论文检索

Deep Learning · Generative Models and Autoencoders

Gabriel Loaiza-Ganem, Kevin Zhang, Wei Cui, Marc Law, Kin Kwan Leung

Conformal prediction (CP) and its extension, conformal risk control (CRC), are established frameworks for quantifying uncertainty in supervised machine learning through formal guarantees. However, recent breakthroughs in artificial intelligence (AI) have been driven by unsupervised generative models, such as large language models (LLMs) and image generators, which are not directly compatible with CP or CRC. In this work we introduce conformal generation (Conf-Gen), a general framework adapting CRC to generative tasks while relaxing its theoretical assumptions. Conf-Gen unifies and generalizes previous attempts to apply CP to LLMs, and extends conformal methodology to entirely new domains. We demonstrate the flexibility of Conf-Gen through some novel applications, including obtaining conformal guarantees on: image generators producing non-memorized images, conversational AI systems having asked enough clarifying questions, and the output of AI agents being correct.

Deep Learning · Large Language Models

Ali Taghibakhshi, Ruisi Cai, Saurav Muralidharan, Sharath Turuvekere Sreenivas, Ameya Mahabaleshwarkar, Marcin Chochowski, Akhiad Bercovich, Ran Zilberstein, Ran El-Yaniv, Yonatan Geifman 等

Training a family of large language models (LLMs), either from scratch or via iterative compression, is prohibitively expensive and inefficient, requiring separate training runs for each model in the family. In this paper, we introduce Star Elastic, a novel LLM post-training method that adds N nested submodels to a given parent reasoning model using the compute of one run (Nx savings) via a single post-training job. Beyond reducing training costs, Star Elastic also addresses a fundamental limitation in efficient reasoning: the rigidity of static architectures, which forces the allocation of constant resources regardless of token difficulty. By unlocking elastic budget control, Star Elastic enables a novel approach that uses different submodels for each reasoning phase (thinking and answering). Star Elastic supports (1) nesting along the SSM, embedding channel, MoE and FFN axes, (2) learning nested submodels via an end-to-end trainable router, and (3) curriculum-based knowledge distillation. We apply Star Elastic to the NVIDIA Nemotron Nano models; in particular, we demonstrate its effectiveness on hybrid MoE architectures with Nemotron Nano v3 (30B/3.6A), generating 23B (2.8A) and 12B (2.0A) variants with 160B training tokens. For Nemotron Nano v2 (12B), we produce 9B and 6B nested models using only 110B training tokens, achieving a 360x reduction versus training from scratch and a 7x reduction over state-of-the-art compression methods. All nested models match or outperform independently trained baselines of comparable size. Crucially, elastic budget control advances the accuracy--latency Pareto frontier, achieving up to 16% higher accuracy and 1.9x lower latency via dynamic per-phase model selection.

General Machine Learning · Evaluation

Yuanzhe Hu, Xiaopeng Wang, Yuxin Wang, Xiaokun Zhong, Haiquan Lu, Tianyu Pang, Michael Mahoney, Yujun Yan, Pu Ren, Yaoqing Yang

Neural networks (NNs) trained under different hyperparameters can fall into distinct training ``regimes'', with models in the same regime showing homogeneous properties and models across regimes differing qualitatively. In this paper, we analyze multi-regime patterns in scientific machine learning (SciML) models by characterizing these regimes and the transitions between them. We show how different regimes affect trainability and generalization, and we demonstrate that loss-landscape analysis enables regime-based diagnostics to understand, evaluate, and improve SciML model training. Our analysis yields three key insights: (1) compared with computer vision (CV) tasks, SciML models exhibit significantly more pathological loss landscapes; (2) optimization methods are regime-specific -- different optimization strategies help in different regimes, but none is uniformly effective; and (3) SciML models exhibit fine-grained failure modes that challenge conventional interpretations of standard loss-landscape metrics. Using this study, we aim to unify our understanding of seemingly different failure modes across SciML tasks and obtain task-oblivious insights and methodologies for addressing these failures. We validate these findings across widely used SciML models, including physics-informed neural networks (PINNs), Fourier neural operators (FNOs), and Neural Ordinary Differential Equations (NeuralODEs), on benchmarks spanning representative ordinary and partial differential equations.

