Hanyu Li

Hanyu Li, Yi-Ping Hsu, Aditya Mantha et al. · stanford

Modern recommendation systems predominantly train retrieval and ranking as separate models despite both increasingly relying on large transformers encoding the same user behavior data, duplicating parameters, compute, and serving cost. Prior work unifies the model architecture but not the full pipeline: input formats, training procedures, and serving stacks remain fragmented across stages. We present UniPinRec, which achieves full-stack unification of retrieval and ranking at Pinterest: one input format, one model, one training stage, deployed within existing serving infrastructure. A shared transformer encodes the user action sequence into candidate-independent representations that branch into retrieval (ANN dot-product) and ranking (cross-attention) via task-specific heads. Three ideas make this work: (1) Masked Action Modeling (MAM) eliminates interleaving, enabling weight sharing without doubling context length; (2) Blended training examples pair action sequences with feedview impression slates to satisfy both objectives jointly; (3) Cross-stage KV cache sharing reuses user-history computation from retrieval for ranking, reducing total FLOPs versus serving two independent models. Deployed in the Pinterest core surfaces, UniPinRec delivers approximately +1% online engagement lift while cutting end-to-end serving latency by 11.1% and lifting QPS by 63.6%. To our knowledge, this is the first full-stack unification of retrieval and ranking, covering inputs, model, training and serving, deployed in a production recommendation system.

Haibao Yu, Yizhen Luo, Mao Shu et al.

Autonomous driving faces great safety challenges for a lack of global perspective and the limitation of long-range perception capabilities. It has been widely agreed that vehicle-infrastructure cooperation is required to achieve Level 5 autonomy. However, there is still NO dataset from real scenarios available for computer vision researchers to work on vehicle-infrastructure cooperation-related problems. To accelerate computer vision research and innovation for Vehicle-Infrastructure Cooperative Autonomous Driving (VICAD), we release DAIR-V2X Dataset, which is the first large-scale, multi-modality, multi-view dataset from real scenarios for VICAD. DAIR-V2X comprises 71254 LiDAR frames and 71254 Camera frames, and all frames are captured from real scenes with 3D annotations. The Vehicle-Infrastructure Cooperative 3D Object Detection problem (VIC3D) is introduced, formulating the problem of collaboratively locating and identifying 3D objects using sensory inputs from both vehicle and infrastructure. In addition to solving traditional 3D object detection problems, the solution of VIC3D needs to consider the temporal asynchrony problem between vehicle and infrastructure sensors and the data transmission cost between them. Furthermore, we propose Time Compensation Late Fusion (TCLF), a late fusion framework for the VIC3D task as a benchmark based on DAIR-V2X. Find data, code, and more up-to-date information at https://thudair.baai.ac.cn/index and https://github.com/AIR-THU/DAIR-V2X.

Xiaoqing Ye, Mao Shu, Hanyu Li et al.

Concurrent perception datasets for autonomous driving are mainly limited to frontal view with sensors mounted on the vehicle. None of them is designed for the overlooked roadside perception tasks. On the other hand, the data captured from roadside cameras have strengths over frontal-view data, which is believed to facilitate a safer and more intelligent autonomous driving system. To accelerate the progress of roadside perception, we present the first high-diversity challenging Roadside Perception 3D dataset- Rope3D from a novel view. The dataset consists of 50k images and over 1.5M 3D objects in various scenes, which are captured under different settings including various cameras with ambiguous mounting positions, camera specifications, viewpoints, and different environmental conditions. We conduct strict 2D-3D joint annotation and comprehensive data analysis, as well as set up a new 3D roadside perception benchmark with metrics and evaluation devkit. Furthermore, we tailor the existing frontal-view monocular 3D object detection approaches and propose to leverage the geometry constraint to solve the inherent ambiguities caused by various sensors, viewpoints. Our dataset is available on https://thudair.baai.ac.cn/rope.

Xiaomi LLM-Core Team, Dong Zhang, Gang Wang et al.

