* Ghosh and Sanyal (2021) Indranil Ghosh and Manas K Sanyal. 2021.  Introspecting predictability of market fear in Indian context during COVID-19 pandemic: An integrated approach of applied predictive modelling and explainable AI.  _International Journal of Information Management Data Insights_ 1, 2 (2021), 100039.
  * Gite et al. (2021) Shilpa Gite, Hrituja Khatavkar, Ketan Kotecha, Shilpi Srivastava, Priyam Maheshwari, and Neerav Pandey. 2021.  Explainable stock prices prediction from financial news articles using sentiment analysis.  _PeerJ Computer Science_ 7 (2021), e340.
  * Goodman and Flaxman (2017) Bryce Goodman and Seth Flaxman. 2017.  European Union regulations on algorithmic decision-making and a “right to explanation”.  _AI magazine_ 38, 3 (2017), 50–57.
  * Gramegna and Giudici (2020) Alex Gramegna and Paolo Giudici. 2020.  Why to buy insurance? An explainable artificial intelligence approach.  _Risks_ 8, 4 (2020), 137.
  * Gramespacher and Posth (2021) Thomas Gramespacher and Jan-Alexander Posth. 2021.  Employing explainable AI to optimize the return target function of a loan portfolio.  _Frontiers in Artificial Intelligence_ 4 (2021), 693022.
  * Grath et al. (2018) Rory Mc Grath, Luca Costabello, Chan Le Van, Paul Sweeney, Farbod Kamiab, Zhao Shen, and Freddy Lecue. 2018.  Interpretable credit application predictions with counterfactual explanations.  _arXiv preprint arXiv:1811.05245_ (2018).
  * Guidotti et al. (2018) Riccardo Guidotti, Anna Monreale, Salvatore Ruggieri, Franco Turini, Fosca Giannotti, and Dino Pedreschi. 2018.  A survey of methods for explaining black box models.  _ACM computing surveys (CSUR)_ 51, 5 (2018), 1–42.
  * Gurumoorthy et al. (2019) Karthik S Gurumoorthy, Amit Dhurandhar, Guillermo Cecchi, and Charu Aggarwal. 2019.  Efficient data representation by selecting prototypes with importance weights. In _2019 IEEE International Conference on Data Mining (ICDM)_. IEEE, 260–269.
  * Han et al. (2022) Sooji Han, Rui Mao, and Erik Cambria. 2022.  Hierarchical Attention Network for Explainable Depression Detection on Twitter Aided by Metaphor Concept Mappings. In _Proceedings of the 29th International Conference on Computational Linguistics (COLING)_. International Committee on Computational Linguistics, Gyeongju, Republic of Korea, 94–104.
  * He et al. (2022a) Kai He, Rui Mao, Tieliang Gong, Erik Cambria, and Chen Li. 2022a.  JCBIE: A Joint Continual Learning Neural Network for Biomedical Information Extraction.  _BMC Bioinformatics_ 23, 549 (2022).  https://doi.org/10.1186/s12859-022-05096-w
  * He et al. (2022b) Kai He, Rui Mao, Tieliang Gong, Chen Li, and Erik Cambria. 2022b.  Meta-based Self-training and Re-weighting for Aspect-based Sentiment Analysis.  _IEEE Transactions on Affective Computing_ (2022).  https://doi.org/10.1109/TAFFC.2022.3202831
  * HLEG (2019) AI HLEG. 2019.  _Ethics guidelines for trustworthy AI_.  Retrieved February 8, 2023 from https://digital-strategy.ec.europa.eu/en/library/ethics-guidelines-trustworthy-ai
  * Hoffman et al. (2018) Robert R Hoffman, Shane T Mueller, Gary Klein, and Jordan Litman. 2018.  Metrics for explainable AI: Challenges and prospects.  _arXiv preprint arXiv:1812.04608_ (2018).
  * Hohman et al. (2019) Fred Hohman, Andrew Head, Rich Caruana, Robert DeLine, and Steven M Drucker. 2019.  Gamut: A design probe to understand how data scientists understand machine learning models. In _Proceedings of the 2019 CHI conference on human factors in computing systems_. 1–13.
  * Islam et al. (2019) Sheikh Rabiul Islam, William Eberle, Sid Bundy, and Sheikh Khaled Ghafoor. 2019.  Infusing domain knowledge in AI-based “ black box” models for better explainability with application in bankruptcy prediction.  _arXiv preprint arXiv:1905.11474_ (2019).
  * Ito et al. (2020) Tomoki Ito, Hiroki Sakaji, Kiyoshi Izumi, Kota Tsubouchi, and Tatsuo Yamashita. 2020.  GINN: gradient interpretable neural networks for visualizing financial texts.  _International Journal of Data Science and Analytics_ 9 (2020), 431–445.
  * Jacovi and Goldberg (2020) Alon Jacovi and Yoav Goldberg. 2020.  Towards faithfully interpretable NLP systems: How should we define and evaluate faithfulness?  _arXiv preprint arXiv:2004.03685_ (2020).
  * Jin et al. (2019) Xisen Jin, Zhongyu Wei, Junyi Du, Xiangyang Xue, and Xiang Ren. 2019.  Towards hierarchical importance attribution: Explaining compositional semantics for neural sequence models.  _arXiv preprint arXiv:1911.06194_ (2019).
  * Kaur et al. (2020) Harmanpreet Kaur, Harsha Nori, Samuel Jenkins, Rich Caruana, Hanna Wallach, and Jennifer Wortman Vaughan. 2020.  Interpreting interpretability: understanding data scientists’ use of interpretability tools for machine learning. In _Proceedings of the 2020 CHI conference on human factors in computing systems_. 1–14.
  * Kolanovic and Krishnamachari (2017) Marko Kolanovic and Rajesh T. Krishnamachari. 2017.  Big Data and AI Strategies, Machine Learning and Alternative Data Approach to Investing.
  * Kuiper et al. (2022) Ouren Kuiper, Martin van den Berg, Joost van der Burgt, and Stefan Leijnen. 2022.  Exploring explainable ai in the financial sector: Perspectives of banks and supervisory authorities. In _Artificial Intelligence and Machine Learning: 33rd Benelux Conference on Artificial Intelligence, BNAIC/Benelearn 2021, Esch-sur-Alzette, Luxembourg, November 10–12, 2021, Revised Selected Papers 33_. Springer, 105–119.
  * Kumar et al. (2017) Devinder Kumar, Graham W Taylor, and Alexander Wong. 2017\.  Opening the black box of financial ai with clear-trade: A class-enhanced attentive response approach for explaining and visualizing deep learning-driven stock market prediction.  _arXiv preprint arXiv:1709.01574_ (2017).
  * Kumar et al. (2022) Satyam Kumar, Mendhikar Vishal, and Vadlamani Ravi. 2022\.  Explainable Reinforcement Learning on Financial Stock Trading using SHAP.  _arXiv preprint arXiv:2208.08790_ (2022).
  * Lachuer and Jabeur (2022) Julien Lachuer and Sami Ben Jabeur. 2022.  Explainable artificial intelligence modeling for corporate social responsibility and financial performance.  _Journal of Asset Management_ 23, 7 (2022), 619–630.
  * Levine (2014) Timothy R Levine. 2014\.  Truth-default theory (TDT) a theory of human deception and deception detection.  _Journal of Language and Social Psychology_ 33, 4 (2014), 378–392.
  * Liang et al. (2022) Bin Liang, Hang Su, Lin Gui, Erik Cambria, and Ruifeng Xu. 2022.  Aspect-based sentiment analysis via affective knowledge enhanced graph convolutional networks.  _Knowledge-Based Systems_ 235 (2022), 107643.
  * Lin et al. (2021) Ting-Wei Lin, Ruei-Yao Sun, Hsuan-Ling Chang, Chuan-Ju Wang, and Ming-Feng Tsai. 2021.  XRR: Explainable Risk Ranking for Financial Reports. In _Machine Learning and Knowledge Discovery in Databases. Applied Data Science Track: European Conference, ECML PKDD 2021, Bilbao, Spain, September 13–17, 2021, Proceedings, Part IV 21_. Springer, 253–268.
  * Lipton (1990) Peter Lipton. 1990\.  Contrastive explanation.  _Royal Institute of Philosophy Supplements_ 27 (1990), 247–266.
  * Lipton (2018) Zachary C Lipton. 2018\.  The mythos of model interpretability: In machine learning, the concept of interpretability is both important and slippery.  _Queue_ 16, 3 (2018), 31–57.
  * Liu et al. (2020) Rong Liu, Feng Mai, Zhe Shan, and Ying Wu. 2020\.  Predicting shareholder litigation on insider trading from financial text: An interpretable deep learning approach.  _Information & Management_ 57, 8 (2020), 103387\.
  * Lundberg et al. (2018) Scott M Lundberg, Gabriel G Erion, and Su-In Lee. 2018\.  Consistent individualized feature attribution for tree ensembles.  _arXiv preprint arXiv:1802.03888_ (2018).
  * Lundberg and Lee (2017) Scott M Lundberg and Su-In Lee. 2017.  A unified approach to interpreting model predictions.  _Advances in neural information processing systems_ 30 (2017).
  * Luo et al. (2018) Ling Luo, Xiang Ao, Feiyang Pan, Jin Wang, Tong Zhao, Ningzi Yu, and Qing He. 2018.  Beyond Polarity: Interpretable Financial Sentiment Analysis with Hierarchical Query-driven Attention. In _IJCAI_. 4244–4250.
  * Ma et al. (2023) Yu Ma, Rui Mao, Qika Lin, Peng Wu, and Erik Cambria. 2023.  Multi-source Aggregated Classification for Stock Price Movement Prediction.  _Information Fusion_ 91 (2023), 515–528.  https://doi.org/10.1016/j.inffus.2022.10.025
  * Mao and Li (2021) Rui Mao and Xiao Li. 2021\.  Bridging Towers of Multi-task Learning with a Gating Mechanism for Aspect-based Sentiment Analysis and Sequential Metaphor Identification.  _Proceedings of the AAAI Conference on Artificial Intelligence_ 35, 15 (2021), 13534–13542.  https://doi.org/10.1609/aaai.v35i15.17596
  * Mao et al. (2022) Rui Mao, Xiao Li, Mengshi Ge, and Erik Cambria. 2022\.  MetaPro: A computational metaphor processing model for text pre-processing.  _Information Fusion_ 86-87 (2022), 30–43.  https://doi.org/10.1016/j.inffus.2022.06.002
  * Maree et al. (2020) Charl Maree, Jan Erik Modal, and Christian W Omlin. 2020\.  Towards responsible AI for financial transactions. In _2020 IEEE Symposium Series on Computational Intelligence (SSCI)_. IEEE, 16–21.
  * Maree and Omlin (2022a) Charl Maree and Christian W Omlin. 2022a.  Can interpretable reinforcement learning manage prosperity your way?  _AI_ 3, 2 (2022), 526–537.
  * Maree and Omlin (2022b) Charl Maree and Christian W Omlin. 2022b.  Understanding Spending Behavior: Recurrent Neural Network Explanation and Interpretation. In _2022 IEEE Symposium on Computational Intelligence for Financial Engineering and Economics (CIFEr)_. IEEE, 1–7.
  * Markowitz (1952) Harry Markowitz. 1952\.  Portfolio Selection.  _The Journal of Finance_ 7, 1 (1952), 77–91.  http://www.jstor.org/stable/2975974
  * Mellon (2021) BNY Mellon. 2021\.  _Why Every Financial Institution Should Consider Explainable AI_.  Retrieved February 8, 2023 from https://www.bnymellon.com/us/en/insights/all-insights/why-every-financial-institution-should-consider-explainable-ai.html
  * Miller (2019) Tim Miller. 2019\.  Explanation in artificial intelligence: Insights from the social sciences.  _Artificial intelligence_ 267 (2019), 1–38.
  * Misheva et al. (2021) Branka Hadji Misheva, Joerg Osterrieder, Ali Hirsa, Onkar Kulkarni, and Stephen Fung Lin. 2021\.  Explainable AI in credit risk management.  _arXiv preprint arXiv:2103.00949_ (2021).
  * Mohseni et al. (2021a) Sina Mohseni, Fan Yang, Shiva Pentyala, Mengnan Du, Yi Liu, Nic Lupfer, Xia Hu, Shuiwang Ji, and Eric Ragan. 2021a.  Machine learning explanations to prevent overtrust in fake news detection. In _Proceedings of the International AAAI Conference on Web and Social Media_ , Vol. 15. 421–431.
  * Mohseni et al. (2021b) Sina Mohseni, Niloofar Zarei, and Eric D Ragan. 2021b.  A multidisciplinary survey and framework for design and evaluation of explainable AI systems.  _ACM Transactions on Interactive Intelligent Systems (TiiS)_ 11, 3-4 (2021), 1–45.
  * Molnar (2020) Christoph Molnar. 2020\.  _Interpretable machine learning_.  Lulu. com.
  * Mroczkowska (2020) Agnieszka Mroczkowska. 2020\.  _What is a Fintech Application?, Definition and Insights for Business Owners_.  Retrieved February 7, 2023 from https://www.thedroidsonroids.com/blog/what-is-a-fintech-application-definition-and-insights-for-business-owners/
  * Mueller et al. (2019) Shane T Mueller, Robert R Hoffman, William Clancey, Abigail Emrey, and Gary Klein. 2019.  Explanation in human-AI systems: A literature meta-review, synopsis of key ideas and publications, and bibliography for explainable AI.  _arXiv preprint arXiv:1902.01876_ (2019).