Reinforcement Learning · Multi-agent

Tian Xia, Tianrun Gao, Wenhao Deng, Long Wei, Xiaowei Qian, Chenglei Yu, Tailin Wu

Engineering construction automation aims to transform natural language specifications into physically viable structures, requiring complex integrated reasoning under strict physical constraints. While modern LLMs possess broad knowledge and strong reasoning capabilities that make them promising candidates for this domain, their construction competencies remain largely unevaluated. To address this gap, we introduce BuildArena, the first physics-aligned interactive benchmark designed for language-driven engineering construction. It takes a first step towards engineering automation using LLMs. Technically, it contributes to the community in two aspects: (1) an extendable task design strategy spanning static and dynamic mechanics across multiple difficulty tiers; (2) a 3D Spatial Geometric Computation Library for supporting construction based on language instructions. On nine frontier LLMs, BuildArena comprehensively evaluates their capabilities for language-driven and physics-grounded construction automation.

Deep Learning · Large Language Models

Jiyeon Kim, Sungik Choi, Yongrae Jo, Moontae Lee, Minjoon Seo

Diffusion-based language models(dLLMs) have emerged as a promising alternative to autoregressive language models, offering the potential for parallel token generation and bidirectional context modeling. However, harnessing this flexibility for fully non-autoregressive decoding remains an open question, particularly for reasoning and planning tasks. In this work, we investigate non-autoregressive decoding in dLLMs by systematically analyzing its inference dynamics along the temporal axis. Specifically, we uncover an inherent failure modes in confidence-based non-autoregressive generation stem from a strong proximity bias—the denoising order tends to concentrate on spatially adjacent tokens. This local dependency leads to spatial error propagation, rendering the entire trajectory critically contingent on the initial unmasking position. Leveraging this insight, we present a minimal-intervention approach that guides early token selection, employing a lightweight planner and end-of-sequence temperature annealing. We thoroughly evaluate our method on various reasoning and planning tasks and observe substantial overall improvement over existing heuristic baselines without significant computational overhead.

Deep Learning · Other Representation Learning

Yeqin Zhang, Yunfei Wang, Jiaxuan Chen, Ke Qin, Yizheng Zhao, Cam-Tu Nguyen

Sentence representations are foundational to many Natural Language Processing (NLP) applications. While recent methods leverage Large Language Models (LLMs) to derive sentence representations, most rely on final-layer hidden states, which are optimized for next-token prediction and thus often fail to capture global, sentence-level semantics. This paper introduces a novel perspective, demonstrating that attention value vectors capture sentence semantics more effectively than hidden states. We propose *Value Aggregation (VA)*, a simple method that pools token values across multiple layers and token indices. In a training-free setting, VA outperforms other LLM-based embeddings, even matches or surpasses the ensemble-based MetaEOL. Furthermore, we demonstrate that when paired with suitable prompts, the layer attention outputs can be interpreted as aligned weighted value vectors. Specifically, the attention scores of the last token function as the weights, while the output projection matrix ($W_O$) aligns these weighted value vectors with the common space of the LLM residual stream. This refined method, termed *Aligned Weighted VA (AlignedWVA)*, achieves state-of-the-art performance among training-free LLM-based embeddings, outperforming the high-cost MetaEOL by a substantial margin. Finally, we highlight the potential of obtaining strong LLM embedding models through fine-tuning Value Aggregation.

Deep Learning · Everything Else

Sangwoo Hwang, Yeeun Hong, Jaeha Kung

The rapid growth of deep neural networks (DNNs) has intensified the demand for efficient hardware acceleration under quantization. While prior research has successfully reduced weight and activation precision, partial sums generated during accumulation often retain high precision, resulting in significant energy overhead. In this work, we analyze psum distributions in tiled architectures and reveal that within-tile outliers are input-dependent. We propose PsumQuant, a post-training, input-aware quantization that predicts psum scales on-the-fly. By leveraging the crest factor of input activations, our learnable scale predictor effectively bounds the psum bit-width while handling the extreme outliers in DNNs. Experimental results on a $128 \times 128$ systolic array demonstrate that PsumQuant compresses psum precision down to 8-bit within only a 1\% accuracy drop on ResNet-18 and a marginal 0.04 perplexity increase on Llama-3.1. Furthermore, bit-width reduction with PsumQuant results in a 45\% reduction in total energy with minimal accuracy loss, demonstrating that PsumQuant provides a highly efficient solution for actual NPU architectures.

