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Zongyi Li

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20 papers

2 author rows

AAAI Conference 2026 Conference Paper

Dual-stream Relation-modeling Disentanglement for Cloth-Changing Person Re-Identification

Shijuan Huang
Hefei Ling
Zongyi Li
Xu Li
Zhao Lv

Cloth-changing person re-identification (CC-ReID) aims to identify individuals across non-overlapping cameras despite clothing variations. Existing methods are often constrained by two primary limitations: approaches using auxiliary modalities typically rely on a single specific cue, limiting their robustness, while feature disentanglement methods struggle with discrete labels that create inconsistencies between ground truth labels and modality semantic similarity. To overcome these limitations, we propose DRDnet, a unified framework that synergistically integrates dual auxiliary cues and advanced relation modeling. Specifically, our Dual-Stream Disentanglement (DSD) module leverages textual descriptions and parsing images to decouple clothing factors through high-level semantic supervision and pixel-level operations, yielding robust clothing-agnostic features. Simultaneously, our Modal Relation Modeling (MRM) module constructs feature memory banks and employs adaptive soft label smoothing, effectively enhancing image-text semantic alignment and reinforcing identity consistency across clothing changes. We evaluate DRDnet on several CC-ReID benchmarks to demonstrate its effectiveness and provide state-of-the-art performance across all benchmarks.

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ICLR Conference 2025 Conference Paper

ARLON: Boosting Diffusion Transformers with Autoregressive Models for Long Video Generation

Zongyi Li
Shujie Hu
Shujie Liu 0001
Long Zhou
Jeongsoo Choi
Lingwei Meng
Xun Guo
Jinyu Li 0001

Text-to-video (T2V) models have recently undergone rapid and substantial advancements. Nevertheless, due to limitations in data and computational resources, achieving efficient generation of long videos with rich motion dynamics remains a significant challenge. To generate high-quality, dynamic, and temporally consistent long videos, this paper presents ARLON, a novel framework that boosts diffusion Transformers with autoregressive (\textbf{AR}) models for long (\textbf{LON}) video generation, by integrating the coarse spatial and long-range temporal information provided by the AR model to guide the DiT model effectively. Specifically, ARLON incorporates several key innovations: 1) A latent Vector Quantized Variational Autoencoder (VQ-VAE) compresses the input latent space of the DiT model into compact and highly quantized visual tokens, bridging the AR and DiT models and balancing the learning complexity and information density; 2) An adaptive norm-based semantic injection module integrates the coarse discrete visual units from the AR model into the DiT model, ensuring effective guidance during video generation; 3) To enhance the tolerance capability of noise introduced from the AR inference, the DiT model is trained with coarser visual latent tokens incorporated with an uncertainty sampling module. Experimental results demonstrate that ARLON significantly outperforms the baseline OpenSora-V1.2 on eight out of eleven metrics selected from VBench, with notable improvements in dynamic degree and aesthetic quality, while delivering competitive results on the remaining three and simultaneously accelerating the generation process. In addition, ARLON achieves state-of-the-art performance in long video generation, outperforming other open-source models in this domain. Detailed analyses of the improvements in inference efficiency are presented, alongside a practical application that demonstrates the generation of long videos using progressive text prompts. Project page: \url{http://aka.ms/arlon}.

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NeurIPS Conference 2025 Conference Paper

Autoregressive Motion Generation with Gaussian Mixture-Guided Latent Sampling

Linnan Tu
Lingwei Meng
Zongyi Li
Hefei Ling
Shijuan Huang

Existing efforts in motion synthesis typically utilize either generative transformers with discrete representations or diffusion models with continuous representations. However, the discretization process in generative transformers can introduce motion errors, while the sampling process in diffusion models tends to be slow. In this paper, we propose a novel text-to-motion synthesis method GMMotion that combines a continuous motion representation with an autoregressive model, using the Gaussian mixture model (GMM) to represent the conditional probability distribution. Unlike autoregressive approaches relying on residual vector quantization, our model employs continuous motion representations derived from the VAE's latent space. This choice streamlines both the training and the inference processes. Specifically, we utilize a causal transformer to learn the distributions of continuous motion representations, which are modeled with a learnable Gaussian mixture model. Extensive experiments demonstrate that our model surpasses existing state-of-the-art models in the motion synthesis task.