  * Müller et al. (2022) Ricardo Müller, Marco Schreyer, Timur Sattarov, and Damian Borth. 2022.  RESHAPE: Explaining Accounting Anomalies in Financial Statement Audits by enhancing SHapley Additive exPlanations. In _Proceedings of the Third ACM International Conference on AI in Finance_. 174–182.
  * Nazemi et al. (2022) Abdolreza Nazemi, Jonas Rauch, and Frank J Fabozzi. 2022\.  Interpretable Machine Learning for Creditor Recovery Rates.  _Available at SSRN 4190345_ (2022).
  * Neyshabur et al. (2020) Behnam Neyshabur, Hanie Sedghi, and Chiyuan Zhang. 2020\.  What is being transferred in transfer learning?  _Advances in neural information processing systems_ 33 (2020), 512–523.
  * Nori et al. (2019) Harsha Nori, Samuel Jenkins, Paul Koch, and Rich Caruana. 2019\.  Interpretml: A unified framework for machine learning interpretability.  _arXiv preprint arXiv:1909.09223_ (2019).
  * of Singapore (2021) Monetary Authority of Singapore. 2021\.  _Veritas Initiative Addresses Implementation Challenges in the Responsible Use of Artificial Intelligence and Data Analytics_.  Retrieved February 8, 2023 from https://www.mas.gov.sg/news/media-releases/2021/veritas-initiative-addresses-implementation-challenges
  * Ong et al. (2023) Keane Ong, Wihan van der Heever, Ranjan Satapathy, Gianmarco Mengaldo, and Erik Cambria. 2023\.  FinXABSA: Explainable Finance through Aspect-Based Sentiment Analysis.  arXiv:2303.02563 [cs.CL]
  * Park and Yang (2022) Sangjin Park and Jae-Suk Yang. 2022.  Interpretable deep learning LSTM model for intelligent economic decision-making.  _Knowledge-Based Systems_ 248 (2022), 108907.
  * Radford et al. (2019) Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei, Ilya Sutskever, et al. 2019\.  Language models are unsupervised multitask learners.  _OpenAI blog_ 1, 8 (2019), 9.
  * Rallis et al. (2022) Ioannis Rallis, Yannis Markoulidakis, Ioannis Georgoulas, George Kopsiaftis, Maria Kaselimi, Nikolaos Doulamis, and Anastasios Doulamis. 2022\.  Interpretation of net promoter score attributes using explainable AI. In _Proceedings of the 15th International Conference on PErvasive Technologies Related to Assistive Environments_. 113–117.
  * Ribeiro et al. (2016) Marco Tulio Ribeiro, Sameer Singh, and Carlos Guestrin. 2016\.  “Why should i trust you?” Explaining the predictions of any classifier. In _Proceedings of the 22nd ACM SIGKDD international conference on knowledge discovery and data mining_. 1135–1144.
  * Rizinski et al. (2022) Maryan Rizinski, Hristijan Peshov, Kostadin Mishev, Lubomir T Chitkushev, Irena Vodenska, and Dimitar Trajanov. 2022.  Ethically Responsible Machine Learning in Fintech.  _IEEE Access_ 10 (2022), 97531–97554.
  * Rojat et al. (2021) Thomas Rojat, Raphaël Puget, David Filliat, Javier Del Ser, Rodolphe Gelin, and Natalia Díaz-Rodríguez. 2021.  Explainable artificial intelligence (xai) on timeseries data: A survey.  _arXiv preprint arXiv:2104.00950_ (2021).
  * Rudin (2019) Cynthia Rudin. 2019\.  Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead.  _Nature machine intelligence_ 1, 5 (2019), 206–215.
  * Rudin and Radin (2019) Cynthia Rudin and Joanna Radin. 2019.  Why are we using black box models in AI when we don’t need to? A lesson from an explainable AI competition.  _Harvard Data Science Review_ 1, 2 (2019), 10–1162.
  * Sahakyan et al. (2021) Maria Sahakyan, Zeyar Aung, and Talal Rahwan. 2021\.  Explainable artificial intelligence for tabular data: A survey.  _IEEE Access_ 9 (2021), 135392–135422.
  * Selvaraju et al. (2017) Ramprasaath R Selvaraju, Michael Cogswell, Abhishek Das, Ramakrishna Vedantam, Devi Parikh, and Dhruv Batra. 2017.  Grad-cam: Visual explanations from deep networks via gradient-based localization. In _Proceedings of the IEEE international conference on computer vision_. 618–626.
  * Serengil et al. (2022) Sefik Ilkin Serengil, Salih Imece, Ugur Gurkan Tosun, Ege Berk Buyukbas, and Bilge Koroglu. 2022\.  A Comparative Study of Machine Learning Approaches for Non Performing Loan Prediction with Explainability.  _International Journal of Machine Learning and Computing_ 12, 5 (2022).
  * Serrano and Smith (2019) Sofia Serrano and Noah A Smith. 2019.  Is attention interpretable?  _arXiv preprint arXiv:1906.03731_ (2019).
  * Shapley et al. (1953) Lloyd S Shapley et al. 1953\.  A value for n-person games.  (1953).
  * Shi et al. (2021) Si Shi, Jianjun Li, Guohui Li, Peng Pan, and Ke Liu. 2021.  XPM: An explainable deep reinforcement learning framework for portfolio management. In _Proceedings of the 30th ACM international conference on information & knowledge management_. 1661–1670.
  * Sokol and Flach (2019) Kacper Sokol and Peter A Flach. 2019.  Counterfactual Explanations of Machine Learning Predictions: Opportunities and Challenges for AI Safety.  _SafeAI@ AAAI_ (2019).
  * Srinivasan et al. (2019) Ramya Srinivasan, Ajay Chander, and Pouya Pezeshkpour. 2019\.  Generating user-friendly explanations for loan denials using GANs.  _arXiv preprint arXiv:1906.10244_ (2019).
  * Sudjianto and Zhang (2021) Agus Sudjianto and Aijun Zhang. 2021.  Designing Inherently Interpretable Machine Learning Models.  _arXiv preprint arXiv:2111.01743_ (2021).
  * Team (2022) CFI Team. 2022.  _Finance Overview: Personal,business and government_.  Retrieved February 7, 2023 from https://corporatefinanceinstitute.com/resources/wealth-management/finance-industry-overview/
  * Tomsett et al. (2018) Richard Tomsett, Dave Braines, Dan Harborne, Alun Preece, and Supriyo Chakraborty. 2018\.  Interpretable to whom? A role-based model for analyzing interpretable machine learning systems.  _arXiv preprint arXiv:1806.07552_ (2018).
  * Tran et al. (2022) Kim Long Tran, Hoang Anh Le, Thanh Hien Nguyen, and Duc Trung Nguyen. 2022.  Explainable Machine Learning for Financial Distress Prediction: Evidence from Vietnam.  _Data_ 7, 11 (2022), 160.
  * Turbé et al. (2023) Hugues Turbé, Mina Bjelogrlic, Christian Lovis, and Gianmarco Mengaldo. 2023.  Evaluation of post-hoc interpretability methods in time-series classification.  _Nature Machine Intelligence_ (2023).  https://doi.org/10.1038/s42256-023-00620-w
  * van den Berg and Kuiper (2020) Martin van den Berg and Ouren Kuiper. 2020.  XAI in the financial sector: a conceptual framework for explainable AI (XAI).  _https://www. hu. nl/-/media/hu/documenten/onderzoek/projecten/_ (2020).
  * Vaswani et al. (2017) Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. 2017.  Attention is all you need.  _Advances in neural information processing systems_ 30 (2017).
  * VINCENT (2018) JAMES VINCENT. 2018\.  _Google ‘fixed’ its racist algorithm by removing gorillas from its image-labeling tech_.  Retrieved February 8, 2023 from https://www.theverge.com/2018/1/12/16882408/google-racist-gorillas-photo-recognition-algorithm-ai
  * Vivek et al. (2022) Yelleti Vivek, Vadlamani Ravi, Abhay Anand Mane, and Laveti Ramesh Naidu. 2022.  Explainable Artificial Intelligence and Causal Inference based ATM Fraud Detection.  _arXiv preprint arXiv:2211.10595_ (2022).
  * Wand et al. (2022) Tobias Wand, Martin Heßler, and Oliver Kamps. 2022\.  Identifying Dominant Industrial Sectors in Market States of the S&P 500 Financial Data.  _arXiv preprint arXiv:2208.14106_ (2022).
  * Weitz et al. (2019) Katharina Weitz, Dominik Schiller, Ruben Schlagowski, Tobias Huber, and Elisabeth André. 2019\.  ” Do you trust me?” Increasing user-trust by integrating virtual agents in explainable AI interaction design. In _Proceedings of the 19th ACM international conference on intelligent virtual agents_. 7–9.
  * Weng et al. (2022) Futian Weng, Jianping Zhu, Cai Yang, Wang Gao, and Hongwei Zhang. 2022.  Analysis of financial pressure impacts on the health care industry with an explainable machine learning method: China versus the USA.  _Expert Systems with Applications_ 210 (2022), 118482.
  * Wexler et al. (2019) James Wexler, Mahima Pushkarna, Tolga Bolukbasi, Martin Wattenberg, Fernanda Viégas, and Jimbo Wilson. 2019.  The what-if tool: Interactive probing of machine learning models.  _IEEE transactions on visualization and computer graphics_ 26, 1 (2019), 56–65.
  * Xie et al. (2022) Binhui Xie, Longhui Yuan, Shuang Li, Chi Harold Liu, Xinjing Cheng, and Guoren Wang. 2022\.  Active learning for domain adaptation: An energy-based approach. In _Proceedings of the AAAI Conference on Artificial Intelligence_ , Vol. 36. 8708–8716.
  * Yan et al. (2019) Han Yan, Sheng Lin, et al. 2019\.  New Trend in FinTech: Research on Artificial Intelligence Model Interpretability in Financial Fields.  _Open Journal of Applied Sciences_ 9, 10 (2019), 761\.
  * Yang et al. (2020) Linyi Yang, Eoin Kenny, Tin Lok James Ng, Yi Yang, Barry Smyth, and Ruihai Dong. 2020\.  Generating Plausible Counterfactual Explanations for Deep Transformers in Financial Text Classification. In _Proceedings of the 28th International Conference on Computational Linguistics_. 6150–6160.
  * Yang et al. (2018) Linyi Yang, Zheng Zhang, Su Xiong, Lirui Wei, James Ng, Lina Xu, and Ruihai Dong. 2018.  Explainable text-driven neural network for stock prediction. In _2018 5th IEEE International Conference on Cloud Computing and Intelligence Systems (CCIS)_. IEEE, 441–445.
  * Yang et al. (2019) Qiang Yang, Yang Liu, Tianjian Chen, and Yongxin Tong. 2019\.  Federated machine learning: Concept and applications.  _ACM Transactions on Intelligent Systems and Technology (TIST)_ 10, 2 (2019), 1–19.
  * Yasodhara et al. (2021) Angeline Yasodhara, Azin Asgarian, Diego Huang, and Parinaz Sobhani. 2021.  On the trustworthiness of tree ensemble explainability methods. In _Machine Learning and Knowledge Extraction: 5th IFIP TC 5, TC 12, WG 8.4, WG 8.9, WG 12.9 International Cross-Domain Conference, CD-MAKE 2021, Virtual Event, August 17–20, 2021, Proceedings 5_. Springer, 293–308.
  * Yeong Zee Kin (2023) Tan Wen Rui Yeong Zee Kin, Lee Wan Sie. 2023\.  _How Singapore is developing trustworthy AI_.  Retrieved February 8, 2023 from https://www.weforum.org/agenda/2023/01/how-singapore-is-demonstrating-trustworthy-ai-davos2023/
  * Yuan and Zhang (2020) Jie Yuan and Zhu Zhang. 2020.  Connecting the dots: forecasting and explaining short-term market volatility. In _Proceedings of the First ACM International Conference on AI in Finance_. 1–8.
  * Zhang et al. (2022c) Chanyuan Abigail Zhang, Soohyun Cho, and Miklos Vasarhelyi. 2022c.  Explainable Artificial Intelligence (XAI) in auditing.  _International Journal of Accounting Information Systems_ 46 (2022), 100572\.
  * Zhang et al. (2020b) Ruoyun Zhang, Chao Yi, and Yixin Chen. 2020b.  Explainable machine learning for regime-based asset allocation. In _2020 IEEE International Conference on Big Data (Big Data)_. IEEE, 5480–5485.
  * Zhang et al. (2022a) Wei Zhang, Brian Barr, and John Paisley. 2022a.  An Interpretable Deep Classifier for Counterfactual Generation. In _Proceedings of the Third ACM International Conference on AI in Finance_. 36–43.
  * Zhang et al. (2022b) Wei Zhang, Brian Barr, and John Paisley. 2022b.  Understanding Counterfactual Generation using Maximum Mean Discrepancy. In _Proceedings of the Third ACM International Conference on AI in Finance_. 44–52.
  * Zhang et al. (2020a) Xiaohui Zhang, Qianzhou Du, and Zhongju Zhang. 2020a.  An explainable machine learning framework for fake financial news detection. In _2020 International Conference on Information Systems-Making Digital Inclusive: Blending the Local and the Global, ICIS 2020_. Association for Information Systems.
  * Zhang et al. (2022d) Zijiao Zhang, Chong Wu, Shiyou Qu, and Xiaofang Chen. 2022d.  An explainable artificial intelligence approach for financial distress prediction.  _Information Processing & Management_ 59, 4 (2022), 102988\.