Theory · Probabilistic Methods

Tom Yates, Yuzhou Cheng, Ignacio Alzugaray, Danyal Akarca, Pedro Mediano, Andrew Davison

Belief Propagation (BP) is a powerful algorithm for distributed inference in probabilistic graphical models, however it quickly becomes infeasible for practical compute and memory budgets. Many efficient, non-parametric forms of BP have been developed, but the most popular is Gaussian Belief Propagation (GBP), a variant that assumes all distributions are locally Gaussian. GBP is widely used due to its efficiency and empirically strong performance in applications like computer vision or sensor networks – even when modelling non-Gaussian problems. In this paper, we seek to provide a theoretical guarantee for when Gaussian approximations are valid in highly non-Gaussian, sparsely-connected factor graphs performing BP (common in Spatial AI). We leverage the Central Limit Theorem to prove mathematically that variables’ beliefs under BP converge to a Gaussian distribution in complex, loopy factor graphs obeying our 4 key assumptions. We then confirm experimentally that variable beliefs become increasingly Gaussian after just a few BP iterations in a stereo depth estimation task.

Applications · Chemistry, Physics, and Earth Sciences

Kevin Han, Haolin Cong, Bowen Deng, Amir Barati Farimani

Machine learning interatomic potentials (MLIPs) have proven to be wildly useful for molecular dynamics simulations, powering countless drug and materials discovery applications. However, MLIPs face two primary bottlenecks preventing them from reaching realistic simulation scales: inference time and memory consumption. In this work, we address both issues by challenging the long-held belief that the cutoff radius for the MLIP must be held to a fixed, constant value. For the first time, we introduce a **dynamic** cutoff formulation that still leads to stable, long timescale molecular dynamics simulation. In introducing the dynamic cutoff, we are able to induce sparsity onto the underlying atom graph by targeting a specific number of neighbors per atom, significantly reducing both memory consumption and inference time. We show the effectiveness of a dynamic cutoff by implementing it onto 4 state of the art MLIPs: MACE, Nequip, Orbv3, and TensorNet, leading to **2.26x** less memory consumption and **2.04x** faster inference time, depending on the model and atomic system. We also perform an extensive error analysis and find that the dynamic cutoff models exhibit minimal accuracy dropoff compared to their fixed cutoff counterparts on both materials and molecular datasets. All model implementations and training code will be fully open sourced.

Social Aspects · Privacy

Aditya Kulkarni, Carlos Soto

In statistical applications it has become increasingly common to encounter data structures that live on non-linear spaces such as manifolds. Classical linear regression, one of the most fundamental methodologies of statistical learning, captures the relationship between an independent variable and a response variable which both are assumed to live in Euclidean space. Thus, geodesic regression emerged as an extension where the response variable lives on a Riemannian manifold. The parameters of geodesic regression, as with linear regression, capture the relationship of sensitive data and hence one should consider the privacy protection practices of said parameters. We consider releasing Differentially Private (DP) parameters of geodesic regression via the K-Norm Gradient (KNG) mechanism for Riemannian manifolds. We derive theoretical bounds for the sensitivity of the parameters showing they are tied to their respective Jacobi fields and hence the curvature of the space. This corroborates, and extends, recent findings of differential privacy for the Fr\'echet mean. We demonstrate the efficacy of our methodology on the sphere, $S_2\subset\mathbb{R}^3$, the space of symmetric positive definite matrices, and Kendall's planar shape space. Our methodology is general to any Riemannian manifold, and thus, it is suitable for data in domains such as medical imaging and computer vision.