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AAAI Conference 2025 Conference Paper

Exploring the Potential of Large Vision-Language Models for Unsupervised Text-Based Person Retrieval

Zongyi Li
Li Jianbo
Yuxuan Shi
Jiazhong Chen
Shijuan Huang
Linnan Tu
Fei Shen
Hefei Ling

The aim of text-based person retrieval is to identify pedestrians using natural language descriptions within a large-scale image gallery. Traditional methods rely heavily on manually annotated image-text pairs, which are resource-intensive to obtain. With the emergence of Large Vision-Language Models (LVLMs), the advanced capabilities of contemporary models in image understanding have led to the generation of highly accurate captions. Therefore, this paper explores the potential of employing Large Vision-Language Models for unsupervised text-based pedestrian image retrieval and proposes a Multi-grained Uncertainty Modeling and Alignment framework (MUMA). Initially, multiple Large Vision-Language Models are employed to generate diverse and hierarchically structured pedestrian descriptions across different styles and granularities. However, the generated captions inevitably introduce noise. To address this issue, an uncertainty-guided sample filtration module is proposed to estimate and filter out unreliable image-text pairs. Additionally, to simulate the diversity of styles and granularities in captions, a multi-grained uncertainty modeling approach is applied to model the distributions of captions, with each caption represented as a multivariate Gaussian distribution. Finally, a multi-level consistency distillation loss is employed to integrate and align the multi-grained captions, aiming to transfer knowledge across different granularities. Experimental evaluations conducted on three widely-used datasets demonstrate the significant advancements achieved by our approach.

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JMLR Journal 2025 Journal Article

Geometric Operator Learning with Optimal Transport

Xinyi Li
Zongyi Li
Nikola Kovachki
Anima Anandkumar

We propose integrating optimal transport (OT) into operator learning for partial differential equations (PDEs) on complex geometries. Classical geometric learning methods typically represent domains as meshes, graphs, or point clouds. Our approach generalizes discretized meshes to mesh density functions, formulating geometry embedding as an OT problem that maps these functions to a uniform density in a reference space. Compared to previous methods relying on interpolation or shared deformation, our OT-based method employs instance-dependent deformation, offering enhanced flexibility and effectiveness. For 3D simulations focused on surfaces, our OT-based neural operator embeds the surface geometry into a 2D parameterized latent space. By performing computations directly on this 2D representation of the surface manifold, it achieves significant computational efficiency gains compared to volumetric simulation. Experiments with Reynolds-averaged Navier-Stokes equations (RANS) on the ShapeNet-Car and DrivAerNet-Car datasets show that our method achieves better accuracy and also reduces computational expenses in terms of both time and memory usage compared to existing machine learning models. Additionally, our model demonstrates significantly improved accuracy on the FlowBench dataset, underscoring the benefits of employing instance-dependent deformation for datasets with highly variable geometries. [abs] [ pdf ][ bib ] &copy JMLR 2025. ( edit, beta )

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NeurIPS Conference 2025 Conference Paper

Incentivizing Reasoning for Advanced Instruction-Following of Large Language Models

Yulei Qin
Gang Li
Zongyi Li
Zihan Xu
Yuchen Shi
Zhekai Lin
Xiao Cui
Ke Li

Existing large language models (LLMs) face challenges of following complex instructions, especially when multiple constraints are present and organized in paralleling, chaining, and branching structures. One intuitive solution, namely chain-of-thought (CoT), is expected to universally improve capabilities of LLMs. However, we find that the vanilla CoT exerts a negative impact on performance due to its superficial reasoning pattern of simply paraphrasing the instructions. It fails to peel back the compositions of constraints for identifying their relationship across hierarchies of types and dimensions. To this end, we propose RAIF, a systematic method to boost LLMs in dealing with complex instructions via incentivizing reasoning for test-time compute scaling. First, we stem from the decomposition of complex instructions under existing taxonomies and propose a reproducible data acquisition method. Second, we exploit reinforcement learning (RL) with verifiable rule-centric reward signals to cultivate reasoning specifically for instruction following. We address the shallow, non-essential nature of reasoning under complex instructions via sample-wise contrast for superior CoT enforcement. We also exploit behavior cloning of experts to facilitate steady distribution shift from fast-thinking LLMs to skillful reasoners. Extensive evaluations on seven comprehensive benchmarks confirm the validity of the proposed method, where a 1. 5B LLM achieves 11. 74% gains with performance comparable to a 8B LLM. Evaluation on OOD constraints also confirms the generalizability of our RAIF.

