arXiv subjects

Tianle Liu

Publications and source records attributed to Tianle Liu.

At least 19 recordsLinked to original sources

arXiv2026

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

We determine the cellular $\mathbb{A}^1$-chain complex of full and partial flag varieties of split semisimple simply connected groups over a perfect field of characteristic different from 2. With compatible integral normal orientations and reindexing by opposite Bruhat cells, the differential in each degree at least two is represented by the motivic Hopf map $η$ times one half of the signed cellular boundary matrix of the corresponding real flag manifold. Consequently, the Smith data of the resulting integer complex give all positive-degree cellular homology sheaves in terms of Milnor--Witt $K$-theory and the kernels and cokernels of integer multiples of $η$. We obtain a uniform computation of the second cellular $\mathbb{A}^1$-homology of $SL_n/B$ for $n\geq3$, together with complete calculations for $SL_3/B$ and the full flags of types $F_4$ and $G_2$.

math.AG↗

arXiv2026

Optimality of the $D_4$ lattice quantizer

We prove that the root lattice $D_4$ uniquely minimizes the normalized second moment among four-dimensional lattices, up to similarity. The minimum is $13/(120\sqrt{2})$. The proof reduces the problem to polynomial inequalities on three secondary cones. We derive the second-moment formulas geometrically and give explicit polynomial certificates that can be checked by coefficient comparison. We also derive a lower bound for lattice quantizers in every dimension at least two from the number of Voronoi facets, using a comparison with circular cones and concavity in the solid angle. This bound strictly improves the sphere bound and recovers the hexagonal value in dimension two.

math.MG↗

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 show that DEPO achieves approximately 1.86x faster rollouts than DAPO in the profiled configuration, while improving benchmark performance over GRPO at comparable training cost. Our approach significantly lowers the computational barrier for training high-performance reasoning models, offering a more sustainable path for reasoning scaling.

cs.AI↗

arXiv2026

Categorified Cluster Structures on Coulomb Branches of Certain Star-Shaped Quivers

We study the quantized $K$-theoretic Coulomb branches of framed star-shaped quivers with central gauge dimension $n\geq2$, $r-1$ one-dimensional outer gauge vertices, and a one-dimensional framing at the central vertex. For $n=2$ and arbitrary $r\geq1$, and for every $n\geq2$ with $1\leq r\leq5$, we prove that the loop-graded Koszul-perverse heart is a monoidal categorification of the associated quantum cluster algebra over $\mathbb{Z}[v^{\pm1}]$. This quantum cluster algebra coincides with its upper quantum cluster algebra. For all $n\geq2$ and $r\geq1$, we construct an explicit initial seed and identify the associated upper quantum cluster algebra with the Grothendieck ring of this heart. Every mutation is realized by a short exact sequence, and every quantum cluster monomial is represented by a simple object.

math.RT↗

arXiv2026

Energy Ring: Stabilizing Feature Energy for Reliable Test-Time Discovery

Test-Time Discovery (TTD) aims to recognize known classes and detect novel ones while the model is already deployed, operating directly on a continual unlabeled test-time stream. Existing TTD methods are overly conservative, making known and unknown features collapse together and difficult to separate during test-time adaptation. In this paper, we revisit TTD from an energy perspective and find that the natural energy gap between known and unknown classes is unstable and easily collapses. We therefore propose Energy Ring to stabilize this gap by anchoring the two types of classes on separate energy rings. A lightweight adapter enlarges the norm-based gap while preserving known-class directions, and a perturbation consistency module forms compact clusters for unknown classes. During testing, energy-guided confidence weighting improves prediction, and a stability-based rectification mechanism corrects unreliable updates. Experimental results show that Energy Ring significantly improves TTD performance against existing TTD methods.

cs.AI↗

arXiv2026

GG-Nav: Goal-oriented Grounding Chain-of-Thought Elicits Reasoning in Navigation Foundation Models

Emerging strong Vision-Language Models (VLMs) have advanced navigation foundation models. However, these models still have unsatisfactory performance on goal-oriented navigation settings. In this paper, we first conduct a motivating study revealing that the key issue lies in the insufficient utilization of streaming observations and imprecise target localization. To tackle these challenges, we propose GG-Nav, a simple yet effective navigation framework that learns a structured chain-of-thought (CoT) mechanism via supervised fine-tuning to explicitly localize goal-relevant instances, accumulate grounding evidence, and support more reliable decision-making. Specifically, the structured CoT is designed with two key stages: (i) Observation-to-Evidence, which reasons over all intermediate frames between adjacent decision steps to retain useful goal-related evidence; and (ii) Evidence-to-Decision, which integrates the current observation with accumulated evidence to ground the target and support the generation of navigation actions or clarification questions. Extensive experiments show that GG-Nav outperforms the state-of-the-art methods in terms of success rates on the HM3D ObjectNav, VL-LN, and CoIN benchmarks by 3.1%, 16.6%, and 5.3%, respectively. Ablation studies further show that accumulating evidence from intermediate observations reduces evidence-missing failures by 11.6%, while explicit grounding reduces stopping errors by 49.4%. We also deploy GG-Nav on a Unitree Go2 robot, demonstrating its zero-shot generalization capability in unseen real-world environments.