Applications · Computer Vision

Manuel Traub, Martin V Butz

Current state-of-the-art segmentation models encode entire images before focusing on specific objects. This wastes computational resources. We introduce FLIP (Fovea-Like Input Patching), a parameter-efficient vision model that realizes object segmentation through biologically-inspired top-down attention. FLIP selectively samples multi-resolution patches centered on objects of interest from the input. As a result, it allocates high-resolution processing to object centers while maintaining coarser peripheral context. This off-grid, scale-invariant design enables FLIP to outperform META's Segment Anything models (SAM, SAM2 and fast variants) by large margins: With more than 440$\times$ fewer parameters, FLIP-Tiny (0.51M parameters) reaches a mean IoU of 78.24\% while SAM2-L reaches 75.87\% IoU (224.45M parameters). FLIP-Large even achieves 80.33\% mean IoU (96.6M parameters), still running about $2.3\times$ faster than SAM2-L. We evaluate on six benchmarks in total. In five established benchmarks (Hypersim, KITTI-360, OpenImages, COCO, LVIS) FLIP consistently outperforms SAM and various variants of it. In our novel ObjaScale dataset, which stress-tests scale invariance with objects ranging from 0.0001\% up to 25\% of the image area, we show that FLIP segments even very small objects accurately, where existing models fail severely. FLIP opens new possibilities for real-time, object-centric vision applications and offers much higher energy efficiency. We believe that FLIP can act as a powerful foundation model, as it is very well-suited to track objects over time, for example, when being integrated into slot-based scene segmentation architectures.

Theory · Online Learning and Bandits

Yu Chen, Yuhao Liu, Jiatai Huang, Yihan Du, Longbo Huang

We investigate episodic Markov Decision Processes with heavy-tailed feedback (HTMDPs). Existing approaches for HTMDPs are conservative in stochastic environments and lack adaptivity in adversarial regimes. In this work, we propose algorithms ```HT-FTRL-OM``` and ```HT-FTRL-UOB``` for HTMDPs that achieve Best-of-Both-Worlds (BoBW) guarantees: instance-independent regret in adversarial environments and logarithmic instance-dependent regret in self-bounding (including the stochastic case) environments. For the known transition setting, ```HT-FTRL-OM``` applies the Follow-The-Regularized-Leader (FTRL) framework over occupancy measures with novel skipping loss estimators, achieving a $\widetilde{\mathcal{O}}(T^{1/\alpha})$ regret bound in adversarial regimes and a $\mathcal{O}(\log T)$ regret in stochastic regimes. Building upon this framework, we develop a novel algorithm ```HT-FTRL-UOB``` to tackle the more challenging unknown-transition setting. This algorithm employs a pessimistic skipping loss estimator and achieves a $\widetilde{\mathcal{O}}(T^{1/\alpha} + \sqrt{T})$ regret in adversarial regimes and a $\mathcal{O}(\log^2(T))$ regret in stochastic regimes. Our analysis overcomes key barriers through several technical insights, including a local control mechanism for heavy-tailed shifted losses, a new suboptimal-mass propagation principle, and a novel regret decomposition that isolates transition uncertainty from heavy-tailed estimation errors and skipping bias.

Applications · Computer Vision

Wen Dong, Haiyang Mei, Yinglian Ji, Zijun Zhang, Wenyuan Zhang, Pengwei Luo, Bo Dong, Shengfeng He, Xin Yang

Depth estimation for transparent objects remains a fundamental challenge, as RGB-based cues often fail in regions affected by refraction and light transmission. Polarization provides physically grounded information related to surface orientation and material properties, offering reliable geometric cues even in the absence of texture. In this work, we introduce PolarDepth, a monocular framework that incorporates both RGB and polarization inputs, including the degree and angle of linear polarization (DoLP and AoLP), to estimate dense depth and localize transparent regions. PolarDepth injects polarization-derived physical priors by estimating the refractive index, zenith angle, and azimuth angle from polarization measurements and embedding them into an implicit geometric representation that constrains depth inference in ambiguous transparent regions. To support model development and evaluation, we introduce PTOD, a dataset with synchronized RGB, polarization, and depth data and manually annotated transparent region masks. Experimental results demonstrate that PolarDepth achieves state-of-the-art performance in transparent object depth estimation. The findings highlight the effectiveness of embedding polarization-derived physical priors into learned representations for robust perception in complex visual environments.