Existing audio language models typically rely on task-specific fine-tuning to accomplish particular audio tasks. In contrast, humans are able to generalize to new audio tasks with only a few examples or simple instructions. GPT-3 has shown that scaling next-token prediction pretraining enables strong generalization capabilities in text, and we believe this paradigm is equally applicable to the audio domain. By scaling MiMo-Audio's pretraining data to over one hundred million of hours, we observe the emergence of few-shot learning capabilities across a diverse set of audio tasks. We develop a systematic evaluation of these capabilities and find that MiMo-Audio-7B-Base achieves SOTA performance on both speech intelligence and audio understanding benchmarks among open-source models. Beyond standard metrics, MiMo-Audio-7B-Base generalizes to tasks absent from its training data, such as voice conversion, style transfer, and speech editing. MiMo-Audio-7B-Base also demonstrates powerful speech continuation capabilities, capable of generating highly realistic talk shows, recitations, livestreaming and debates. At the post-training stage, we curate a diverse instruction-tuning corpus and introduce thinking mechanisms into both audio understanding and generation. MiMo-Audio-7B-Instruct achieves open-source SOTA on audio understanding benchmarks (MMSU, MMAU, MMAR, MMAU-Pro), spoken dialogue benchmarks (Big Bench Audio, MultiChallenge Audio) and instruct-TTS evaluations, approaching or surpassing closed-source models. Model checkpoints and full evaluation suite are available at https://github.com/XiaomiMiMo/MiMo-Audio.

Xiaotie Deng, Hanyu Li, Ningyuan Li · pku

In this work, a convergence lemma for function $f$ being finite compositions of analytic mappings and the maximum operator is proved. The lemma shows that the set of $δ$-stationary points near an isolated local minimum point $x^*$ is shrinking to $x^*$ as $δ\to 0$. It is a natural extension of the version for strongly convex $C^1$ functions. However, the correctness of the lemma is subtle. Analytic mappings are necessary for the lemma in the sense that replacing it with differentiable or $C^\infty$ mappings makes the lemma false. The proof is based on stratification theorems of semi-analytic sets by Łojasiewicz. An extension of this proof presents a geometric characterization of the set of stationary points of $f$. Finally, a notion of stability on stationary points, called convergence stability, is proposed. It asks, under small numerical errors, whether a reasonable convergent optimization method started near a stationary point should eventually converge to the same stationary point. The concept of convergence stability becomes nontrivial qualitatively only when the objective function is both nonsmooth and nonconvex. Via the convergence lemma, an intuitive equivalent condition for convergence stability of $f$ is proved. These results together provide a new geometric perspective to study the problem of "where-to-converge" in nonsmooth nonconvex optimization.

Qianxin Qu, Yijin Xiong, Guipeng Zhang et al.

Cooperative LiDAR systems integrating vehicles and road infrastructure, termed V2I calibration, exhibit substantial potential, yet their deployment encounters numerous challenges. A pivotal aspect of ensuring data accuracy and consistency across such systems involves the calibration of LiDAR units across heterogeneous vehicular and infrastructural endpoints. This necessitates the development of calibration methods that are both real-time and robust, particularly those that can ensure robust performance in urban canyon scenarios without relying on initial positioning values. Accordingly, this paper introduces a novel approach to V2I calibration, leveraging spatial association information among perceived objects. Central to this method is the innovative Overall Intersection over Union (oIoU) metric, which quantifies the correlation between targets identified by vehicle and infrastructure systems, thereby facilitating the real-time monitoring of calibration results. Our approach involves identifying common targets within the perception results of vehicle and infrastructure LiDAR systems through the construction of an affinity matrix. These common targets then form the basis for the calculation and optimization of extrinsic parameters. Comparative and ablation studies conducted using the DAIR-V2X dataset substantiate the superiority of our approach. For further insights and resources, our project repository is accessible at https://github.com/MassimoQu/v2i-calib.

Shaoxin Wang, Hanyu Li, Hu Yang

In this paper, we consider the explicit expressions of the normwise condition number for the scaled total least squares problem. Some techniques are introduced to simplify the expression of the condition number, and some new results are derived. Based on these new results, new expressions of the condition number for the total least squares problem can be deduced as a special case. New forms of the condition number enjoy some storage and computational advantages. We also proposed three different methods to estimate the condition number. Some numerical experiments are carried out to illustrate the effectiveness of our results.

Hanyu Li, Shaoxin Wang

In this paper, the normwise condition number of a linear function of the equality constrained linear least squares solution called the partial condition number is considered. Its expression and closed formulae are first presented when the data space and the solution space are measured by the weighted Frobenius norm and the Euclidean norm, respectively. Then, we investigate the corresponding structured partial condition number when the problem is structured. To estimate these condition numbers with high reliability, the probabilistic spectral norm estimator and the small-sample statistical condition estimation method are applied and two algorithms are devised. The obtained results are illustrated by numerical examples.