cs.RO↗

arXiv2026

Relational Semantic Grounding for Test-Time Discovery

Test-Time Discovery (TTD) aims to recognize known classes and detect novel ones from a continuous unlabeled test stream after deployment. Existing TTD methods mainly rely on visual clustering or prototype comparison, and can only assign anonymous labels to discovered classes, leaving them difficult to interpret in open-world applications. In this paper, we propose SEGA, a Semantic Grounding with Attributes framework for TTD. SEGA leverages large language models to generate attribute-rich descriptions for known classes and constructs an attribute-anchored visual-semantic space during training. At test time, we introduce Relational Semantic Grounding (RSG), which grounds unknown samples with similar and dissimilar known-class anchors, prompts the LLM to generate candidate concepts, and reranks them in the learned semantic space. The selected concept is further used for confidence-guided test-time semantic stabilization. By coupling attribute-level semantic alignment with relational concept grounding, SEGA transforms anonymous test-time clusters into interpretable semantic concepts. Experiments show that SEGA improves TTD baselines with the promise of semantic grounding for open-world.

cs.AI↗

arXiv2026

SparkDiffusion: Mitigating the High-Sparsity Trap --- A Unified Framework for up to $265\times$ Single-GPU Acceleration of Visual Generation

Video diffusion transformers are expensive because attention dominates long spatiotemporal token sequences. We identify the \emph{high-sparsity trap}: at extreme attention sparsity, step-local training losses keep decreasing while terminal generation quality stagnates or degrades. The trap is one of supervision: the dominant terminal errors originate in the high-noise structure-generation stage, and terminal-aligned training corrects terminal errors that substantially extended step-local training cannot. This yields a simple staging principle: \emph{first adapt the sparse architecture into a coarse prior, then correct the terminal distribution}. We instantiate the principle as \method, a unified acceleration framework for visual generation that combines a short sparse warm-up, few-step trajectory-mixed distillation, and FP8 quantization with fused kernels. \method sustains $97\%$ attention sparsity with strong visual quality on long-sequence 720P generation across Wan2.1/Wan2.2 backbones and T2V/I2V tasks, and $90\%$ sparsity on Wan2.1-T2V-1.3B-480P. With 3-step CFG-free inference, \method achieves a $265\times$ end-to-end speedup over the 50-step CFG dense baseline for Wan2.1-T2V-14B-720P on a single RTX~5090 ($220\times$ on H100), and denoises a Wan2.1-T2V-1.3B-480P video in $1.3$s.

cs.CV↗

arXiv2026

Algebraic Vector Bundles on Cluster Varieties

We study algebraic vector bundles on cluster varieties. Over a perfect field of characteristic different from two, every algebraic vector bundle is trivial on a cluster variety with principal coefficients at an acyclic seed of irreducible finite Dynkin type. The proof is based on integral contractions of cellular $\mathbb A^1$-chain complexes and successive freezing. We also construct smooth cluster varieties admitting indecomposable rank-two vector bundles, whose nontriviality is detected by quadratic Euler classes. The cellular calculation also gives a criterion for stable freeness and computes integral real cohomology and the images of cycle class maps.

math.AG↗

arXiv2026

Data-driven measurements of the beauty hadron production in p-Pb collisions at \sqrt{s_{\mathrm{NN}}} = 5.02 and 8.16 TeV

Beauty-hadron production in proton-nucleus collisions is a particularly powerful probe of cold nuclear matter (CNM) effects: thanks to the large beauty-quark mass, the production is initiated in early hard scatterings and is calculable in perturbative QCD, while the modifications observed in nuclear collisions directly constrain the nuclear parton distribution functions (nPDFs) in a kinematic region complementary to that accessed by charm. Precisely quantifying these effects is a prerequisite for isolating possible signatures of quark-gluon plasma formation in heavy-ion collisions. In this work, the production of open beauty hadrons in p-Pb collisions at $\sqrt{s_{\mathrm{NN}}} = 5.02$ and $8.16$~TeV is studied via a data-driven Bayesian unfolding of ALICE and LHCb measurements of non-prompt $D^{0}$ and J/$ψ$ mesons. The unfolded $p_{T}$- and $y$-differential cross sections are mutually consistent across decay channels and compatible with directly reconstructed beauty-hadron measurements. The nuclear modification factor $R_{pPb}$ is reported as a function of transverse momentum and rapidity, providing the first unfolding-based result spanning $-4.0 < y < 4.0$ of beauty-hadron $R_{pPb}$ covering the full rapidity range, with a precision comparable to or better than that of existing data. The results are compared with calculations based on nPDFs and can provide valuable constraints for future global nPDF fits. The total inclusive B hadron production cross section in p-Pb collisions is also reported. A weaker CNM suppression is observed for beauty hadrons than for charm hadrons, consistent with the expected mass hierarchy of cold-nuclear-matter effects.