# The Alon-Tarsi number of $K_{3,3}$-minor-free graphs

Leyou Xu111Email<EMAIL_ADDRESS>Bo Zhou222Email<EMAIL_ADDRESS>
School of Mathematical Sciences, South China Normal University
Guangzhou 510631, P.R. China

###### Abstract

The well known Wagner’s theorem states that a graph is a planar graph if and
only if it is $K_{5}$-minor-free and $K_{3,3}$-minor-free. Denote by $AT(G)$
the Alon-Tarsi number of a graph $G$. We show that for any $K_{3,3}$-minor-
free graph $G$, $AT(G)\leq 5$, there exists a matching $M$ and a forest $F$
such that $AT(G-M)\leq 4$ and $AT(G-E(F))\leq 3$, extending the result on the
Alon-Tarsi number of $K_{5}$-minor-free graphs due to Abe, Kim and Ozeki.

Keywords:  Alon-Tarsi number, $K_{3,3}$-minor-free graph, planar graph

AMS Classifications: 05C15

##  1 Introduction

We consider simple, finite and undirected graphs. For a graph $G$, denote by
$V(G)$ the vertex set and $E(G)$ the edge set of $G$. A graph $H$ is a minor
of a connected graph $G$ if we may obtain $H$ from $G$ by repeatedly deleting
vertices and edges and contracting edges. A graph $G$ is $H$-minor-free if $H$
is not a minor of $G$. For $\emptyset\neq V_{1}\subseteq V(G)$, $G[V_{1}]$
denotes the subgraph of $G$ induced by $V_{1}$. Denote by $K_{n}$ the complete
graph on $n$ vertices and $K_{s,t}$ the complete bipartite graph on $s$ and
$t$ vertices in its color classes, respectively. A clique $S$ of a graph $G$
is a nonempty subset of $V(G)$ such that $G[S]$ is complete.

Let $G$ be a graph and $D$ an orientation of $G$. For $v\in V(G)$, denote by
$d_{D}^{+}(v)$ ($d_{D}^{-}(v)$, respectively) the out-degree (in-degree,
respectively) of $v$ in $D$. The maximum out-degree of $D$ is denoted by
$\Delta^{+}(D)$. An Eulerian subdigraph (or circulation) $H$ of $D$ is a
spanning subdigraph of $D$ such that $d_{H}^{+}(v)=d_{H}^{-}(v)$ for each
vertex $v\in V(G)$. Denote by $EE(D)$ ($OE(D)$, respectively) the set of all
Eulerian subdigraphs of $D$ with even (odd, respectively) number of edges,
respectively. We say that $D$ is acyclic if $D$ does not contain any directed
cycle. Use the Combinatorial Nullstellensatz, Alon and Tarsi [2] obtained a
remarkable relationship between a special orientation of a graph and a certain
graph polynomial, which is known as Alon-Tarsi Theorem. The Alon-Tarsi number
$AT(G)$ of a graph $G$, introduced by Jensen and Toft [8], is the minimum
integer $k$ such that there exists an orientation $D$ of $G$ with
$|EE(D)|\neq|OE(D)|$ and $\Delta^{+}(D)<k$.

A coloring of a graph $G$ is a mapping $c:V(G)\rightarrow\mathbb{R}$. For
$d\geq 0$, a $d$-defective coloring of $G$ is a coloring such that each color
class induces a subgraph of maximum degree at most $d$. Particularly, a
$0$-defective coloring is said to be proper. A graph is $k$-colorable if there
exists a proper coloring $c$ with $|c(V(G))|\leq k$. The chromatic number
$\chi(G)$ of $G$ is the least $k$ such that $G$ is $k$-colorable.

A list assignment of $G$ is a function $L$ assigning to every vertex $v\in
V(G)$ a set $L(v)\subset\mathbb{R}$. Given a list assignment $L$, a
$d$-defective-$L$-coloring of $G$ is a $d$-defective coloring $c$ such that
$c(v)\in L(v)$ for each vertex $v\in V(G)$. A graph $G$ is $d$-defective
$k$-choosable if there exists an $d$-defective-$L$-coloring of $G$ for each
assignment $L$ with $|L(v)|\geq k$ for every $v\in V(G)$. Particularly, we say
$G$ is $k$-choosable if $G$ is $0$-defective $k$-choosable. The list chromatic
number $\chi_{\ell}(G)$ of $G$ is the least integer $k$ such that $G$ is
$k$-choosable.

By Alon-Tarsi Theorem [2], $\chi(G)\leq\chi_{\ell}(G)\leq AT(G)$ for any $G$.

The famous Four-Color Theorem states that for any planar graph $G$,
$\chi(G)\leq 4$. Vizing [13] asked whether every planar graph is
$5$-choosable. Erdős, Rubin and Taylor [5] conjectured that every planar graph
is $5$-choosable but not necessarily $4$-choosable. Thomassen [11] confirmed
the former and Voigt [12] confirmed the latter. Cushing and Kierstead [3]
proved that every planar graph is $1$-defective $4$-choosable. Let $G$ be a
planar graph. Zhu [16] improved Thomassen’s upper bound by proving that
$AT(G)\leq 5$, solving a natural open problem [7]. Grytczuk and Zhu [6] showed
that there exists a matching $M$ of $G$ such that $AT(G-M)\leq 4$.
Furthermore, Kim, Kim and Zhu [9] showed that there exists a forest $F$ in $G$
such that $AT(G-E(F))\leq 3$.

By the well known Wagner’s theorem [14], a graph is a planar graph if and only
if it is $K_{5}$-minor-free and $K_{3,3}$-minor-free. Recently, Abe, Kim and
Ozeki [1] extended the above results on the Alon-Tarsi number from planar
graphs to $K_{5}$-minor-free graphs.

######  Theorem 1.

[1, Theorem 1.6, Corollary 1.8] Let $G$ be a $K_{5}$-minor-free graph. The
following statements are true.

  1. (i)

$AT(G)\leq 5$.

  2. (ii)

There exists a matching $M$ of $G$ such that $AT(G-M)\leq 4$.

  3. (iii)

There exists a forest $F$ in $G$ such that $AT(G-E(F))\leq 3$.

In this note, we extend these results in another direction to $K_{3,3}$-minor-
free graphs.

######  Theorem 2.

Let $G$ be a $K_{3,3}$-minor-free graph. Then the following results hold.

  1. (i)

$AT(G)\leq 5$.

  2. (ii)

There exists a matching $M$ of $G$ such that $AT(G-M)\leq 4$.

  3. (iii)

There exists a forest $F$ of $G$ such that $AT(G-E(F))\leq 3$.

As mentioned above, there exist planar graphs which are not $4$-choosable
[12], which are surely $K_{3,3}$-minor-free, so the bound in Theorem 2 (i) is
tight. Theorem 2 (ii) implies that a $K_{3,3}$-minor-free graph is
$1$-defective $4$-choosable.

######  Corollary 1.

[15] Let $G$ be a $K_{3,3}$-minor-free graph. Then $\chi_{\ell}(G)\leq 5$.

The study of list coloring of $K_{s,t}$-minor-free graphs has received much
attention [15, 10]. Steiner [10] proved that for every pair of constants
$\epsilon>0$ and $C>1$, there exists a positive integer $N=N(\epsilon,C)$ such
that for all integers $s$ and $t$ satisfying $N\leq s\leq t\leq Cs$, there
exists a $K_{s,t}$-minor-free graph $G$ such that
$\chi_{\ell}(G)>(1-\epsilon)(2s+t)$. For any such graph $G$,
$AT(G)>(1-\epsilon)(2s+t)$. So, in general, it is impossible that $AT(G)\leq
s+t-1$ for a $K_{s,t}$-minor-free graph $G$.

##  2 Proof of Theorem 2

We need the following important result due to Zhu and his coauthors [16, 6,
9]. In a plane graph every face is bounded by a closed walk (not necessarily a
cycle), which is called a boundary walk. An orientation $D$ of a graph $G$ is
said to be an AT-orientation if $|EE(D)|\neq|OE(D)|$.

######  Lemma 1.

Let $G$ be a nontrivial plane graph with a boundary walk $v_{1}\dots v_{m}$ of
the infinite face. The following statements are true.

  1. (a)

[16] There exists an AT-orientation $D$ of $G$ such that $d_{D}^{+}(v_{1})=0$,
$d_{D}^{+}(v_{2})=1$, $d_{D}^{+}(v_{i})\leq 2$ for each $i=3,\dots,m$ and
$\Delta^{+}(D)\leq 4$.

  2. (b)

[6] There exists a matching $M$ of $G$ and an AT-orientation $D$ of $G-M$ such
that $d_{D}^{+}(v_{1})=d_{D}^{+}(v_{2})=0$, $d_{D}^{+}(v_{i})\leq
2-d_{M}(v_{i})$ for each $i=3,\dots,m$ and $\Delta^{+}(D)\leq 3$.

  3. (c)

[9] There exists a forest $F$ of $G$ and an acyclic orientation $D$ of
$G-E(F)$ such that $d_{D}^{+}(v_{1})=d_{D}^{+}(v_{2})=0$, $d_{D}^{+}(v_{i})=1$
for each $i=3,\dots,m$ and $\Delta^{+}(D)\leq 2$.

Let $G_{1}$ and $G_{2}$ be two vertex disjoint graphs. Suppose that
$X_{i}\subset V(G_{i})$ is a clique of $G_{i}$ for $i=1,2$ with
$|X_{1}|=|X_{2}|=k$. Let $f:X_{1}\rightarrow X_{2}$ be a bijection. A graph
$G$ obtained from $G_{1}$ and $G_{2}$ by identifying $x$ and $f(x)$ for every
$x\in X_{1}$ and possibly deleting some edges of the clique is called a
$k$-clique-sum of $G_{1}$ and $G_{2}$. Evidently, a $0$-sum of $G_{1}$ and
$G_{2}$ is $G_{1}\cup G_{2}$.

######  Lemma 2.

[14] A graph $G$ is $K_{3,3}$-minor-free if and only if $G$ is a planar graph
or $K_{5}$, or $G$ can be obtained from planar graphs and $K_{5}$ by $0$-,
$1$-, and $2$-sums.

######  Lemma 3.

Let $G$ be a graph obtained by the $k$-clique-sum of $G_{1}$ and $G_{2}$,
where $k\geq 1$. Let $K$ be their common clique with
$K=\\{v_{1},\dots,v_{k}\\}$. Let $G_{i}^{\prime}=G_{i}\cap G$ for $i=1,2$.
Suppose that $G_{1}^{\prime}$ has an AT-orientation $D_{1}^{\prime}$ with
$\Delta^{+}(D_{1}^{\prime})\leq\ell$ and that $G_{2}$ has an AT-orientation
$D_{2}$ with $\Delta^{+}(D_{2})\leq\ell$ and $d^{+}_{D_{2}}(v_{i})=i-1$ for
$i=1,\dots,k$. Then $G$ has an AT-orientation $D$ such that
$\Delta^{+}(D)\leq\ell$ and $d_{D}^{+}(v)=d_{D_{1}^{\prime}}^{+}(v)$ for each
$v\in V(G_{1})$.

###### Proof.

Let $D_{2}^{\prime}$ be restriction of $D_{2}$ on $G_{2}-E(G[K])$. Then
$D=D_{1}^{\prime}\cup D_{2}^{\prime}$ is an orientation of $G$. As
$d_{D_{2}}^{+}(v_{i})=i-1$ for $i=1,\dots,k$ and $D_{2}[K]$ is a tournament,
$D_{2}[K]$ is transitive and there is no arc from $K$ to $V(G_{2})\setminus K$
in $D_{2}$. Then $d_{D_{2}^{\prime}}^{+}(u)=0$ for $u\in K$. So
$d_{D}^{+}(v)=d_{D_{1}^{\prime}}^{+}(v)$ for each $v\in V(G_{1})$. Note that
$d_{D}^{+}(v)=d_{D_{2}^{\prime}}^{+}(v)=d_{D_{2}}^{+}(v)$ for each $v\in
V(G_{2})\setminus K$. Thus
$\Delta^{+}(D)=\max\\{\Delta^{+}(D_{1}^{\prime}),\Delta^{+}(D_{2}^{\prime})\\}\leq\ell$.
We are left to show that $D$ is an AT-orientation of $G$.

Let $H$ be an Eulerian subdigraph of $D$ and $H_{1}=H[V(G_{1})]$.

Claim 1. $H$ contains no arcs from $V(G_{2})\setminus K$ to $K$.