Haolong Hu, Hanyu Li, Tiancheng He et al.

MLLMs are increasingly deployed in multi-turn settings, where attackers can escalate unsafe intent through the evolving visual-text history and exploit long-context safety decay. Yet safety alignment is still dominated by single-turn data and fixed-template dialogues, leaving a mismatch between training and deployment.To bridge this gap, we propose SaFeR-Steer, a progressive multi-turn alignment framework that combines staged synthetic bootstrapping with tutor-in-the-loop GRPO to train a single student under adaptive, on-policy attacks. We also introduce TCSR, which uses trajectory minimum/average safety to propagate late-turn failures to earlier turns.I. Dataset. We release STEER, a multi-turn multimodal safety dataset with STEER-SFT (12,934), STEER-RL (2,000), and STEER-Bench (3,227) dialogues spanning 2~10 turns.II. Experiment. Starting from Qwen2.5-VL-3B/7B, SaFeR-Steer substantially improves Safety/Helpfulness on both single-turn (48.30/45.86 -> 81.84/70.77 for 3B; 56.21/60.32 -> 87.89/77.40 for 7B) and multi-turn benchmarks (12.55/27.13 -> 55.58/70.27 for 3B; 24.66/46.48 -> 64.89/72.35 for 7B), shifting failures to later turns and yielding robustness beyond scaling alone.Codes are available at https://github.com/Ed-Bg/SaFeR-Steer

Hanyu Li, Yichi Zhang, Speed Zhu et al.

Code agents are currently having skillful performance on repository-level software engineering benchmarks, but it remains unclear whether success on end-to-end tasks such as issue resolution truly reflects repository context reasoning, the ability to identify the task-relevant information across multiple files and reason over the relations among them. To investigate this question, we introduce RepoMirage, a two-stage evaluation suite built on SWE-Bench Verified that adopts perturbation as a diagnostic tool to increase the demand for context reasoning by transforming how the repository is exposed. First, RepoMirage-Perturb applies three types of semantics-preserving repository-level perturbations, revealing a clear performance drop when correct solving requires broader context access. RepoMirage-Extend further turns perturbation-targeted structural bottlenecks into explicit tasks beyond issue resolution, where the average performance declines from 66.8% in the original setting to 25.3%, indicating a significant deficiency in repository context reasoning. Further trajectory analysis reveals an exploration drift, where agents access broader repository context but fail to turn it into effective structure information. Motivated by this observation, we propose RepoAnchor, a structure-first prototype workflow that separates repository exploration from downstream problem solving, and show that explicit structural scaffolding yields notable gains. These results uncover an previously overlooked gap in repository context reasoning for code agents and suggest that stronger structure-aware methods are potential to improve them.

Hanyu Li, Yanfei Yang

Some new rigorous perturbation bounds for the generalized Cholesky factorization with normwise or componentwise perturbations in the given matrix are obtained, where the componentwise perturbation has the form of backward rounding error for the generalized Cholesky factorization algorithm. These bounds can be much tighter than some existing ones while the conditions for them to hold are simple and moderate.

Hanyu Li, Jiangshan Duo, Bofei Gao et al.

Chain-of-thought reasoning in large language models can trigger an "overthinking trap": longer rollouts raise cost and latency yet often yield unreliable accuracy gains. Existing methods use global, static controls that may suppress needed reasoning. We propose mastery-gated, sample-level, soft reinforcement learning compression that penalizes long rollouts only when the model already solves the problem and has produced a shorter rollout. Across benchmarks, it cuts response length by 20-40% with comparable or higher accuracy and generalizes across domains: a model trained on math spontaneously shortens unseen tasks (code, instruction following, general-knowledge QA) without hurting accuracy. We further show two-way transfer between non-agent CoT and tool-use agents: non-agent training reduces SWE-Bench Verified rounds by 13%, while compressing a thinking agent cuts SWE trajectories by 67% tokens and 52% rounds and shortens non-agent outputs by up to 44%. Compression is thus not cosmetic brevity, but an inherent computation policy -- what to keep, and what to forget.