nucl-ex↗

arXiv2026

F4R: Failure-Driven Recognition, Reconstruction, Refinement, and Redeployment for Continual Robot Self-Improvement

The real-world performance of current vision-language-action models is fundamentally constrained by the limited coverage of expert demonstrations and their insufficient understanding of physical interactions. A common remedy is to collect additional real-world demonstrations of newly encountered failures. However, this process is costly, inefficient, potentially unsafe, and difficult to scale. To address this challenge, we propose Failure for Rising (F4R), a failure-driven real-to-sim-to-real closed-loop learning framework that converts real-world failures into targeted policy improvement. F4R first uses an agent to automatically identify and diagnose failures from rollouts. It reconstructs each failure as an interactive, object-centric table-top environment that preserves the task-relevant spatial and physical conditions. The policy is then refined through failure-conditioned sim-real co-training followed by targeted reinforcement learning in the reconstructed environments. The improved policy is subsequently redeployed, while newly observed failures are continuously fed back into the next reconstruction and learning cycle. Real-world evaluations on four manipulation tasks show that F4R achieves 93.75% In-Distribution and 90.0% Out-of-Distribution (OOD) success, outperforming the budget-matched Targeted BC baseline by 18.75 percentage points under OOD conditions without collecting additional real-world corrective demonstrations.

cs.RO↗

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

A Heavily Right Strategy for Statistical Inference with Dependent Studies in Arbitrary Dimensions

We leverage recent advances in heavy-tail approximations for global hypothesis testing with dependent studies to construct approximate confidence regions without modeling or estimating their dependence structures. A non-rejection region is a confidence region but it may not be convex. Convexity is appealing because it ensures any one-dimensional linear projection of the region is a confidence interval, easy to compute and interpret. We show why convexity fails for nearly all heavy-tail combination tests proposed in recent years, including the influential Cauchy combination test. These insights motivate a heavily right strategy: truncating the left half of the Cauchy distribution to obtain the Half-Cauchy combination test. The harmonic mean test also corresponds to a heavily right distribution with a Cauchy-like tail, namely a Pareto distribution with unit power. We prove that both approaches guarantee convexity when individual studies are summarized by Hotelling $T^2$ or $χ^{2}$ statistics (regardless of the validity of this summary) and provide efficient, exact algorithms for implementation. Applying these methods, we develop a divide-and-combine strategy for mean estimation in any dimension and construct simultaneous confidence intervals in a network meta-analysis for treatment effect comparisons across multiple clinical trials. We also present many open problems and conclude with epistemic reflections.

stat.ME↗

arXiv2026

Wasserstein-p Bounds via Cumulant-Based Edgeworth Expansion for $α$-Mixing Random Fields

We establish normal-approximation bounds in the Wasserstein-$p$ distance for $α$-mixing random fields, extending results previously available mainly for independent or locally dependent variables. For standardized sums indexed by $T\subset\mathbb{Z}^d$, we obtain rates $O(|T|^{-β})$, where $β\in(0,1/2]$ depends on $p$, the moment order, the dimension, and the mixing decay; the optimal square-root rate holds under sufficiently fast polynomial mixing. For integer $p\geq2$, a refined bound retains the stronger mixing power produced by the expansion, covers the endpoint of finite $(p+2)$-th moments, and identifies exact logarithmic corrections at critical decay. Under finite-range dependence, the endpoint moment condition gives the optimal rate for every real $p\geq1$. The proof combines a cumulant-based Edgeworth expansion with Stein's method. Its main technical tool is a constructive graph calculus that organizes high-order dependence and preserves cancellations before mixing bounds are applied. The resulting Wasserstein estimates also imply two-regime non-uniform normal-approximation bounds with sharper sample-size decay in the far tail. A Lean 4/Mathlib companion machine-checks the principal theorem interfaces and the complete paper-internal genogram argument for the expansions, relative to seven explicitly identified literature and standard inputs.

math.PR↗

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 $π_0$, 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

Wasserstein-p Bounds in the Central Limit Theorem Under Weak Dependence

The central limit theorem is one of the most fundamental results in probability and has been successfully extended to locally dependent data and strongly-mixing random fields. In this paper, we establish its rate of convergence for transport distances, namely for arbitrary $p\ge1$ we obtain an upper bound for the Wasserstein-$p$ distance for locally dependent random variables and strongly mixing stationary random fields. Our proofs adapt the Stein dependency neighborhood method to the Wasserstein-$p$ distance and as a by-product we establish high-order local expansions of the Stein equation for dependent random variables. Finally, we demonstrate how our results can be used to obtain tail bounds that are asymptotically tight, and decrease polynomially fast, for the empirical average of weakly dependent random variables.

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

Kac's Walk on Rotation Matrices Mixes in $\boldsymbol{Θ(n^2)}$ 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 $0<\varepsilon<1$, \[ t_{\mathrm{mix}}^{(n)}(\varepsilon)=Θ_\varepsilon(n^2). \] 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↗