###### Proof.

If there exists some vertex $v\in K$ and $w\in V(G_{2})\setminus K$ such that
$(w,v)$ is an arc of $H$, then $d_{H_{1}}^{-}(v)<d_{H}^{-}(v)$. As there is no
arcs from $K$ to $V(G_{2})\setminus K$ in $D_{2}$, we have
$d_{H_{1}}^{+}(u)=d_{H}^{+}(u)$ for each $u\in K$. Thus
$d_{H_{1}}^{+}(u)=d_{H}^{+}(u)$ for each $u\in V(G_{1})$, so

$\sum_{u\in V(H_{1})}d_{H}^{+}(u)=\sum_{u\in
V(H_{1})}d_{H_{1}}^{+}(u)=|A(H_{1})|=\sum_{u\in
V(H_{1})}d_{H_{1}}^{-}(u)<\sum_{u\in V(H_{1})}d_{H}^{-}(u),$

a contradiction. ∎

By Claim 1, $H$ contains no arcs either from $K$ to $V(G_{2})\setminus K$ or
from $V(G_{2})\setminus K$ to $K$. Therefore, $H$ has an edge-disjoint
decomposition $H=H_{1}\cup H_{2}$, where $H_{1}$ and $H_{2}$ are Eulerian
subdigraphs of $D_{1}^{\prime}$ and $D_{2}^{\prime}$, respectively.

If $H\in EE(D)$, then either $H_{1}\in EE(D_{1}^{\prime})$ and $H_{2}\in
EE(D_{2}^{\prime})$ or $H_{1}\in OE(D_{1}^{\prime})$ and $H_{2}\in
OE(D_{2}^{\prime})$. On the other hand, $H_{1}^{\prime}\cup H_{2}^{\prime}\in
EE(D)$ for any $H_{i}^{\prime}\in EE(D_{i}^{\prime})$ with $i=1,2$ or
$H_{i}^{\prime}\in OE(D_{1}^{\prime})$ with $i=1,2$. Thus, there is a
bijection between $EE(D)$ and $(EE(D_{1}^{\prime})\times
EE(D_{2}^{\prime}))\cup(OE(D_{1}^{\prime})\times OE(D_{2}^{\prime}))$.
Similarly, there is a bijection between $OE(D)$ and $(EE(D_{1}^{\prime})\times
OE(D_{2}^{\prime}))\cup(OE(D_{1}^{\prime})\times EE(D_{2}^{\prime}))$. Note
that any Eulerian subdigraph of $D_{2}$ contains no arc incident to vertices
in $K$ as $d_{D_{2}}^{+}(v_{i})=i-1$ for $i=1,\dots,k$. Thus
$EE(D_{2}^{\prime})=EE(D_{2})$ and $OE(D_{2}^{\prime})=OE(D_{2})$. Recall that
$D_{1}^{\prime}$ and $D_{2}$ are AT-orientation. Thus
$|EE(D_{1}^{\prime})|-|OE(D_{1}^{\prime})|\neq 0$ and
$|EE(D_{2}^{\prime})|-|OE(D_{2}^{\prime})|\neq 0$. It hence follows that

$\displaystyle|EE(D)|-|OE(D)|$
$\displaystyle=\left(|EE(D_{1}^{\prime})|\times|EE(D_{2}^{\prime})|+|OE(D_{1}^{\prime})|\times|OE(D_{2}^{\prime})|\right)$
$\displaystyle\quad-\left(|EE(D_{1}^{\prime})|\times|OE(D_{2}^{\prime})|+|OE(D_{1}^{\prime})|\times|EE(D_{2}^{\prime})|\right)$
$\displaystyle=\left(|EE(D_{1}^{\prime})|-|OE(D_{1}^{\prime})|\right)\left(|EE(D_{2}^{\prime})|-|OE(D_{2}^{\prime})|\right)$
$\displaystyle\neq 0,$

which implies that $D$ is an AT-orientation of $G$. ∎

Now we are ready to prove Theorem 2.

###### Proof of Theorem 2.

It suffices to show that for each $uv\in E(G)$, we have

  1. (a)

There exists an AT-orientation $D$ of $G$ such that $\Delta^{+}(D)\leq 4$,
$d_{D}^{+}(u)=0$ and $d_{D}^{+}(v)=1$.

  2. (b)

There exists a matching $M$ of $G$ and an AT-orientation $D$ of $G-M$ such
that $\Delta^{+}(D)\leq 3$ and $d_{D}^{+}(u)=d_{D}^{+}(v)=0$.

  3. (c)

There exists a forest $F$ of $G$ and an acyclic orientation $D$ of $G-E(F)$
such that $\Delta^{+}(D)\leq 2$ and $d_{D}^{+}(u)=d_{D}^{+}(v)=0$.

If $G$ is a planar graph, then we may assume that $G$ is an plane graph so
that edge $uv$ lies on the boundary of the infinite face, so Item (a) ((b),
(c), respectively) follows from (a) ((b), (c), respectively) of Lemma 1.

Suppose that $G\cong K_{5}$. Let $V(G)=\\{v_{1},\dots,v_{5}\\}$ with
$u=v_{1},v=v_{2}$.

Let $D$ be the orientation of $G$ such that $(v_{i},v_{j})$ is an arc of $D$
if and only if $i>j$. It is obvious that $d_{D}^{+}(v_{1})=0$,
$d_{D}^{+}(v_{2})=1$ and $\Delta^{+}(D)=4$. As $D$ is an acyclic orientation,
it is an AT-orientation. So (a) follows.

Let $M=\\{v_{1}v_{2},v_{4}v_{5}\\}$ and let $D$ be the orientation of $G-M$
such that $(v_{i},v_{j})$ is an arc of $D$ if and only if $i>j$. Then
$d_{D}^{+}(v_{1})=d_{D}^{+}(v_{2})=0$ and $\Delta^{+}(D)=3$. As $D$ is an
acyclic orientation, it is an AT-orientation. So (b) follows.

Let $F$ be a forest with
$E(F)=\\{v_{1}v_{2},v_{3}v_{5},v_{2}v_{4},v_{4}v_{5}\\}$ and let $D$ be the
orientation of $G-E(F)$ such that $(v_{i},v_{j})$ is an arc of $D$ if and only
if $i>j$. Then $d_{D}^{+}(v_{1})=d_{D}^{+}(v_{2})=0$ and $\Delta^{+}(D)=2$. It
is easy to see that $D$ is acyclic, so (c) follows.

Suppose that $G$ is not planar and $G\ncong K_{5}$. Suppose by contradiction
that $G$ is a minimum counterexample with respect to the order. Then $G$ is
connected by the minimality of $G$. As $G$ is $K_{3,3}$-minor-free, we have by
Lemma 2 that there exists two $K_{3,3}$-minor-free graphs $G_{1}$ and $G_{2}$
such that $G$ is a $k$-clique-sum of $G_{1}$ and $G_{2}$ with $k=1,2$. Let $K$
be the common clique of $G_{1}$ and $G_{2}$ with $K=\\{x\\}$ if $k=1$ and
$K=\\{x,y\\}$ if $k=2$. Let $G_{i}^{\prime}=G\cap G_{i}$ for $i=1,2$. Let
$uv\in E(G_{1}^{\prime})$. By the minimality of $G$, we have

  1. (a1)

There exists an AT-orientation $D_{1}$ of $G_{1}^{\prime}$ such that
$\Delta^{+}(D_{1})\leq 4$, $d_{D_{1}}^{+}(u)=0$ and $d_{D_{1}}^{+}(v)=1$.

  2. (b1)

There exists a matching $M_{1}$ of $G_{1}^{\prime}$ and an AT-orientation
$D_{1}$ of $G_{1}^{\prime}-M_{1}$ such that $\Delta^{+}(D_{1})\leq 3$ and
$d_{D_{1}}^{+}(u)=d_{D_{1}}^{+}(v)=0$.

  3. (c1)

There exists a forest $F_{1}$ of $G_{1}^{\prime}$ and an acyclic orientation
$D_{1}$ of $G_{1}^{\prime}-E(F_{1})$ such that $\Delta^{+}(D_{1})\leq 2$ and
$d_{D_{1}}^{+}(u)=d_{D_{1}}^{+}(v)=0$.

Firstly, we show (a). By the minimality of $G$, there exists an AT-orientation
$D_{2}$ of $G_{2}$ such that $\Delta^{+}(D_{2})\leq 4$, $d_{D_{2}}^{+}(x)=0$
and if $k=2$, then $d_{D_{2}}^{+}(y)=1$. So, by (a1) and Lemma 3, $G$ has an
AT-orientation $D$ such that $\Delta^{+}(D)\leq 4$, $d_{D}^{+}(u)=0$, and
$d_{D}^{+}(v)=1$, contradicting the choice of $G$.

Secondly, we show (b). Denote by $y$ a neighbor of $x$ in $G_{2}$ if $k=1$. By
the minimality of $G$, there exists a matching $M_{2}$ of $G_{2}$ and an AT-
orientation $D_{2}$ of $G_{2}-M_{2}$ such that $\Delta^{+}(D_{2})\leq 3$,
$d_{D_{2}}^{+}(x)=0$ and $d_{D_{2}}^{+}(y)=0$. Then $xy\in M_{2}$ and neither
$x$ nor $y$ is $M_{2}\setminus\\{xy\\}$-saturated. Let $D_{2}^{\prime}$ be an
orientation of $G_{2}-(M_{2}\setminus\\{xy\\})$ with
$D_{2}^{\prime}=D_{2}\cup\\{(y,x)\\}$. Then $d_{D_{2}^{\prime}}^{+}(y)=1$. So
$D_{2}^{\prime}$ is an AT-orientation with $\Delta^{+}(D_{2}^{\prime})\leq 3$,
$d_{D_{2}^{\prime}}^{+}(x)=0$ and $d_{D_{2}^{\prime}}^{+}(y)=1$. Let
$M=M_{1}\cup(M_{2}\setminus\\{xy\\})$. Then $M$ is a matching of $G$. By (b1)
and Lemma 3, $G-M$ has an AT-orientation $D$ such that $\Delta^{+}(D)\leq 3$
and $d_{D}^{+}(u)=d_{D}^{+}(v)=0$, contradicting the choice of $G$.

Now we show (c). Denote by $y$ a neighbor of $x$ in $G_{2}$ if $k=1$. By the
minimality of $G$, there exists a forest $F_{2}$ of $G_{2}$ and an acyclic
orientation of $G_{2}-E(F_{2})$ such that $\Delta^{+}(D_{2})\leq 2$,
$d_{D_{2}}^{+}(x)=0$ and $d_{D_{2}}^{+}(y)=0$. Then $xy\in E(F_{2})$. Let
$D_{2}^{\prime}$ be an orientation of $G_{2}-(E(F_{2})\setminus\\{xy\\})$ with
$D_{2}^{\prime}=D_{2}\cup\\{(y,x)\\}$. Then $d_{D_{2}^{\prime}}^{+}(y)=1$. So
$D_{2}^{\prime}$ is an acyclic orientation with
$\Delta^{+}(D_{2}^{\prime})\leq 2$, $d_{D_{2}^{\prime}}^{+}(x)=0$ and
$d_{D_{2}^{\prime}}^{+}(y)=1$. Let $F=F_{1}\cup(F_{2}-xy)$. Then $F$ is a
forest of $G$. By (c1) and Lemma 3, $G-E(F)$ has an acyclic orientation $D$
such that $\Delta^{+}(D)\leq 2$ and $d_{D}^{+}(u)=d_{D}^{+}(v)=0$,
contradicting the choice of $G$. ∎

A graph $G$ is $k$-degenerate if each subgraph of $G$ contains a vertex of
degree at most $k$, equivalently, $G$ has an acyclic orientation $D$ with
$\Delta^{+}(D)\leq k$.

######  Corollary 2.

Let $G$ be a $K_{3,3}$-minor-free graph. Then there exists a forest $F$ such
that $G-E(F)$ is $2$-degenerate.

###### Proof.

By the proof of Theorem 2, there exists a forest $F$ and an acyclic AT-
orientation $D$ such that $\Delta^{+}(D)\leq 2$. We may label the vertices in
$G$ such that $(v_{i},v_{j})$ is an arc of $D$ if and only if $i>j$. Let
$H\subseteq G-E(F)$ and $H^{\prime}$ be the orientation of $H$ on $D$. Let
$t=\max\\{v_{i}:v_{i}\in V(H)\\}$. Then $d_{H^{\prime}}^{+}(v_{t})\leq 2$ and
$d_{H^{\prime}}^{-}(v_{t})=0$ and so the degree of $v_{t}$ is at most $2$ in
$H$. Therefore, the result follows. ∎

Remark 1. Theorem 1 may be extended to $K_{5}^{\bot}$-minor-free graphs, where
$K_{5}^{\bot}$ is the graph obtained from $K_{4}$ by adding two adjacent
vertices of degree three with no common neighbors. From [4, Lemmas 2.4 and
4.1], a graph $G$ is $K_{5}^{\bot}$-minor free if and only if $G$ a
$K_{5}$-minor-free graph or $K_{5}$, or $G$ can be obtained from
$K_{5}$-minor-free graphs and $K_{5}$ by $0$-, $1$-, and $2$-sums. The result
follows from a similar argument as in the proof in Theorem 2 (that $K_{5}$
satisfies (a), (b) and (c) there) and [1, Lemma 3.1].

Remark 2. Theorem 2 may be extended to $(K_{3,3}+e)$-minor-free graphs. By [4,
Lemmas 2.4 and 3.7] a graph $G$ is $(K_{3,3}+e)$-minor free graph if and only
if $G$ is a $K_{3,3}$-minor-graph or $K_{3,3}$, or $G$ can be obtained from
$K_{3,3}$-minor-free graphs and $K_{3,3}$ by $0$-, $1$\- and $2$-sums. So, by
similar argument as in the proof of Theorem 2, we only need to check the
validity of (a)–(c) there if $G=K_{3,3}$. This is verified as follows: Let
$\\{u_{1},u_{2},u_{3}\\}\cup\\{v_{1},v_{2},v_{3}\\}$ be the bipartition of $G$
with $u=u_{1}$ and $v=v_{1}$.