Hanyu Li, Zhengqi Sun, Xiaotie Deng

Synthetic data becomes crucial for large language model training, but its effectiveness is highly inconsistent. We provide an information-theoretic account of this inconsistency: synthetic data improves a model only when the generation-training loop is information-open, i.e., shaped by external signals (verifiers, environments, or rubrics) that inject task-relevant information beyond the model's current distribution. When the loop is information-closed (relying on the model's own outputs without such signals), the data processing inequality ensures that task-relevant information can only decrease, making collapse a predicted outcome. Among information-open pipelines, both efficiency and generalization hinge on the meta-level of supervision: a coarser signal such as binary correctness treats all acceptable outputs as equivalent, so the behavior it teaches is not tied to any particular domain or surface form and generalizes naturally across tasks and domains. These observations lead to a guiding thesis: learning preferentially converges to the most information-efficient signal component available, which accelerates learning when that component is the intended one, but causes reward hacking when a spurious pattern happens to be simpler.

Xiaotie Deng, Dongchen Li, Hanyu Li

AI in Math deals with mathematics in a constructive manner so that reasoning becomes automated, less laborious, and less error-prone. For algorithms, the question becomes how to automate analyses for specific problems. For the first time, this work provides an automatic method for approximation analysis on a well-studied problem in theoretical computer science: computing approximate Nash equilibria in two-player games. We observe that such algorithms can be reformulated into a search-and-mix paradigm, which involves a search phase followed by a mixing phase. By doing so, we are able to fully automate the procedure of designing and analyzing the mixing phase. For example, we illustrate how to perform our method with a program to analyze the approximation bounds of all the algorithms in the literature. Same approximation bounds are computed without any hand-written proof. Our automatic method heavily relies on the LP-relaxation structure in approximate Nash equilibria. Since many approximation algorithms and online algorithms adopt the LP relaxation, our approach may be extended to automate the analysis of other algorithms.

Jiangshan Duo, Hanyu Li, Hailin Zhang et al.

Reinforcement Learning with Verifiable Rewards (RLVR) has become a standard paradigm for reasoning in Large Language Models. However, optimizing solely for final-answer correctness often drives models into aimless, verbose exploration, where they rely on exhaustive trial-and-error tactics rather than structured planning to reach solutions. While heuristic constraints like length penalties can reduce verbosity, they often truncate essential reasoning steps, creating a difficult trade-off between efficiency and verification. In this paper, we argue that discriminative capability is a prerequisite for efficient generation: by learning to distinguish valid solutions, a model can internalize a guidance signal that prunes the search space. We propose JudgeRLVR, a two-stage judge-then-generate paradigm. In the first stage, we train the model to judge solution responses with verifiable answers. In the second stage, we fine-tune the same model with vanilla generating RLVR initialized from the judge. Compared to Vanilla RLVR using the same math-domain training data, JudgeRLVR achieves a better quality--efficiency trade-off for Qwen3-30B-A3B: on in-domain math, it delivers about +3.7 points average accuracy gain with -42\% average generation length; on out-of-domain benchmarks, it delivers about +4.5 points average accuracy improvement, demonstrating enhanced generalization.

Hanyu Li, Wenhan Huang, Zhijian Duan et al.

Nash equilibrium is one of the most influential solution concepts in game theory. With the development of computer science and artificial intelligence, there is an increasing demand on Nash equilibrium computation, especially for Internet economics and multi-agent learning. This paper reviews various algorithms computing the Nash equilibrium and its approximation solutions in finite normal-form games from both theoretical and empirical perspectives. For the theoretical part, we classify algorithms in the literature and present basic ideas on algorithm design and analysis. For the empirical part, we present a comprehensive comparison on the algorithms in the literature over different kinds of games. Based on these results, we provide practical suggestions on implementations and uses of these algorithms. Finally, we present a series of open problems from both theoretical and practical considerations.

Xiangyi Chen, Kousik Rajesh, Matthew Lawhon et al.

User activity sequences have emerged as one of the most important signals in recommender systems. We present a foundational model, PinFM, for understanding user activity sequences across multiple applications at a billion-scale visual discovery platform. We pretrain a transformer model with 20B+ parameters using extensive user activity data, then fine-tune it for specific applications, efficiently coupling it with existing models. While this pretraining-and-fine-tuning approach has been popular in other domains, such as Vision and NLP, its application in industrial recommender systems presents numerous challenges. The foundational model must be scalable enough to score millions of items every second while meeting tight cost and latency constraints imposed by these systems. Additionally, it should capture the interactions between user activities and other features and handle new items that were not present during the pretraining stage. We developed innovative techniques to address these challenges. Our infrastructure and algorithmic optimizations, such as the Deduplicated Cross-Attention Transformer (DCAT), improved our throughput by 600% on Pinterest internal data. We demonstrate that PinFM can learn interactions between user sequences and candidate items by altering input sequences, leading to a 20% increase in engagement with new items. PinFM is now deployed to help improve the experience of more than half a billion users across various applications.