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IJCAI Conference 2024 Conference Paper

Cross-modal Generation and Alignment via Attribute-guided Prompt for Unsupervised Text-based Person Retrieval

Zongyi Li
Jianbo Li
Yuxuan Shi
Hefei Ling
Jiazhong Chen
Runsheng Wang
Shijuan Huang

Text-based Person Search aims to retrieve a specified person using a given text query. Current methods predominantly rely on paired labeled image-text data to train the cross-modality retrieval model, necessitating laborious and time-consuming labeling. In response to this challenge, we present the Cross-modal Generation and Alignment via Attribute-guided Prompt framework (GAAP) for fully unsupervised text-based person search, utilizing only unlabeled images. Our proposed GAAP framework consists of two key parts: Attribute-guided Prompt Caption Generation and Attribute-guided Cross-modal Alignment module. The Attribute-guided Prompt Caption Generation module generates pseudo text labels by feeding the attribute prompts into a large-scale pre-trained vision-language model. These synthetic texts are then meticulously selected through a sample selection, ensuring the reliability for subsequent fine-tuning. The Attribute-guided Cross-modal Alignment module encompasses three sub-modules for feature alignment across modalities. Firstly, Cross-Modal Center Alignment (CMCA) aligns the samples with different modality centroids. Subsequently, to address ambiguity arising from local attribute similarities, an Attribute-guided Image-Text Contrastive Learning module (AITC) is proposed to facilitate the alignment of relationships among different pairs by considering local attribute similarities. Lastly, the Attribute-guided Image-Text Matching (AITM) module is introduced to mitigate noise in pseudo captions by using the image-attribute matching score to soften the hard matching labels. Empirical results showcase the effectiveness of our method across various text-based person search datasets under the fully unsupervised setting.

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TMLR Journal 2024 Journal Article

Incremental Spatial and Spectral Learning of Neural Operators for Solving Large-Scale PDEs

Robert Joseph George
Jiawei Zhao
Jean Kossaifi
Zongyi Li
Anima Anandkumar

Fourier Neural Operators (FNO) offer a principled approach to solving challenging partial differential equations (PDE) such as turbulent flows. At the core of FNO is a spectral layer that leverages a discretization-convergent representation in the Fourier domain, and learns weights over a fixed set of frequencies. However, training FNO presents two significant challenges, particularly in large-scale, high-resolution applications: (i) Computing Fourier transform on high-resolution inputs is computationally intensive but necessary since fine-scale details are needed for solving many PDEs, such as fluid flows, (ii) selecting the relevant set of frequencies in the spectral layers is challenging, and too many modes can lead to overfitting, while too few can lead to underfitting. To address these issues, we introduce the Incremental Fourier Neural Operator (iFNO), which progressively increases both the number of frequency modes used by the model as well as the resolution of the training data. We empirically show that iFNO reduces total training time while maintaining or improving generalization performance across various datasets. Our method demonstrates a 38% lower testing error, using 20% fewer frequency modes compared to the existing FNO, while also achieving up to 46% faster training and a 2.8x reduction in model size.

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NeurIPS Conference 2024 Conference Paper

Pretraining Codomain Attention Neural Operators for Solving Multiphysics PDEs

Ashiqur Rahman
Robert J. George
Mogab Elleithy
Daniel Leibovici
Zongyi Li
Boris Bonev
Colin White
Julius Berner

Existing neural operator architectures face challenges when solving multiphysics problems with coupled partial differential equations (PDEs) due to complex geometries, interactions between physical variables, and the limited amounts of high-resolution training data. To address these issues, we propose Codomain Attention Neural Operator (CoDA-NO), which tokenizes functions along the codomain or channel space, enabling self-supervised learning or pretraining of multiple PDE systems. Specifically, we extend positional encoding, self-attention, and normalization layers to function spaces. CoDA-NO can learn representations of different PDE systems with a single model. We evaluate CoDA-NO's potential as a backbone for learning multiphysics PDEs over multiple systems by considering few-shot learning settings. On complex downstream tasks with limited data, such as fluid flow simulations, fluid-structure interactions, and Rayleigh-Bénard convection, we found CoDA-NO to outperform existing methods by over 36%.