(a) Let $D$ be the orientation of $G$ with arc set
$\\{(v_{i},u_{j}):i=2,3,j=1,2,3\\}\cup\\{(v_{1},u_{1}),(u_{2},v_{1}),(u_{3},v_{1})\\}$.
Then $d_{D}^{+}(u_{1})=0$, $d_{D}^{+}(v_{1})=1$ and $\Delta^{+}(D)=3$. As $D$
is an acyclic orientation, it is an AT-orientation.

(b) Let $M=\\{v_{i}u_{i}:i=1,2,3\\}$. Let $D$ be the orientation of $G-M$ with
arc set
$\\{(v_{2},u_{1}),(v_{3},u_{1}),(u_{2},v_{1}),(u_{2},v_{3}),(u_{3},v_{1}),(u_{3},v_{2})\\}$.
Then $d_{D}^{+}(u_{1})=d_{D}^{+}(v_{1})=0$ and $\Delta^{+}(D)=2$. As $D$ is an
acyclic orientation, it is an AT-orientation.

(c) Let $F$ be a forest with
$E(F)=\\{u_{1}v_{1},u_{1}v_{2},u_{2}v_{2},u_{2}v_{3},u_{3}v_{3}\\}$ and let
$D$ be the orientation of $G-E(F)$ with arc set
$\\{(v_{3},u_{1}),(u_{2},v_{1}),(u_{3},v_{1}),(u_{3},v_{2})\\}$. Then
$d_{D}^{+}(u_{1})=d_{D}^{+}(v_{1})=0$ and $\Delta^{+}(D)=2$. As $D$ is an
acyclic orientation, it is an AT-orientation.

##  3 Future research direction

As mentioned above, it is impossible that $AT(G)\leq s+t-1$ for a
$K_{s,t}$-minor-free graph $G$ in general. It is of interest to find a tight
upper bound of the Alon-Tarsi number for any $K_{s,t}$-minor-free graph.
Steiner [10] proposed an open problem: Is it true that for all integers $1\leq
s\leq t$, every $K_{s,t}$-minor-free graph $G$ satisfies $\chi_{\ell}(G)\leq
2s+t$? Similarly, one may ask whether it is true that for all integers $1\leq
s\leq t$, every $K_{s,t}$-minor-free graph $G$ satisfies $AT(G)\leq 2s+t$.

Acknowledgement. This work was supported by the National Natural Science
Foundation of China (No. 12071158).

## References

  * [1] T. Abe, S.-J. Kim, K. Ozeki, The Alon-Tarsi number of $K_{5}$-minor-free graphs, Discrete Math. 345 (2022) 112764.
  * [2] N. Alon, M. Tarsi, Colorings and orientations of graphs, Combinatorica 12 (1992) 125–134.
  * [3] W. Cushing, H. A. Kirestread, Planar graphs are $1$-relaxed, $4$-choosable, Eur. J. Combin. 31 (2010) 1395–1397.
  * [4] G. Ding, C. Liu, Excluding a small minor, Discrete Appl. Math. 161 (2013) 355–368.
  * [5] P. Erdős, A.L. Rubin, H. Taylor, Choosability in graphs, in: Proceedings of the West Coast Conference on Combinatorics, Graph Theory and Computing, Arcata, CA, 1979; Congress. Numer. 26 (1980) 125–157.
  * [6] J. Grytczuk, X. Zhu, The Alon-Tarsi number of a planar graph minus a matching, J. Combin. Theory Ser. B 145 (2020) 511–520.
  * [7] D. Hefetz, On two generalizations of the Alon-Tarsi polynomial method, J. Combin. Theory Ser. B 101 (6) (2011) 403–414.
  * [8] T. Jensen, B. Toft, Graph Coloring Problems, Wiley, New York, 1995.
  * [9] R. Kim, S. Kim, X. Zhu, The Alon-Tarsi number of subgraphs of a planar graph, arXiv:1906.01506.
  * [10] R. Steiner, Disproof of a conjecture by Woodall on the choosability of $K_{s,t}$-minor-free graphs, Electron. J. Combin. 29 (3) (2022) Paper 3.2, 11 pp.
  * [11] C. Thomassen, Every planar graph is $5$-choosable, J. Combin. Theory Ser. B 62 (1994) 180–181.
  * [12] M. Viogt, List colourings of planar graphs, Discrete Math. 120 (1993) 215–219.
  * [13] V.G. Vizing, Coloring the vertices of a graph in prescribed colors (in Russian), Diskretn. Anal. 29 (1976) 3–10, 101.
  * [14] K. Wagner, Uber eine Eigenschaft der ebenen Komplexe, Math. Ann. 114 (1937) 570–590.
  * [15] D.R. Woodall, List colourings of graphs, in: J.W.P. Hirschfeld (Ed.), Surveys in Combinatorics, 2001, London Math. Soc. Lecture Note Ser., 288, Cambridge Univ. Press, Cambridge, 2001, pp. 269–301.
  * [16] X. Zhu, The Alon-Tarsi number of planar graphs, J. Combin. Theory Ser. B 134 (2019) 354–358.


# Object Recognition in Atmospheric Turbulence Scenes
††thanks: This work was supported by the UKRI MyWorld Strength in Places
Programme (SIPF00006/1).

Disen Hu  Visual Information Laboratory
University of Bristol
Bristol, UK
<EMAIL_ADDRESS>Nantheera Anantrasirichai  Visual Information
Laboratory
University of Bristol
Bristol, UK
<EMAIL_ADDRESS>
###### Abstract

The influence of atmospheric turbulence on acquired surveillance imagery poses
significant challenges in image interpretation and scene analysis.
Conventional approaches for target classification and tracking are less
effective under such conditions. While deep-learning-based object detection
methods have shown great success in normal conditions, they cannot be directly
applied to atmospheric turbulence sequences. In this paper, we propose a novel
framework that learns distorted features to detect and classify object types
in turbulent environments. Specifically, we utilise deformable convolutions to
handle spatial turbulent displacement. Features are extracted using a feature
pyramid network, and Faster R-CNN is employed as the object detector.
Experimental results on a synthetic VOC dataset demonstrate that the proposed
framework outperforms the benchmark with a mean Average Precision (mAP) score
exceeding 30%. Additionally, subjective results on real data show significant
improvement in performance.

###### Index Terms:

atmospheric turbulence, object detection, deep learning, deformable
convolution, object recognition

##  I Introduction

Atmospheric turbulence consistently degrades visual quality and has a negative
impact on the performance of automated target recognition and tracking in a
scene. These distortions occur when there is a temperature difference between
the ground and the air, causing rapid upward movement of air layers and
resulting in changes to the interference pattern of light refraction. This
leads to visible ripples and waves in both spatial and temporal directions in
images and videos. Mitigating this effect is an ill-posed problem with non-
stationary distortions varying across time and space, and a degree of
distortions is unknown. The restoration process is thus complex and time-
consuming. Objects behind the distorting layers are almost impossible to
recognise by machines leading to a failure of automatic detection and tracking
processes. Examples of applications directly affected with atmospheric
turbulence are video surveillance, security and defence.

Object detection methods on natural and clean images have been greatly
developed achieving high performance in term of both detection accuracy and
computational speed. The state of the arts are based on deep learning with
convolutional neural networks (CNN) (see recent techniques for object
detection in [1]). However, the performance of these methods declines when the
features are corrupted by noise or distorted by blur. This degradation is even
more pronounced in the case of atmospheric turbulence, where distortions
appear randomly and are spatially and temporally invariant [2]. Up to now,
only face recognition in atmospheric turbulence has been developed [3]. They
have also proved that training deblurring and detection models together gives
better results than separating the models.

In this paper, we tackle the multi-class object detection in atmospheric
turbulence scenes without image restoration process. Our framework is hence
fast and straightforward. The method is developed based on the Faster R-CNN
detector [4], but this should not be limited to. The features are extracted
with a feature pyramid network (FPN) [5], which can deal with different
resolutions, different sizes of the objects, and different amounts of
distortions. To mitigate the effects of atmospheric turbulence, we incorporate
deformable convolutions [6], which help reduce the impact of visible ripples
along object edges caused by atmospheric turbulence. A key contribution is
that, as in the atmospheric turbulent environments, the objects exhibit visual
distortions within small ranges of pixel displacement appearing randomly at
all directions. The use of deformable convolutions provides flexibility in
capturing the shapes of the objects and assists the FPN in extracting the
appropriate features from the distorted objects. As there is no ground truth
available for this problem, we trained the model using a synthetic dataset and
evaluated its performance using both synthetic and real datasets. Our code is
available at https://github.com/disen-hu/FasterRcnn_FPN_DCN.

##  II Related work

Image restoration techniques for atmospheric turbulence have been extensively
studied [7, 8, 9], with some methods specifically addressing moving objects in
distorted scenes [2, 10, 11, 12]. Deep learning technologies have also gained
attention in atmospheric turbulence mitigation, although they are still in the
early stages of development. Existing architectures have been employed and
retrained using synthetic datasets in various methods [13, 14, 15].
Additionally, the use of Complex-Valued CNN was explored in [16],
demonstrating significant improvements over traditional CNN-based approaches.

For object detection in atmospheric turbulence scenes, most traditional
methods were proposed to detect long-distance target objects. The main
limitation of these is that those objects are at sufficiently low fidelity to
exhibit little or no detail, instead appearing as blurred silhouettes or blobs
[2, 17, 18, 19, 20]. To the best of our knowledge, the deep learning-based
object detection has been applied for only face recognition [3]. This method
is based on Generative Adversarial Networks (GAN). However, the features on
human faces are very clear and distinguish from surroundings. This method is
therefore not suitable for complicated scenes with many types of objects. For
general object recognition, there is not any deep learning methods
specifically proposed for that in atmospheric turbulent environment. Authors
in [21] has tested three state-of-the-art object detection methods, retrained
with the synthetic thermal imagery. They reported that among VfNet [22], YOLOR
[23], and TOOD [24], YOLOR gave the best performance in both mean average
precision and speed.

##  III Methodology

###  III-A Dealing with atmospheric turbulent distortion

A turbulent medium causes phase fluctuations, which exhibit in the image as a
phase shift and its amount depends approximately linearly on spatial
displacement [25]. Following quasi-periodic property, the phase of each pixel
is altered randomly, whilst the magnitude of high frequency is generally
decreased due to mixing of the signals leading to a blur. This causes images
look like ripples across time – the pixels spatially shift from their actual
locations in random directions. In term of image degradation model, the
distorted image $I_{\text{atmos}}$ is described as
$I_{\text{atmos}}=h*I_{\text{ideal}}+n$, where $n$ is noise, and $h$ is an
unknown spatially variant point spread function, comprising geometric
distortion and blur.

The clean or ideal image $I_{\text{ideal}}$ is altered by $h$ more than $n$
significantly. That is, there are two aspects of atmospheric turbulence to
concern: i) ripple effect (geometric distortion and blur) and ii) unknown
amount of this effect locally. In this paper, We exploit deformable
convolutions to deal with the ripple effect. Mathematically, the deformable
convolution can be expressed as Eq. 1. Following [6], the output feature map
$y$ at pixel $p$ is the result of convolution between a learnable weight
$w(p_{n})$, where $p_{n}$ are elements of regular grid $R$. The irregular
positions of an input feature map $x$ are a combination of the regular grid
$R$ and the offsets $\Delta p_{n}$.

$y(p)=\sum_{p_{n}\in R}w(p_{n})\cdot x(p+p_{n}+\Delta p_{n})$ (1)

This can be interpreted that the deformable convolutions allow the pixels
associated to the current kernel to spatially locate outside the regular grid
search. So the same object in the different images, affected by different
turbulent distortions, could then have the same features, as shown in Fig. 1.
This fully benefits in object recognition as gaining more accurate features.

To deal with unknown amount of distortions, we exploit multiple scales of
features so that they cover all possible ranges of displacement duo to pixels
in $I_{\text{atmos}}$ shifted from those in $I_{\text{ideal}}$.

###  III-B Network architectures

Figure 1: Visualisation of the effect of 3$\times$3 deformable convolutions on
turbulence distortion. The red dots correspond to the same feature.  Figure 2:
A diagram of the proposed object detection in atmospheric turbulence scenes.

The proposed framework adapted from Faster R-CNN [4] is shown in Fig. 2,
comprising three modules: i) feature extraction, ii) region proposal network
(RPN) and iii) classification network.

For the feature extraction, we replace the single-scale feature extractor
originally used in the Faster R-CNN with a feature pyramid network (FPN) [5].
FPN offers multi-scale features which benefit i) different sizes of targeting
objects, ii) deal with different amount of atmospheric turbulence, and iii)
improve classification accuracy. We initial the FPN with pretrained model on
COCO segmentation dataset [26]. We tested two backbone architectures, VGG16
[27] and ResNet50 [28], and found the ResNet50-based model outperforms the
VGG16-based model more than double in term of mAP. Therefore, we employ
ResNet50. We then replace regular grid convolutions with deformable
convolutions following the reasons stated in Section III-A.