Hanyu Li, Haoyu Liu, Tingyu Zhu et al.

Large Language Models (LLMs) show promise as data analysis agents, but existing benchmarks overlook the iterative nature of the field, where experts' decisions evolve with deeper insights of the dataset. To address this, we introduce IDA-Bench, a novel benchmark evaluating LLM agents in multi-round interactive scenarios. Derived from complex Kaggle notebooks, tasks are presented as sequential natural language instructions by an LLM-simulated user. Agent performance is judged by comparing its final numerical output to the human-derived baseline. Initial results show that even state-of-the-art coding agents (like Claude-3.7-thinking) succeed on < 50% of the tasks, highlighting limitations not evident in single-turn tests. This work underscores the need to improve LLMs' multi-round capabilities for building more reliable data analysis agents, highlighting the necessity of achieving a balance between instruction following and reasoning.

Hanyu Li, Xiaotie Deng

Classic no-trade theorems attribute trade to heterogeneous beliefs. We re-examine this conclusion for AI agents, asking if trade can arise from computational limitations, under common beliefs. We model agents' bounded computational rationality within an unfolding game framework, where computational power determines the complexity of its strategy. Our central finding inverts the classic paradigm: a stable no-trade outcome (Nash equilibrium) is reached only when "almost rational" agents have slightly different computational power. Paradoxically, when agents possess identical power, they may fail to converge to equilibrium, resulting in persistent strategic adjustments that constitute a form of trade. This instability is exacerbated if agents can strategically under-utilize their computational resources, which eliminates any chance of equilibrium in Matching Pennies scenarios. Our results suggest that the inherent computational limitations of AI agents can lead to situations where equilibrium is not reached, creating a more lively and unpredictable trade environment than traditional models would predict.

Lin Lin, Jiefeng Long, Zhihe Wan et al. · pku

Multimodal embedding models aim to yield informative unified representations that empower diverse cross-modal tasks. Despite promising developments in the evolution from CLIP-based dual-tower architectures to large vision-language models, prior works still face unavoidable challenges in real-world applications and business scenarios, such as the limited modality support, unstable training mechanisms, and industrial domain gaps. In this work, we introduce SAIL-Embedding, an omni-modal embedding foundation model that addresses these issues through tailored training strategies and architectural design. In the optimization procedure, we propose a multi-stage training scheme to boost the multifaceted effectiveness of representation learning. Specifically, the content-aware progressive training aims to enhance the model's adaptability to diverse downstream tasks and master enriched cross-modal proficiency. The collaboration-aware recommendation enhancement training further adapts multimodal representations for recommendation scenarios by distilling knowledge from sequence-to-item and ID-to-item embeddings while mining user historical interests. Concurrently, we develop the stochastic specialization and dataset-driven pattern matching to strengthen model training flexibility and generalizability. Experimental results show that SAIL-Embedding achieves SOTA performance compared to other methods in different retrieval tasks. In online experiments across various real-world scenarios integrated with our model, we observe a significant increase in Lifetime (LT), which is a crucial indicator for the recommendation experience. For instance, the model delivers the 7-day LT gain of +0.5% in the Douyin-Selected scenario. For the Douyin feed rank model, the match features produced by SAIL-Embedding yield a +0.1% AUC gain.

Xiaotie Deng, Hanyu Li

We argue that AI's future requires more than scaling. To unlock genuine discovery, large language models need a compass: human-crafted symbols to guide their powerful but blind intuition.

Hanyu Li, Dongchen Li, Xiaotie Deng

Algorithm design and analysis is a cornerstone of computer science, but it confronts a major challenge. Proving an algorithm's performance guarantee across all inputs has traditionally required extensive and often error-prone human effort. While AI has shown great success in finding solutions to specific problem instances, automating the discovery of general algorithms with such provable guarantees has remained a significant barrier. This challenge stems from the difficulty of integrating the creative process of algorithm design with the rigorous process of formal analysis. To address this gap, we propose LegoNE, a framework that tightly fuses these two processes for the fundamental and notoriously difficult problem of computing approximate Nash equilibria. LegoNE automatically translates any algorithm written by a simple Python-like language into a constrained optimization problem. Solving this problem derives and proves the algorithm's approximation bound. Using LegoNE, a state-of-the-art large language model rediscovered the state-of-the-art algorithm for two-player games within hours, a feat that had taken human researchers 15 years to achieve. For three-player games, the model discovered a novel algorithm surpassing all existing human-designed ones. This work demonstrates a new human-machine collaborative paradigm for theoretical science: humans reason at a higher-abstract level, using symbols to compress the search space, and AI explores within it, achieving what neither could alone.