Theory · Online Learning and Bandits

Sifan Yang, Wenhao Yang, Wei Jiang, Lijun Zhang

We investigate decentralized online convex optimization with compressed communication, where $n$ learners connected by a network collaboratively minimize a sequence of global loss functions using only local information and compressed data from neighbors. Prior work has established regret bounds of $O(\max\\{\omega^{-2}\rho^{-4}n^{1/2},\omega^{-4}\rho^{-8}\\}n\sqrt{T})$ and $O(\max\\{\omega^{-2}\rho^{-4}n^{1/2},\omega^{-4}\rho^{-8}\\}n\ln{T})$ for convex and strongly convex functions, respectively, where $\omega\in(0,1]$ is the compression quality factor and $\rho<1$ is the spectral gap of the communication matrix. However, these regret bounds suffer from a prohibitively high quadratic or even quartic dependence on $\omega^{-1}$. Moreover, the super-linear dependence on $n$ is also undesirable. To overcome these shortcomings, we propose a novel algorithm that achieves improved regret bounds of $\tilde{O}(\omega^{-1/2}\rho^{-1}n\sqrt{T})$ and $\tilde{O}(\omega^{-1}\rho^{-2}n\ln{T})$ for convex and strongly convex functions, respectively. The primary idea is to design a two-level blocking update framework incorporating two novel ingredients: an online gossip strategy and an error compensation scheme, which work together to promote a better consensus among learners. Furthermore, we establish the first lower bounds for this problem, justifying the optimality of our results with respect to both $\omega$ and $T$. Additionally, we consider the bandit feedback scenario and extend our method with the classical gradient estimators to enhance existing regret bounds.

Deep Learning · Generative Models and Autoencoders

Shuchen Xue, Chongjian GE, Shilong Zhang, Yichen Li, Zhi-Ming Ma

Reinforcement Learning (RL) has emerged as a central paradigm for advancing Large Language Models (LLMs), where both pre-training and RL post-training stages are grounded in the same log-likelihood formulation. In contrast, recent RL approaches for diffusion models, most notably Denoising Diffusion Policy Optimization (DDPO), optimize an objective different from the pretraining objectives--score/flow matching loss. In this work, we establish a novel theoretical analysis: DDPO is an implicit form of score/flow matching with noisy targets, which increases variance and slows convergence. Building on this analysis, we introduce Advantage Weighted Matching (AWM), a policy-gradient method for diffusion. It uses the score/flow-matching loss and reweights each sample by its advantage. In effect, AWM raises the influence of high-reward samples and suppresses low-reward ones while keeping the modeling objective identical to pretraining. This unifies pretraining and RL conceptually and practically and reduces variance, yielding faster convergence. This simple yet effective design yields substantial benefits: on the GenEval, OCR, and PickScore benchmarks, AWM delivers up to a $34\times$ speedup over Flow-GRPO (which builds on DDPO), when applied to Stable Diffusion 3.5 Medium and FLUX, without compromising generation quality. Code is provided in the supplementary material.

Reinforcement Learning · Deep RL

Haowen Sun, Liqi Huang, Mingyang Li, Sihua Ren, Xinzhe Chen, Chengzhong Ma, Zeyang Liu, Xingyu Chen, Xuguang Lan

Reinforcement learning from demonstrations (RLfD) offers a promising method for robotic manipulation with sparse rewards. However, limited demonstrations often cause agents to encounter out-of-distribution states where world models produce poor predictions. In multi-stage tasks, jointly optimizing a learned reward function and policy introduces a moving target problem, and the resulting non-stationarity intensifies the impact of uncertainty on policy learning. In this work, we propose QUEST, a model-based RL framework that adaptively switches between exploration and exploitation guided by uncertainty to achieve stable and efficient learning. Specifically, our approach employs intrinsic rewards to capture environmental stochasticity, leverages ensemble dynamics for uncertainty-guided planning, and introduces a hybrid sampling strategy to prioritize rare successful stage transitions. We evaluate QUEST on challenging sparse-reward manipulation tasks with limited expert demonstrations. Results show that QUEST outperforms state-of-the-art methods by 17\% on average, with gains increasing to 60\% on difficult tasks. We further demonstrate successful zero-shot sim-to-real transfer on three real-world tasks.