RPN generates bounding box proposals in different sizes using shallow
architecture (3$\times$3 Conv + ReLU + 1$\times$1 Conv). The region proposals
are sent to the RoI Pooling layer together with feature maps for
classification. We employ pretrained ResNet50 model from the official PyTorch,
which has achieved 81% accuracy on VOC datasets. Using a pretrained model can
reduce training duration and achieve better results. Similar to the feature
extractor, we apply deformable convolutions in RPN and in the ResNet blocks.
At the end, the feature vectors are passed to two parallel fully connected
layers. One branch uses Softmax function to calculate the probability of being
each category, and the other branch outputs the coordinate offset of the
proposal box and uses the frame regression to correct the coordinate. Non-
maximum Suppression (NMS) is employed to eliminate the redundant boxes.

###  III-C Training procedure

Since the Faster R-CNN combines both classification and bounding box
regression, the loss function $L$ of the proposed framework consists of two
parts: RPN loss and regression loss of Faster R-CNN as shown in the first and
the second parts of Eq. 2, respectively.

$\begin{split}L&=-\frac{1}{N_{f}}\sum\nolimits_{i=1}^{N_{f}}\text{log}[p_{i}^{*}p_{i}+(1-p_{i}^{*})(1-p_{i})]\\\
&\ \
+\alpha\frac{1}{N_{r}}\sum\nolimits_{i=1}^{N_{r}}p_{i}^{*}\text{smooth}_{L_{1}}(t_{i}-t_{i}^{*}),\end{split}$
(2)

where $p_{i}$ represents a probability that anchor forecast is the object $i$,
$p^{*}$ is equal 0 or 1 when ground truth is negative or positive,
respectively. $t_{i}$ represents the prediction in RPN training stage, of
which the ground truth is $t^{*}$. The smooth L1 loss is defined as Eq. 3,
where $\sigma$ is empirically set to 3.

$\text{smooth}_{L_{1}}(x)=\begin{cases}0.5x^{2}\frac{1}{\sigma^{2}},\
\text{if}\ |x|<\frac{1}{\sigma^{2}}\\\ |x|-0.5,\ \text{otherwise}\end{cases}$
(3)

We employ Adam optimizer with an initial learning rate of 0.0001. Three ratios
of RPN anchor are used, i.e. 0.5 , 1 and 2. The IoU thresholds of RPN and RCNN
for positive detection are set to 0.7 and 0.5, respectively.

##  IV Results and discussion

Our framework was trained with synthetic dataset, generated following the
procedure in [13], where nine point spread functions of atmospheric turbulence
were locally applied with random size and strength. We used VOC dataset [29],
containing 20 classes111VOC dataset inclues person, bird, cat, cow, dog,
horse, sheep, aeroplane, bicycle, boat, bus, car, motorbike, train, bottle,
chair, dining table, potted plant, sofa, tv/monitor labelled as bounding
boxes. There are 16,551 images for training and 4,952 images for testing, with
total 52,090 annotated objects. The results of this synthetic testing dataset
were presented in Section IV-A, and the inference of training with this
synthetic training dataset was then used to detect objects in the real
dataset, as shown in Section IV-B.

###  IV-A Synthetic datasets

We compared our methods with state-of-the-are object detection, YOLOv4 [30].
YOLOv4 employs Path Aggregation Network (PAN) which is the same concept of
FPN, so we only implemented deformable convolutions on YOLOv4 framework
(YOLOv4+DC). The results are shown in Table I, and our proposed method (Faster
R-CNN+FPN+DC) achieves the best performance in all three metrics used. Both
Faster R-CNN and YOLOv4 are very successful in common object detection
datasets. However, for the object detection problem with atmospheric
turbulence effect, the background is blurred, and objects are distorted due to
atmospheric turbulence, which makes them insufficient to solve this problem.

We examined the influence of FPN and deformable convolutions. Fig. 3 reveals a
significant improvement on the detection performance. The highest precision
was achieved when detecting horses. Detecting the bottles is the most
improvement – FPN alone improved mAP by 55%, and FPN and DC together improved
as high as 145%. The most missed objects contain thin structures, e.g.
bottles, chair, and boat, as these parts could be distorted, blurred and their
features are mixed with the background. The subjective results in Fig. 4 show
examples of i) easy case, where all models can detect the house and the person
correctly; ii) medium case, where only the proposed method achieves accurate
results, whilst others can only detect the right bird correctly; iii) the hard
case, where all models can detect the person but most models missed out the
bottles.

TABLE I: Detection results on the synthetic datasets with mean Average Precision (mAP), mean Average Recall (mAR) and f1-score  Method | mAP | mAR | f1-score
---|---|---|---
YOLOv4 | 0.598 | 0.610 | 0.562
YOLOv4 + DC | 0.654 | 0.672 | 0.609
Faster R-CNN | 0.563 | 0.576 | 0.542
Faster R-CNN + FPN | 0.630 | 0.650 | 0.583
Faster R-CNN + FPN + DC | 0.779 | 0.773 | 0.703
Figure 3: An improvement of mAP in each category when FPN and deformable
convolutions (DC) are employed. Figure 4: Subjective results of different
models and ground truth. The top to bottom rows show the easy, medium and
difficult cases, respectively. Figure 5: Subjective results of 1100$\times$440
‘Truck’ sequence. (Left-Right) The incorrect detection, correct detected,
mixed results, respectively.

###  IV-B Real datasets

We explored the performance of our proposed model on real atmospheric
turbulence datasets, detailed in [16]. The objective assessment was however
not applicable to the real distortion datasets due to the absence of ground
truth. Fig. 6 shows that our method correctly detected the aeroplane and the
car with the probabilities of 0.89 and 1.0, respectively. Fig. 7 shows the
frame-by-frame detection of a ‘Van’ sequence. The vans in the top row and the
bottom-left frames were detected as a car, but the bottom-right frame shows
the incorrect detection. Notably, despite being trained on synthetic datasets,
as described above, the model detected a white building, visually similar to a
part of the van, as a car. Fig. 5 presents a challenging sequence example. A
group of objects, including a container, several rectangular sign posts, in
many frames was consistently detected as a car as shown in Fig. 5 left.
However, the probabilities were not high ($<$0.5). Fig. 5 right shows that the
van was detected as a car and also a train with probabilities of 0.74 and
0.33, respectively. The incorrect detection may be attributed to the lack of
color information in this sequence and the strong distortion effect.

Figure 6: Detection results of 640$\times$360 ‘Airport’ and ‘Car’ sequences.
Figure 7: Detection results of 480$\times$384 ‘Van’ sequence.

##  V Conclusions

This paper presents a novel framework for object recognition in the
atmospheric turbulence environment, where image quality is severely degraded
due to random signal interference. The framework is based on Faster R-CNN and
utilizes deformable convolutions to handle spatial displacement caused by
turbulence. The feature extraction is performed in a pyramid manner,
considering different object sizes and levels of distortion. Experimental
results demonstrate that the proposed method outperforms state-of-the-art
object detection methods significantly.

## References

  * [1] N. Anantrasirichai and David Bull,  “Artificial intelligence in the creative industries: a review,”  Artifcial Intelligence Review, 2021.
  * [2] A. S. Deshmukh, S. S. Medasani, and G. R. Reddy,  “Moving object detection from images distorted by atmospheric turbulence,”  in International Conference on Intelligent Systems and Signal Processing, March 2013, pp. 122–127.
  * [3] Chun Pong Lau, Hossein Souri, and Rama Chellappa,  “ATFaceGAN: Single face image restoration and recognition from atmospheric turbulence,”  in IEEE International Conference on Automatic Face and Gesture Recognition, 2020, pp. 32–39.
  * [4] S. Ren, K. He, R. Girshick, and J. Sun,  “Faster R-CNN: Towards real-time object detection with region proposal networks,”  IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 39, no. 6, pp. 1137–1149, 2017.
  * [5] T. Lin, P. Dollár, R. Girshick, K. He, B. Hariharan, and S. Belongie,  “Feature pyramid networks for object detection,”  in CVPR, 2017, pp. 936–944.
  * [6] J. Dai, H. Qi, Y. Xiong, Y. Li, G. Zhang, H. Hu, and Y. Wei,  “Deformable convolutional networks,”  in ICCV, Oct 2017, pp. 764–773.
  * [7] N. Anantrasirichai, Alin Achim, David Bull, and Nick Kingsbury,  “Mitigating the effects of atmospheric distortion using dt-cwt fusion,”  in ICIP, 2012, pp. 3033–3036.
  * [8] N. Anantrasirichai, A. Achim, N.G. Kingsbury, and D.R. Bull,  “Atmospheric turbulence mitigation using complex wavelet-based fusion,”  Image Processing, IEEE Transactions on, vol. 22, no. 6, pp. 2398–2408, 2013.
  * [9] Akshay Patel, Dippal Israni, Nerella Arun Mani Kumar, and Chintan Bhatt,  “An adaptive image registration technique to remove atmospheric turbulence,”  Statistics, Optimization and Information Computing, vol. 7, no. 2, pp. 439–446, May 2019.
  * [10] Kalyan Kumar Halder, Murat Tahtali, and Sreenatha G. Anavatti,  “Geometric correction of atmospheric turbulence-degraded video containing moving objects,”  Optics Express, vol. 23, pp. 5091–5101, 2015.
  * [11] N. Anantrasirichai, A. Achim, and D. Bull,  “Atmospheric turbulence mitigation for sequences with moving objects using recursive image fusion,”  in ICIP, 2018, pp. 2895–2899.
  * [12] Zhiyuan Mao, Nicholas Chimitt, and Stanley H. Chan,  “Image reconstruction of static and dynamic scenes through anisoplanatic turbulence,”  IEEE Transactions on Computational Imaging, vol. 6, pp. 1415–1428, 2020.
  * [13] Jing Gao, N. Anantrasirichai, and David Bull,  “Atmospheric turbulence removal using convolutional neural network,”  in arXiv:1912.11350, 2019.
  * [14] Zhiyuan Mao, Nicholas Chimitt, and Stanley H. Chan,  “Accelerating atmospheric turbulence simulation via learned phase-to-space transform,”  in ICCV, 2021.
  * [15] Wai Ho Chak, Chun Pong Lau, and Lok Ming Lui,  “Subsampled turbulence removal network,”  Mathematics, Computation and Geometry of Data, vol. 1, no. 1, pp. 1–33, 2021.
  * [16] Nantheera Anantrasirichai,  “Atmospheric turbulence removal with complex-valued convolutional neural network,”  arXiv:2204.06989, 2022.
  * [17] Omar Oreifej, Xin Li, and Mubarak Shah,  “Simultaneous video stabilization and moving object detection in turbulence,”  IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 35, no. 2, pp. 450–462, 2013.
  * [18] Eli Chen, Oren Haik, and Yitzhak Yitzhaky,  “Detecting and tracking moving objects in long-distance imaging through turbulent medium,”  Applied Optics, vol. 53, pp. 1181–1190, 2014.
  * [19] J. Gilles, F. Alvarez, N. Ferrante, M. Fortman, L. Tahir, A. Tarter, and A. von Seeger,  “Detection of moving objects through turbulent media. decomposition of oscillatory vs non-oscillatory spatio-temporal vector fields,”  Image and Vision Computing, vol. 73, pp. 40–55, 2018.
  * [20] Chao Zhang, Fugen Zhou, Bindang Xue, and Wenfang Xue,  “Stabilization of atmospheric turbulence-distorted video containing moving objects using the monogenic signal,”  Signal Processing: Image Communication, vol. 63, pp. 19–29, 2018\.
  * [21] Engin Uzun, Ahmet Anıl Dursun, and Erdem Akagündüz,  “Augmentation of atmospheric turbulence effects on thermal adapted object detection models,”  in CVPRW, June 2022, pp. 241–248.
  * [22] Haoyang Zhang, Ying Wang, Feras Dayoub, and Niko Sünderhauf,  “VarifocalNet: An IoU-aware dense object detector,”  in CVPR, 2021, pp. 8510–8519.
  * [23] Chien-Yao Wang, I-Hau Yeh, and Hong-Yuan Mark Liao,  “You only learn one representation: Unified network for multiple tasks,”  arXiv preprint arXiv:2105.04206, 2021.
  * [24] Chengjian Feng, Yujie Zhong, Yu Gao, Matthew R. Scott, and Weilin Huang,  “TOOD: Task-aligned One-stage Object Detection,”  in ICCV, 2021, pp. 3490–3499.
  * [25] P.R. Hill, N. Anantrasirichai, A. Achim, M.E. Al-Mualla, and D. Bull,  “Undecimated dual tree complex wavelet transforms,”  Signal Processing: Image Communication, vol. 35, pp. 61–70, 2015\.
  * [26] Tsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva Ramanan, Piotr Dollár, and C. Lawrence Zitnick,  “Microsoft COCO: Common objects in context,”  in ECCV, 2014, pp. 740–755.
  * [27] Karen Simonyan and Andrew Zisserman,  “Very deep convolutional networks for large-scale image recognition,”  in International Conference on Learning Representations, 2015.
  * [28] K. He, X. Zhang, S. Ren, and J. Sun,  “Deep residual learning for image recognition,”  in CVPR, 2016, pp. 770–778.
  * [29] M. Everingham, S. M. A. Eslami, L. Van Gool, C. K. I. Williams, J. Winn, and A. Zisserman,  “The pascal visual object classes challenge: A retrospective,”  International Journal of Computer Vision, vol. 111, no. 1, pp. 98–136, Jan. 2015.
  * [30] Alexey Bochkovskiy, Chien-Yao Wang, and Hong-Yuan Mark Liao,  “YOLOv4: Optimal speed and accuracy of object detection,”  ArXiv, vol. abs/2004.10934, 2020.


