geophysical and engineering flows. In large-eddy simulations (LES), parameterizing subgrid-scale (SGS) stresses remains a central challenge, as unresolved physical processes strongly influence turbulent transport. Traditional SGS models, such as the Smagorinsky-type models and deep neural networks (DNNs), are deterministic and cannot capture the stochastic nature of turbulence. Despite its wide application in computer vision and natural language processing, generative artificial intelligence (AI) has not previously been applied to directly compute SGS stresses in three-dimensional turbulent boundary layers at high Reynolds numbers. Here we introduce a denoising diffusion probabilistic model (DDPM) to reconstruct SGS stresses from coarse-grained velocity fields in direct numerical simulations of the atmospheric boundary layer. The DDPM consistently outperforms Smagorinsky-type models and previous deep neural networks in terms of spatial correlations and probability distributions for deviatoric stresses, and can be applied to unseen convective stability conditions and resolutions. By learning conditional distributions rather than pointwise values, this generative approach opens a new direction for SGS turbulence modeling at high Reynolds numbers.

physics.flu-dyn↗