Donglai Wei, Kisuk Lee, Hanyu Li et al.

Electron microscopy (EM) enables the reconstruction of neural circuits at the level of individual synapses, which has been transformative for scientific discoveries. However, due to the complex morphology, an accurate reconstruction of cortical axons has become a major challenge. Worse still, there is no publicly available large-scale EM dataset from the cortex that provides dense ground truth segmentation for axons, making it difficult to develop and evaluate large-scale axon reconstruction methods. To address this, we introduce the AxonEM dataset, which consists of two 30x30x30 um^3 EM image volumes from the human and mouse cortex, respectively. We thoroughly proofread over 18,000 axon instances to provide dense 3D axon instance segmentation, enabling large-scale evaluation of axon reconstruction methods. In addition, we densely annotate nine ground truth subvolumes for training, per each data volume. With this, we reproduce two published state-of-the-art methods and provide their evaluation results as a baseline. We publicly release our code and data at https://connectomics-bazaar.github.io/proj/AxonEM/index.html to foster the development of advanced methods.

Hanyu Li, Jingjing Li, Wei Wang

Underwater image enhancement algorithms have attracted much attention in underwater vision task. However, these algorithms are mainly evaluated on different data sets and different metrics. In this paper, we set up an effective and pubic underwater test dataset named U45 including the color casts, low contrast and haze-like effects of underwater degradation and propose a fusion adversarial network for enhancing underwater images. Meanwhile, the well-designed the adversarial loss including Lgt loss and Lfe loss is presented to focus on image features of ground truth, and image features of the image enhanced by fusion enhance method, respectively. The proposed network corrects color casts effectively and owns faster testing time with fewer parameters. Experiment results on U45 dataset demonstrate that the proposed method achieves better or comparable performance than the other state-of-the-art methods in terms of qualitative and quantitative evaluations. Moreover, an ablation study demonstrates the contributions of each component, and the application test further shows the effectiveness of the enhanced images.

Wushi Dong, Murat Keceli, Rafael Vescovi et al.

Mapping all the neurons in the brain requires automatic reconstruction of entire cells from volume electron microscopy data. The flood-filling network (FFN) architecture has demonstrated leading performance for segmenting structures from this data. However, the training of the network is computationally expensive. In order to reduce the training time, we implemented synchronous and data-parallel distributed training using the Horovod library, which is different from the asynchronous training scheme used in the published FFN code. We demonstrated that our distributed training scaled well up to 2048 Intel Knights Landing (KNL) nodes on the Theta supercomputer. Our trained models achieved similar level of inference performance, but took less training time compared to previous methods. Our study on the effects of different batch sizes on FFN training suggests ways to further improve training efficiency. Our findings on optimal learning rate and batch sizes agree with previous works.

Hanyu Li, Shaoxin Wang

The condition number of a linear function of the indefinite least squares solution is called the partial condition number for the indefinite least squares problem. In this paper, based on a new and very general condition number which can be called the unified condition number, the expression of the partial unified condition number is first presented when the data space is measured by the general weighted product norm. Then, by setting the specific norms and weight parameters, we obtain the expressions of the partial normwise, mixed and componentwise condition numbers. Moreover, the corresponding structured partial condition numbers are also taken into consideration when the problem is structured, whose expressions are given. Considering the connections between the indefinite and total least squares problems, we derive the (structured) partial condition numbers for the latter, which generalize the ones in the literature. To estimate these condition numbers effectively and reliably, the probabilistic spectral norm estimator and the small-sample statistical condition estimation method are applied and three related algorithms are devised. Finally, the obtained results are illustrated by numerical experiments.

Hanyu Li, Shaoxin Wang, Chan Zheng

In this paper, we consider the perturbation analysis for the periodic generalized coupled Sylvester (PGCS) equation. The normwise backward error for this equation is first obtained. Then, we present its normwise and componentwise perturbation bounds, from which the normwise and effective condition numbers are derived. Moreover, the mixed and componentwise condition numbers for the PGCS equation are also given. To estimate these condition numbers with high reliability, the probabilistic spectral norm estimator and the statistical condition estimation method are applied. The obtained results are illustrated by numerical examples.