This is the final formula that we were after. Notice that in the isotropic
limit it coincides with the formula for the butterfly velocity derived in
Mezei:2016wfz . It is also consistent with previous results for anisotropic
theories, where _only one_ direction is anisotropic, including the scaling
solutions of Giataganas:2017koz  and various other models Jahnke:2017iwi ;
Avila:2018sqf ; Fischler:2018kwt ; Inkof:2019gmh . Finally, specializing to
our metric (4) we obtain the formulas reported in (50).

###  B.4 Results for $c=0.4$

In section 5 we showed that our results for the string tensions strongly
favoured $c=0.25$ after comparing with the lattice data of Bonati:2014ksa .
However, it is known that our model can exhibit IMC for even slightly higher
values of $c$. It is then interesting to explore the effects of this parameter
on the behavior of the other observables of interest, and investigate how
robust our findings and predictions are. Throughout this appendix we will
study in detail this issue for $c=0.4$, which is also known to display both
IMC Gursoy:2016ofp  and IAC Gursoy:2018ydr .

In fig. 11 we plot the results for the quark-antiquark potentials in a
situation where both $a$ and $B$ are turned on and can compete with each
other. In general, we observe the same qualitative features between the two
values of $c$: the tendency to push the transition to shorter distances when
both effects are present while increasing the tensions, and the two special
cases — $V_{z}$ in the case where $a$ and $B$ are perpendicular/parallel to
each other — where $a$ and $B$ act completely constructively/destructively in
the IR. Perhaps the only distinction that is worth pointing out is the slight
increase in the transition distance as we increase $c$, which make the bound
states a bit more stable in comparison to the $c=0.25$ case.

Figure 11: Quark-antiquark potential in the presence of anisotropy and
magnetic field, for $a=1$ and $B=1$ (in units of $\Lambda\sim 1$ GeV). The
magnetic field $B$ is aligned along the $z$-direction while the anisotropic
deformation is introduced along the $z$\- or $y$-direction (parallel and
perpendicular configurations, respectively). The solid lines depict results
for $c=0.25$, which we have kept for comparison, while dashed lines show the
corresponding results for $c=0.4$. We find the same qualitative features in
both cases, perhaps with the only difference that the increase in $c$ pushes
the transition to slightly higher values of the separation, making the bound
states a bit more stable.  Figure 12: Ratio between the components of the
shear viscosity tensor and the entropy density for the case where the magnetic
field (aligned along the $z$-direction) is perpendicular to the anisotropic
deformation (introduced along the $y$-direction). All the plots are given as a
function of temperature, in units of $\Lambda\sim 1$ GeV, and cut off at the
chiral transition temperature so they correspond to the chirally symmetric
phase. The solid lines depict results for $c=0.25$, which we have kept for
comparison, while dashed lines show the corresponding results for $c=0.4$. We
observe all the same qualitative features for the two values of $c$, with some
small visual distinction along the flow from the UV to the IR: while
$\eta_{xy}$ generally decreases with the increase of $c$, both $\eta_{xz}$ and
$\eta_{yz}$ increase at an intermediate regime of energies, while approaching
roughly the same constants in the IR.  Figure 13: Components of the butterfly
velocity as a function of temperature (in units of $\Lambda\sim 1$ GeV) for
the perpendicular configuration. The magnetic field $B$ is aligned along the
$z$-direction while the anisotropic deformation is introduced along the
$y$-direction. The solid lines depict results for $c=0.25$, which we have kept
for comparison, while dashed lines show the corresponding results for $c=0.4$.
In both cases we observe that the component $v_{B}^{z}$ is enhanced in the IR
and exceeds the bound found in Giataganas:2017koz , although its maximum value
decreases with the increase of $c$. On the other hand, the other components
$v_{B}^{x}$ and $v_{B}^{y}$ show a slight increase with respect to the results
for $c=0.25$.  Figure 14: Components of the butterfly velocity as a function
of temperature (in units of $\Lambda\sim 1$ GeV) for the parallel
configuration. In this case both the magnetic field $B$ and the anisotropic
deformation are introduced along the $z$-direction. The solid lines depict
results for $c=0.25$, which we have kept for comparison, while dashed lines
show the corresponding results for $c=0.4$. The component $v_{B}^{z}$ is also
enhanced in the IR and exceeds the bound found in Giataganas:2017koz ,
although it is slightly smaller than in the perpendicular case. In this case,
the increase in $c$ also makes $v_{B}^{z}$ smaller, while there is no clear
pattern for the other component, $v_{B}^{x}$.

In fig. 12 we plot the components of the shear viscosity in the case where the
magnetic field is perpendicular to the anisotropic deformation. We observe all
the same qualitative features for the two values of $c$: in all cases, the
plots exhibit a monotonic flow from $\eta_{ij}/s=1/4\pi$ in the UV to a lower,
non-zero value in the IR, before the chiral transition kicks in. Moreover, the
three main conclusions reached for $c=0.25$ hold true for $c=0.4$ as well: i)
The increase in magnetic field $B$ raises $\eta_{xy}$ and lowers $\eta_{xz}$
and $\eta_{yz}$. ii) The increase in the anisotropic parameter $a$ lowers
$\eta_{xy}$ and $\eta_{yz}$ and raises $\eta_{xz}$. And iii) $B$ affects
mostly $\eta_{xz}$, while $a$ affects mostly $\eta_{xy}$. Finally, the two
effects seem equally strong for $\eta_{yz}$ so there is no clear pattern for
this component. The only visual differences between the two values of $c$ are
small differences in values along the flow from the UV to the IR: while
$\eta_{xy}$ generally decreases with the increase of $c$, both $\eta_{xz}$ and
$\eta_{yz}$ increase at an intermediate regime of energies, while approaching
roughly the same constants in the IR.

Finally, in figures 13 and 14 we plot the components of the butterfly velocity
in various situations of interest. Both $c=0.25$ and $c=0.4$ show
qualitatively similar results; crucially, that the component $v_{B}^{z}$ is
enhanced in the IR and exceeds the bound found in Giataganas:2017koz .
However, the maximum value decreases with the increase of $c$,
$v_{B}^{z}\approx 0.907$ for $c=0.4$ while $v_{B}^{z}\approx 0.942$ for
$c=0.25$. In both cases, parallel and perpendicular configurations, the
remaining components show a weak dependence on $c$, with a tendency of
slightly increasing the values of $v_{B}^{x}$ and $v_{B}^{y}$. As explained in
the main text, the violation of the bound on the butterfly velocity does not
contradict the results of Giataganas:2017koz  because the IR theory in our
case does not belong to the same universality class of those in
Giataganas:2017koz . We will present a more thorough analysis of these bounds
in a more generic class of backgrounds in our upcoming publication
Giataganas:progress .