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JMLR Journal 2023 Journal Article

Fourier Neural Operator with Learned Deformations for PDEs on General Geometries

Zongyi Li
Daniel Zhengyu Huang
Burigede Liu
Anima Anandkumar

Deep learning surrogate models have shown promise in solving partial differential equations (PDEs). Among them, the Fourier neural operator (FNO) achieves good accuracy, and is significantly faster compared to numerical solvers, on a variety of PDEs, such as fluid flows. However, the FNO uses the Fast Fourier transform (FFT), which is limited to rectangular domains with uniform grids. In this work, we propose a new framework, viz., Geo-FNO, to solve PDEs on arbitrary geometries. Geo-FNO learns to deform the input (physical) domain, which may be irregular, into a latent space with a uniform grid. The FNO model with the FFT is applied in the latent space. The resulting Geo-FNO model has both the computation efficiency of FFT and the flexibility of handling arbitrary geometries. Our Geo-FNO is also flexible in terms of its input formats, viz., point clouds, meshes, and design parameters are all valid inputs. We consider a variety of PDEs such as the Elasticity, Plasticity, Euler's, and Navier-Stokes equations, and both forward modeling and inverse design problems. Comprehensive cost-accuracy experiments show that Geo-FNO is $10^5$ times faster than the standard numerical solvers and twice more accurate compared to direct interpolation on existing ML-based PDE solvers such as the standard FNO. [abs] [ pdf ][ bib ] [ code ] &copy JMLR 2023. ( edit, beta )

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NeurIPS Conference 2023 Conference Paper

Geometry-Informed Neural Operator for Large-Scale 3D PDEs

Zongyi Li
Nikola Kovachki
Chris Choy
Boyi Li
Jean Kossaifi
Shourya Otta
Mohammad Amin Nabian
Maximilian Stadler

We propose the geometry-informed neural operator (GINO), a highly efficient approach for learning the solution operator of large-scale partial differential equations with varying geometries. GINO uses a signed distance function (SDF) representation of the input shape and neural operators based on graph and Fourier architectures to learn the solution operator. The graph neural operator handles irregular grids and transforms them into and from regular latent grids on which Fourier neural operator can be efficiently applied. We provide an efficient implementation of GINO using an optimized hashing approach, which allows efficient learning in a shared, compressed latent space with reduced computation and memory costs. GINO is discretization-invariant, meaning the trained model can be applied to arbitrary discretizations of the continuous domain and applies to any shape or resolution. To empirically validate the performance of our method on large-scale simulation, we generate the industry-standard aerodynamics dataset of 3D vehicle geometries with Reynolds numbers as high as five million. For this large-scale 3D fluid simulation, numerical methods are expensive to compute surface pressure. We successfully trained GINO to predict the pressure on car surfaces using only five hundred data points. The cost-accuracy experiments show a 26, 000x speed-up compared to optimized GPU-based computational fluid dynamics (CFD) simulators on computing the drag coefficient. When tested on new combinations of geometries and boundary conditions (inlet velocities), GINO obtains a one-fourth reduction in error rate compared to deep neural network approaches.

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JMLR Journal 2023 Journal Article

Neural Operator: Learning Maps Between Function Spaces With Applications to PDEs

Nikola Kovachki
Zongyi Li
Burigede Liu
Kamyar Azizzadenesheli
Kaushik Bhattacharya
Andrew Stuart
Anima Anandkumar

The classical development of neural networks has primarily focused on learning mappings between finite dimensional Euclidean spaces or finite sets. We propose a generalization of neural networks to learn operators, termed neural operators, that map between infinite dimensional function spaces. We formulate the neural operator as a composition of linear integral operators and nonlinear activation functions. We prove a universal approximation theorem for our proposed neural operator, showing that it can approximate any given nonlinear continuous operator. The proposed neural operators are also discretization-invariant, i.e., they share the same model parameters among different discretization of the underlying function spaces. Furthermore, we introduce four classes of efficient parameterization, viz., graph neural operators, multi-pole graph neural operators, low-rank neural operators, and Fourier neural operators. An important application for neural operators is learning surrogate maps for the solution operators of partial differential equations (PDEs). We consider standard PDEs such as the Burgers, Darcy subsurface flow, and the Navier-Stokes equations, and show that the proposed neural operators have superior performance compared to existing machine learning based methodologies, while being several orders of magnitude faster than conventional PDE solvers. [abs] [ pdf ][ bib ] [ code ] &copy JMLR 2023. ( edit, beta )