Ilan Doron-Arad, Elchanan Mossel

Theoretical analyses of Empirical Risk Minimization (ERM) are standardly framed within the Real-RAM model of computation. In this setting, training even simple neural networks is known to be $\exists \mathbb{R}$-complete---a complexity class believed to be harder than NP, that characterizes the difficulty of solving systems of polynomial inequalities over the real numbers. However, this algebraic framework diverges from the reality of digital computation with finite-precision hardware. In this work, we analyze the theoretical complexity of ERM under a realistic bit-level model ($ERM_{bit}$), where network parameters and inputs are constrained to be rational numbers with polynomially bounded bit-lengths. Under this model, we reveal a sharp dichotomy in tractability governed by the network's activation function. We prove that for deep networks with any polynomial activations with rational coefficients and degree at least $2$, the bit-complexity of training is severe: deciding $ERM_{bit}$ (with a regularized loss) is #$P$-hard, hence believed to be strictly harder than NP-complete problems. Furthermore, we show that determining the sign of a single partial derivative of the empirical loss function is intractable (unlikely in BPP), and deciding a specific bit in the gradient is #$P$-hard. This provides a complexity-theoretic perspective for the phenomenon of exploding and vanishing gradients. In contrast, we show that for piecewise-linear activations such as ReLU, the precision requirements remain manageable: $ERM_{bit}$ is contained within NP (specifically NP-complete), and standard backpropagation runs in polynomial time. Our results demonstrate that finite-precision constraints are not merely implementation details but fundamental determinants of learnability.

Applications · Health / Medicine

ZIYU XU, zijian zhang, Liang Wang, Zhiyuan Liu, Qiang Liu, Shu Wu, Liang Wang

When reliable target structures are unavailable at scale or phenotypes arise from dysregulated pathways, transcriptomic perturbations provide a system-level functional readout for drug action. In this work, we formalize Transcriptome-based Drug Design (TBDD) as a generative inverse problem: designing drug molecules conditioned on desired transcriptomic state transitions. We analyze the inherently ill-posed nature of this task, which is further complicated by the profound domain gap between biology and chemistry and by the sparsity of transcriptomic signals. To address these challenges, we propose CURE (A CellUlar Response Engine), a multi-resolution transcriptome-guided diffusion framework. CURE features a specialized Transcriptome Perturbation Functional Feature Extractor (TFE) that (1) distills function-oriented perturbation embeddings from pre/post states, (2) aligns these signatures to dual chemical views to bridge the cross-modal gap, and (3) performs heterogeneity-aware aggregation to extract robust state-specific signals from noisy transcriptomic data. Extensive evaluations on both standard benchmarks and rigorous out-of-distribution protocols demonstrate that CURE consistently outperforms strong baselines in structural quality and functional consistency. Furthermore, we validate its practical utility via a zero-shot gene-inhibitor design task, highlighting the potential of phenotype-driven generative discovery.

Social Aspects · Accountability, Transparency, and Interpretability

Yifan Zhang, Wei Bi, Kechi Zhang, Dongming Jin, Jie Fu, Zhi Jin

Algorithm extraction aims to synthesize executable programs directly from models trained on algorithmic tasks, enabling *de novo* algorithm discovery without relying on human-written code. However, applying this paradigm to Transformer is hindered by representation entanglement (e.g., superposition), where entangled features encoded in overlapping directions obstruct the recovery of symbolic expressions. We propose the Discrete Transformer, an architecture explicitly designed to bridge the gap between continuous representations and discrete symbolic logic. By injecting discreteness through temperature-annealed sampling, our framework effectively leverages hypothesis testing and symbolic regression to extract human-readable programs. Empirically, the Discrete Transformer achieves performance comparable to RNN-based methods while extending interpretability to continuous variable domains, and the annealing dynamics exhibit a clear exploration-to-exploitation transition. Finally, we show that architectural inductive biases provide fine-grained control over synthesized programs, establishing the Discrete Transformer as a robust framework for demonstration-free algorithm discovery and Transformer interpretability.

Deep Learning · Generative Models and Autoencoders

Ning Lin, Luxi Chen, Huaguan Chen, Jiacheng Cen, Chongxuan Li, Wenbing Huang, Hao Sun

Generating objects with specific symmetries is essential in various real-world scenarios, with the creation of patterns exhibiting planar group symmetries being a representative task. However, adapting existing 2D continuous representations to enforce symmetry remains a significant challenge, as the transformation of non-reflective group elements may disrupt continuity. To overcome this limitation, we propose a novel symmetric continuous representation framework for arbitrary planar groups. Our method transforms any underlying 2D continuous representation into a symmetric one while strictly preserving continuity. We provide the mathematical formulation of this representation, demonstrate its universal approximation capability for symmetric functions, and detail the construction methodology. We validate our approach through three distinct generation tasks: general symmetric pattern design, connectivity-preserving paper-cutting design, and mechanically-constrained geometric stylized patterns. Experimental results confirm that our representation enables precise and effective symmetry control in pattern generation.