## References

  * (1) W. Busza, K. Rajagopal, and W. van der Schee, “Heavy Ion Collisions: The Big Picture, and the Big Questions,” Ann. Rev. Nucl. Part. Sci. 68 (2018) 339–376, 1802.04801.
  * (2) U. Heinz and R. Snellings, “Collective flow and viscosity in relativistic heavy-ion collisions,” Ann. Rev. Nucl. Part. Sci. 63 (2013) 123–151, 1301.2826.
  * (3) D. Teaney, J. Lauret, and E. V. Shuryak, “Flow at the SPS and RHIC as a quark gluon plasma signature,” Phys. Rev. Lett. 86 (2001) 4783–4786, nucl-th/0011058.
  * (4) P. Romatschke and U. Romatschke, “Viscosity Information from Relativistic Nuclear Collisions: How Perfect is the Fluid Observed at RHIC?,” Phys. Rev. Lett. 99 (2007) 172301, 0706.1522.
  * (5) J. E. Bernhard, J. S. Moreland, S. A. Bass, J. Liu, and U. Heinz, “Applying Bayesian parameter estimation to relativistic heavy-ion collisions: simultaneous characterization of the initial state and quark-gluon plasma medium,” Phys. Rev. C 94 (2016), no. 2 024907, 1605.03954.
  * (6) J. E. Bernhard, J. S. Moreland, and S. A. Bass, “Bayesian estimation of the specific shear and bulk viscosity of quark–gluon plasma,” Nature Phys. 15 (2019), no. 11 1113–1117.
  * (7) J. S. Moreland, J. E. Bernhard, and S. A. Bass, “Bayesian calibration of a hybrid nuclear collision model using p-Pb and Pb-Pb data at energies available at the CERN Large Hadron Collider,” Phys. Rev. C 101 (2020), no. 2 024911, 1808.02106.
  * (8) G. Policastro, D. T. Son, and A. O. Starinets, “The Shear viscosity of strongly coupled N=4 supersymmetric Yang-Mills plasma,” Phys. Rev. Lett. 87 (2001) 081601, hep-th/0104066.
  * (9) K. Tuchin, “Particle production in strong electromagnetic fields in relativistic heavy-ion collisions,” Adv. High Energy Phys. 2013 (2013) 490495, 1301.0099.
  * (10) V. Skokov, A. Yu. Illarionov, and V. Toneev, “Estimate of the magnetic field strength in heavy-ion collisions,” Int. J. Mod. Phys. A24 (2009) 5925–5932, 0907.1396.
  * (11) K. Fukushima, D. E. Kharzeev, and H. J. Warringa, “The Chiral Magnetic Effect,” Phys. Rev. D78 (2008) 074033, 0808.3382.
  * (12) D. Kharzeev, J. Liao, S. Voloshin, and G. Wang, “Chiral magnetic and vortical effects in high-energy nuclear collisions—A status report,” Prog. Part. Nucl. Phys. 88 (2016) 1–28, 1511.04050.
  * (13) S. Shi, Y. Jiang, E. Lilleskov, and J. Liao, “Anomalous Chiral Transport in Heavy Ion Collisions from Anomalous-Viscous Fluid Dynamics,” Annals Phys. 394 (2018) 50–72, 1711.02496.
  * (14) Q. Li, D. E. Kharzeev, C. Zhang, Y. Huang, I. Pletikosic, A. Fedorov, R. Zhong, J. Schneeloch, G. Gu, and T. Valla, “Observation of the chiral magnetic effect in ZrTe5,” Nature Phys. 12 (2016) 550–554, 1412.6543.
  * (15) V. A. Miransky and I. A. Shovkovy, “Quantum field theory in a magnetic field: From quantum chromodynamics to graphene and Dirac semimetals,” Phys. Rept. 576 (2015) 1–209, 1503.00732.
  * (16) G. S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S. D. Katz, and A. Schafer, “QCD quark condensate in external magnetic fields,” Phys. Rev. D86 (2012) 071502, 1206.4205.
  * (17) U. Gursoy, M. Järvinen, G. Nijs, and J. F. Pedraza, “Inverse Anisotropic Catalysis in Holographic QCD,” JHEP 04 (2019) 071, 1811.11724.
  * (18) D. Giataganas, U. Gursoy, and J. F. Pedraza, “Strongly-coupled anisotropic gauge theories and holography,” Phys. Rev. Lett. 121 (2018), no. 12 121601, 1708.05691.
  * (19) U. Gursoy and E. Kiritsis, “Exploring improved holographic theories for QCD: Part I,” JHEP 02 (2008) 032, 0707.1324.
  * (20) U. Gursoy, E. Kiritsis, and F. Nitti, “Exploring improved holographic theories for QCD: Part II,” JHEP 02 (2008) 019, 0707.1349.
  * (21) M. Jarvinen and E. Kiritsis, “Holographic Models for QCD in the Veneziano Limit,” JHEP 03 (2012) 002, 1112.1261.
  * (22) F. Bigazzi, R. Casero, A. L. Cotrone, E. Kiritsis, and A. Paredes, “Non-critical holography and four-dimensional CFT’s with fundamentals,” JHEP 10 (2005) 012, hep-th/0505140.
  * (23) R. Casero, E. Kiritsis, and A. Paredes, “Chiral symmetry breaking as open string tachyon condensation,” Nucl. Phys. B787 (2007) 98–134, hep-th/0702155.
  * (24) D. Mateos and D. Trancanelli, “The anisotropic N=4 super Yang-Mills plasma and its instabilities,” Phys. Rev. Lett. 107 (2011) 101601, 1105.3472.
  * (25) D. Mateos and D. Trancanelli, “Thermodynamics and Instabilities of a Strongly Coupled Anisotropic Plasma,” JHEP 07 (2011) 054, 1106.1637.
  * (26) D. Areán, I. Iatrakis, M. Järvinen, and E. Kiritsis, “The discontinuities of conformal transitions and mass spectra of V-QCD,” JHEP 11 (2013) 068, 1309.2286.
  * (27) D. Arean, I. Iatrakis, M. Jarvinen, and E. Kiritsis, “CP-odd sector and $\theta$ dynamics in holographic QCD,” Phys. Rev. D96 (2017), no. 2 026001, 1609.08922.
  * (28) U. Gursoy, E. Kiritsis, L. Mazzanti, and F. Nitti, “Improved Holographic Yang-Mills at Finite Temperature: Comparison with Data,” Nucl. Phys. B820 (2009) 148–177, 0903.2859.
  * (29) U. Gursoy, I. Iatrakis, E. Kiritsis, F. Nitti, and A. O’Bannon, “The Chern-Simons Diffusion Rate in Improved Holographic QCD,” JHEP 02 (2013) 119, 1212.3894.
  * (30) T. Alho, M. Järvinen, K. Kajantie, E. Kiritsis, and K. Tuominen, “On finite-temperature holographic QCD in the Veneziano limit,” JHEP 01 (2013) 093, 1210.4516.
  * (31) M. Jarvinen, “Massive holographic QCD in the Veneziano limit,” JHEP 07 (2015) 033, 1501.07272.
  * (32) N. Jokela, M. Järvinen, and J. Remes, “Holographic QCD in the Veneziano limit and neutron stars,” 1809.07770.
  * (33) T. Drwenski, U. Gursoy, and I. Iatrakis, “Thermodynamics and CP-odd transport in Holographic QCD with Finite Magnetic Field,” JHEP 12 (2016) 049, 1506.01350.
  * (34) T. Alho, M. Järvinen, K. Kajantie, E. Kiritsis, C. Rosen, and K. Tuominen, “A holographic model for QCD in the Veneziano limit at finite temperature and density,” JHEP 04 (2014) 124, 1312.5199. [Erratum: JHEP02,033(2015)].
  * (35) U. Gursoy, I. Iatrakis, M. Järvinen, and G. Nijs, “Inverse Magnetic Catalysis from improved Holographic QCD in the Veneziano limit,” JHEP 03 (2017) 053, 1611.06339.
  * (36) M. Panero, “Thermodynamics of the QCD plasma and the large-N limit,” Phys. Rev. Lett. 103 (2009) 232001, 0907.3719.
  * (37) U. Gursoy, M. Jarvinen, and G. Nijs, “Holographic QCD in the Veneziano Limit at a Finite Magnetic Field and Chemical Potential,” Phys. Rev. Lett. 120 (2018), no. 24 242002, 1707.00872.
  * (38) M. Ammon, V. G. Filev, J. Tarrio, and D. Zoakos, “D3/D7 Quark-Gluon Plasma with Magnetically Induced Anisotropy,” JHEP 09 (2012) 039, 1207.1047.
  * (39) I. Papadimitriou, “Holographic Renormalization of general dilaton-axion gravity,” JHEP 08 (2011) 119, 1106.4826.
  * (40) G. S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S. D. Katz, S. Krieg, A. Schafer, and K. K. Szabo, “The QCD phase diagram for external magnetic fields,” JHEP 02 (2012) 044, 1111.4956.
  * (41) M. D’Elia, “Lattice QCD Simulations in External Background Fields,” Lect. Notes Phys. 871 (2013) 181–208, 1209.0374.
  * (42) A. Ballon-Bayona, M. Ihl, J. P. Shock, and D. Zoakos, “A universal order parameter for Inverse Magnetic Catalysis,” JHEP 10 (2017) 038, 1706.05977.
  * (43) C. Bonati, M. D’Elia, M. Mariti, M. Mesiti, F. Negro, and F. Sanfilippo, “Anisotropy of the quark-antiquark potential in a magnetic field,” Phys. Rev. D89 (2014), no. 11 114502, 1403.6094.
  * (44) E. D’Hoker and P. Kraus, Quantum Criticality via Magnetic Branes, vol. 871, pp. 469–502.  2013\.  1208.1925.
  * (45) I. Aref’eva and K. Rannu, “Holographic Anisotropic Background with Confinement-Deconfinement Phase Transition,” JHEP 05 (2018) 206, 1802.05652.
  * (46) I. Aref’eva, K. Rannu, and P. Slepov, “Orientation Dependence of Confinement-Deconfinement Phase Transition in Anisotropic Media,” 1808.05596.
  * (47) D. Bak, A. Karch, and L. G. Yaffe, “Debye screening in strongly coupled N=4 supersymmetric Yang-Mills plasma,” JHEP 08 (2007) 049, 0705.0994.
  * (48) J. M. Maldacena, “Wilson loops in large N field theories,” Phys. Rev. Lett. 80 (1998) 4859–4862, hep-th/9803002.
  * (49) S.-J. Rey and J.-T. Yee, “Macroscopic strings as heavy quarks in large N gauge theory and anti-de Sitter supergravity,” Eur. Phys. J. C22 (2001) 379–394, hep-th/9803001.
  * (50) Y. Kinar, E. Schreiber, and J. Sonnenschein, “Q anti-Q potential from strings in curved space-time: Classical results,” Nucl. Phys. B566 (2000) 103–125, hep-th/9811192.
  * (51) P. Kovtun, D. T. Son, and A. O. Starinets, “Holography and hydrodynamics: Diffusion on stretched horizons,” JHEP 10 (2003) 064, hep-th/0309213.
  * (52) J. Casalderrey-Solana, H. Liu, D. Mateos, K. Rajagopal, and U. A. Wiedemann, Gauge/String Duality, Hot QCD and Heavy Ion Collisions.  Cambridge University Press, 2014.
  * (53) M. Brigante, H. Liu, R. C. Myers, S. Shenker, and S. Yaida, “Viscosity Bound Violation in Higher Derivative Gravity,” Phys. Rev. D 77 (2008) 126006, 0712.0805.
  * (54) M. Brigante, H. Liu, R. C. Myers, S. Shenker, and S. Yaida, “The Viscosity Bound and Causality Violation,” Phys. Rev. Lett. 100 (2008) 191601, 0802.3318.
  * (55) X. O. Camanho, J. D. Edelstein, and M. F. Paulos, “Lovelock theories, holography and the fate of the viscosity bound,” JHEP 05 (2011) 127, 1010.1682.
  * (56) S. Cremonini, “The Shear Viscosity to Entropy Ratio: A Status Report,” Mod. Phys. Lett. B 25 (2011) 1867–1888, 1108.0677.
  * (57) S. Cremonini, U. Gursoy, and P. Szepietowski, “On the Temperature Dependence of the Shear Viscosity and Holography,” JHEP 08 (2012) 167, 1206.3581.
  * (58) P. Burikham and N. Poovuttikul, “Shear viscosity in holography and effective theory of transport without translational symmetry,” Phys. Rev. D 94 (2016), no. 10 106001, 1601.04624.
  * (59) S. A. Hartnoll, D. M. Ramirez, and J. E. Santos, “Entropy production, viscosity bounds and bumpy black holes,” JHEP 03 (2016) 170, 1601.02757.
  * (60) L. Alberte, M. Baggioli, and O. Pujolas, “Viscosity bound violation in holographic solids and the viscoelastic response,” JHEP 07 (2016) 074, 1601.03384.
  * (61) Y. Ling, Z.-Y. Xian, and Z. Zhou, “Holographic Shear Viscosity in Hyperscaling Violating Theories without Translational Invariance,” JHEP 11 (2016) 007, 1605.03879.
  * (62) M. Baggioli and W.-J. Li, “Universal Bounds on Transport in Holographic Systems with Broken Translations,” SciPost Phys. 9 (2020), no. 1 007, 2005.06482.
  * (63) J. Erdmenger, P. Kerner, and H. Zeller, “Non-universal shear viscosity from Einstein gravity,” Phys. Lett. B 699 (2011) 301–304, 1011.5912.
  * (64) A. Rebhan and D. Steineder, “Violation of the Holographic Viscosity Bound in a Strongly Coupled Anisotropic Plasma,” Phys. Rev. Lett. 108 (2012) 021601, 1110.6825.
  * (65) K. A. Mamo, “Holographic RG flow of the shear viscosity to entropy density ratio in strongly coupled anisotropic plasma,” JHEP 10 (2012) 070, 1205.1797.
  * (66) S. Jain, N. Kundu, K. Sen, A. Sinha, and S. P. Trivedi, “A Strongly Coupled Anisotropic Fluid From Dilaton Driven Holography,” JHEP 01 (2015) 005, 1406.4874.
  * (67) S. Jain, R. Samanta, and S. P. Trivedi, “The Shear Viscosity in Anisotropic Phases,” JHEP 10 (2015) 028, 1506.01899.
  * (68) R. Samanta, R. Sharma, and S. P. Trivedi, “Shear viscosity in an anisotropic unitary Fermi gas,” Phys. Rev. A 96 (2017), no. 5 053601, 1607.04799.
  * (69) R. Samanta, R. Sharma, and S. P. Trivedi, “A Proposal for measuring Anisotropic Shear Viscosity in Unitary Fermi Gases,” 1611.02720.
  * (70) S. Ryu and T. Takayanagi, “Holographic derivation of entanglement entropy from AdS/CFT,” Phys. Rev. Lett. 96 (2006) 181602, hep-th/0603001.
  * (71) I. R. Klebanov, D. Kutasov, and A. Murugan, “Entanglement as a probe of confinement,” Nucl. Phys. B796 (2008) 274–293, 0709.2140.
  * (72) D. Dudal and S. Mahapatra, “Confining gauge theories and holographic entanglement entropy with a magnetic field,” JHEP 04 (2017) 031, 1612.06248.
  * (73) A. I. Larkin and Y. N. Ovchinnikov, “Quasiclassical method in the theory of superconductivity,” JETP 28, 6 (1969) 1200–1205.
  * (74) J. Maldacena, S. H. Shenker, and D. Stanford, “A bound on chaos,” JHEP 08 (2016) 106, 1503.01409.
  * (75) J. de Boer, E. Llabrés, J. F. Pedraza, and D. Vegh, “Chaotic strings in AdS/CFT,” Phys. Rev. Lett. 120 (2018), no. 20 201604, 1709.01052.
  * (76) K. Murata, “Fast scrambling in holographic Einstein-Podolsky-Rosen pair,” JHEP 11 (2017) 049, 1708.09493.
  * (77) A. Banerjee, A. Kundu, and R. R. Poojary, “Strings, Branes, Schwarzian Action and Maximal Chaos,” 1809.02090.
  * (78) S. H. Shenker and D. Stanford, “Black holes and the butterfly effect,” JHEP 03 (2014) 067, 1306.0622.
  * (79) S. H. Shenker and D. Stanford, “Stringy effects in scrambling,” JHEP 05 (2015) 132, 1412.6087.
  * (80) J. Maldacena and D. Stanford, “Remarks on the Sachdev-Ye-Kitaev model,” Phys. Rev. D 94 (2016), no. 10 106002, 1604.07818.
  * (81) Y. Gu, X.-L. Qi, and D. Stanford, “Local criticality, diffusion and chaos in generalized Sachdev-Ye-Kitaev models,” JHEP 05 (2017) 125, 1609.07832.
  * (82) R. A. Davison, W. Fu, A. Georges, Y. Gu, K. Jensen, and S. Sachdev, “Thermoelectric transport in disordered metals without quasiparticles: The Sachdev-Ye-Kitaev models and holography,” Phys. Rev. B 95 (2017), no. 15 155131, 1612.00849.
  * (83) D. A. Roberts and B. Swingle, “Lieb-Robinson Bound and the Butterfly Effect in Quantum Field Theories,” Phys. Rev. Lett. 117 (2016), no. 9 091602, 1603.09298.
  * (84) V. Jahnke, “Delocalizing entanglement of anisotropic black branes,” JHEP 01 (2018) 102, 1708.07243.
  * (85) D. Avila, V. Jahnke, and L. Patiño, “Chaos, Diffusivity, and Spreading of Entanglement in Magnetic Branes, and the Strengthening of the Internal Interaction,” JHEP 09 (2018) 131, 1805.05351.
  * (86) D. Giataganas, U. Gursoy, and J. F. Pedraza. Work in progress.
  * (87) N. Jokela and J. G. Subils, “Is entanglement a probe of confinement?,” 2010.09392.
  * (88) G. S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S. D. Katz, S. Krieg, A. Schafer, and K. K. Szabo, “The finite temperature QCD transition in external magnetic fields,” PoS LATTICE2011 (2011) 192, 1111.5155.
  * (89) U. Gursoy, E. Kiritsis, L. Mazzanti, and F. Nitti, “Holographic Techniques For Asymptotically-free Gauge Theories,” in 43rd Rencontres de Moriond on Electroweak Interactions and Unified Theories, (Paris, France), pp. 143–150, Moriond, 2008.
  * (90) U. Gursoy, “Improved Holographic QCD and the Quark-gluon Plasma,” Acta Phys. Polon. B 47 (2016) 2509, 1612.00899.
  * (91) I. Y. Aref’eva, K. Rannu, and P. Slepov, “Holographic Anisotropic Model for Heavy Quarks in Anisotropic Hot Dense QGP with External Magnetic Field,” 2011.07023.
  * (92) U. Gursoy, E. Kiritsis, L. Mazzanti, and F. Nitti, “Holography and Thermodynamics of 5D Dilaton-gravity,” JHEP 05 (2009) 033, 0812.0792.
  * (93) S. S. Gubser, S. S. Pufu, and F. D. Rocha, “Bulk viscosity of strongly coupled plasmas with holographic duals,” JHEP 08 (2008) 085, 0806.0407.
  * (94) M. Mezei and D. Stanford, “On entanglement spreading in chaotic systems,” JHEP 05 (2017) 065, 1608.05101.
  * (95) W. Fischler, V. Jahnke, and J. F. Pedraza, “Chaos and entanglement spreading in a non-commutative gauge theory,” JHEP 11 (2018) 072, 1808.10050.
  * (96) G. A. Inkof, J. M. Küppers, J. M. Link, B. Goutéraux, and J. Schmalian, “Quantum critical scaling and holographic bound for transport coefficients near Lifshitz points,” 1907.05744.