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ICLR Conference 2022 Conference Paper

Efficient Token Mixing for Transformers via Adaptive Fourier Neural Operators

John Guibas
Morteza Mardani
Zongyi Li
Andrew Tao
Anima Anandkumar
Bryan Catanzaro

Vision transformers have delivered tremendous success in representation learning. This is primarily due to effective token mixing through self attention. However, this scales quadratically with the number of pixels, which becomes infeasible for high-resolution inputs. To cope with this challenge, we propose Adaptive Fourier Neural Operator (AFNO) as an efficient token mixer that learns to mix in the Fourier domain. AFNO is based on a principled foundation of operator learning which allows us to frame token mixing as a continuous global convolution without any dependence on the input resolution. This principle was previously used to design FNO, which solves global convolution efficiently in the Fourier domain and has shown promise in learning challenging PDEs. To handle challenges in visual representation learning such as discontinuities in images and high resolution inputs, we propose principled architectural modifications to FNO which results in memory and computational efficiency. This includes imposing a block-diagonal structure on the channel mixing weights, adaptively sharing weights across tokens, and sparsifying the frequency modes via soft-thresholding and shrinkage. The resulting model is highly parallel with a quasi-linear complexity and has linear memory in the sequence size. AFNO outperforms self-attention mechanisms for few-shot segmentation in terms of both efficiency and accuracy. For Cityscapes segmentation with the Segformer-B3 backbone, AFNO can handle a sequence size of 65k and outperforms other efficient self-attention mechanisms.

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NeurIPS Conference 2022 Conference Paper

Learning Chaotic Dynamics in Dissipative Systems

Zongyi Li
Miguel Liu-Schiaffini
Nikola Kovachki
Kamyar Azizzadenesheli
Burigede Liu
Kaushik Bhattacharya
Andrew Stuart
Anima Anandkumar

Chaotic systems are notoriously challenging to predict because of their sensitivity to perturbations and errors due to time stepping. Despite this unpredictable behavior, for many dissipative systems the statistics of the long term trajectories are governed by an invariant measure supported on a set, known as the global attractor; for many problems this set is finite dimensional, even if the state space is infinite dimensional. For Markovian systems, the statistical properties of long-term trajectories are uniquely determined by the solution operator that maps the evolution of the system over arbitrary positive time increments. In this work, we propose a machine learning framework to learn the underlying solution operator for dissipative chaotic systems, showing that the resulting learned operator accurately captures short-time trajectories and long-time statistical behavior. Using this framework, we are able to predict various statistics of the invariant measure for the turbulent Kolmogorov Flow dynamics with Reynolds numbers up to $5000$.

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AAAI Conference 2022 Conference Paper

Reliability Exploration with Self-Ensemble Learning for Domain Adaptive Person Re-identification

Zongyi Li
Yuxuan Shi
Hefei Ling
Jiazhong Chen
Qian Wang
Fengfan Zhou

Person re-identification (Re-ID) based on unsupervised domain adaptation (UDA) aims to transfer the pre-trained model from one labeled source domain to an unlabeled target domain. Existing methods tackle this problem by using clustering methods to generate pseudo labels. However, pseudo labels produced by these techniques may be unstable and noisy, substantially deteriorating models’ performance. In this paper, we propose a Reliability Exploration with Self-ensemble Learning (RESL) framework for domain adaptive person Re- ID. First, to increase the feature diversity, multiple branches are presented to extract features from different data augmentations. Taking the temporally average model as a mean teacher model, online label refining is conducted by using its dynamic ensemble predictions from different branches as soft labels. Second, to combat the adverse effects of unreliable samples in clusters, sample reliability is estimated by evaluating the consistency of different clusters’ results, followed by selecting reliable instances for training and re-weighting sample contribution within Re-ID losses. A contrastive loss is also utilized with cluster-level memory features which are updated by the mean feature. The experiments demonstrate that our method can significantly surpass the state-of-the-art performance on the unsupervised domain adaptive person Re- ID.

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ICLR Conference 2021 Conference Paper

Fourier Neural Operator for Parametric Partial Differential Equations

Zongyi Li
Nikola Borislavov Kovachki
Kamyar Azizzadenesheli
Burigede Liu
Kaushik Bhattacharya
Andrew M. Stuart
Anima Anandkumar

The classical development of neural networks has primarily focused on learning mappings between finite-dimensional Euclidean spaces. Recently, this has been generalized to neural operators that learn mappings between function spaces. For partial differential equations (PDEs), neural operators directly learn the mapping from any functional parametric dependence to the solution. Thus, they learn an entire family of PDEs, in contrast to classical methods which solve one instance of the equation. In this work, we formulate a new neural operator by parameterizing the integral kernel directly in Fourier space, allowing for an expressive and efficient architecture. We perform experiments on Burgers' equation, Darcy flow, and Navier-Stokes equation. The Fourier neural operator is the first ML-based method to successfully model turbulent flows with zero-shot super-resolution. It is up to three orders of magnitude faster compared to traditional PDE solvers. Additionally, it achieves superior accuracy compared to previous learning-based solvers under fixed resolution.

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NeurIPS Conference 2020 Conference Paper

Multipole Graph Neural Operator for Parametric Partial Differential Equations

Zongyi Li
Nikola Kovachki
Kamyar Azizzadenesheli
Burigede Liu
Andrew Stuart
Kaushik Bhattacharya
Anima Anandkumar

One of the main challenges in using deep learning-based methods for simulating physical systems and solving partial differential equations (PDEs) is formulating physics-based data in the desired structure for neural networks. Graph neural networks (GNNs) have gained popularity in this area since graphs offer a natural way of modeling particle interactions and provide a clear way of discretizing the continuum models. However, the graphs constructed for approximating such tasks usually ignore long-range interactions due to unfavorable scaling of the computational complexity with respect to the number of nodes. The errors due to these approximations scale with the discretization of the system, thereby not allowing for generalization under mesh-refinement. Inspired by the classical multipole methods, we purpose a novel multi-level graph neural network framework that captures interaction at all ranges with only linear complexity. Our multi-level formulation is equivalent to recursively adding inducing points to the kernel matrix, unifying GNNs with multi-resolution matrix factorization of the kernel. Experiments confirm our multi-graph network learns discretization-invariant solution operators to PDEs and can be evaluated in linear time.

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AAAI Conference 2018 Short Paper

Conditional Linear Regression

Diego Calderon
Brendan Juba
Zongyi Li
Lisa Ruan

Previous work in machine learning and statistics commonly focuses on building models that capture the vast majority of data, possibly ignoring a segment of the population as outliers. By contrast, we may be interested in ﬁnding a segment of the population for which we can ﬁnd a linear rule capable of achieving more accurate predictions. We give an efﬁcient algorithm for the conditional linear regression task, which is the joint task of identifying a signiﬁcant segment of the population, described by a k-DNF, along with its linear regression ﬁt.

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AAAI Conference 2018 Short Paper

Learning Abduction Under Partial Observability

Brendan Juba
Zongyi Li
Evan Miller

Our work extends Juba’s formulation of learning abductive reasoning from examples, in which both the relative plausibility of various explanations, as well as which explanations are valid, are learned directly from data. We extend the formulation to consider partially observed examples, along with declarative background knowledge about the missing data. We show that it is possible to use implicitly learned rules together with the explicitly given declarative knowledge to support hypotheses in the course of abduction. We observe that when a small explanation exists, it is possible to obtain a much-improved guarantee in the challenging exceptiontolerant setting.

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AAAI Conference 2018 Conference Paper

Learning Abduction Using Partial Observability

Brendan Juba
Zongyi Li
Evan Miller

Juba recently proposed a formulation of learning abductive reasoning from examples, in which both the relative plausibility of various explanations, as well as which explanations are valid, are learned directly from data. The main shortcoming of this formulation of the task is that it assumes access to full-information (i. e. , fully speciﬁed) examples; relatedly, it offers no role for declarative background knowledge, as such knowledge is rendered redundant in the abduction task by complete information. In this work we extend the formulation to utilize such partially speciﬁed examples, along with declarative background knowledge about the missing data. We show that it is possible to use implicitly learned rules together with the explicitly given declarative knowledge to support hypotheses in the course of abduction. We also show how to use knowledge in the form of graphical causal models to re- ﬁne the proposed hypotheses. Finally, we observe that when a small explanation exists, it is possible to obtain a muchimproved guarantee in the challenging exception-tolerant setting. Such small, human-understandable explanations are of particular interest for potential applications of the task.

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