@inproceedings{treewasserstein,
 author = {Le, Tam and Yamada, Makoto and Fukumizu, Kenji and Cuturi, Marco},
 booktitle = {Advances in Neural Information Processing Systems},
 editor = {H. Wallach and H. Larochelle and A. Beygelzimer and F. d\textquotesingle Alch\'{e}-Buc and E. Fox and R. Garnett},
 pages = {},
 publisher = {Curran Associates, Inc.},
 title = {Tree-Sliced Variants of Wasserstein Distances},
 url = {https://proceedings.neurips.cc/paper_files/paper/2019/file/2d36b5821f8affc6868b59dfc9af6c9f-Paper.pdf},
 volume = {32},
 year = {2019}
}

@article{bonneel:hal-00881872,
  TITLE = {{Sliced and Radon Wasserstein Barycenters of Measures}},
  AUTHOR = {Bonneel, Nicolas and Rabin, Julien and Peyr{\'e}, Gabriel and Pfister, Hanspeter},
  URL = {https://hal.science/hal-00881872},
  JOURNAL = {{Journal of Mathematical Imaging and Vision}},
  PUBLISHER = {{Springer Verlag}},
  VOLUME = {1},
  NUMBER = {51},
  PAGES = {22-45},
  YEAR = {2015},
  DOI = {10.1007/s10851-014-0506-3},
  KEYWORDS = {Wasserstein distance ; Radon transform ; Optimal transport ; Barycenter of measures},
  PDF = {https://hal.science/hal-00881872v1/file/WassersteinSliced-JMIV.pdf},
  HAL_ID = {hal-00881872},
  HAL_VERSION = {v1},
}

@InProceedings{factoredcouplings,
  title = 	 {Statistical Optimal Transport via Factored Couplings},
  author =       {Forrow, Aden and H\"{u}tter, Jan-Christian and Nitzan, Mor and Rigollet, Philippe and Schiebinger, Geoffrey and Weed, Jonathan},
  booktitle = 	 {Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics},
  pages = 	 {2454--2465},
  year = 	 {2019},
  editor = 	 {Chaudhuri, Kamalika and Sugiyama, Masashi},
  volume = 	 {89},
  series = 	 {Proceedings of Machine Learning Research},
  month = 	 {16--18 Apr},
  publisher =    {PMLR},
  pdf = 	 {http://proceedings.mlr.press/v89/forrow19a/forrow19a.pdf},
  url = 	 {https://proceedings.mlr.press/v89/forrow19a.html}
}

@inproceedings{sinkhorn,
 author = {Cuturi, Marco},
 booktitle = {Advances in Neural Information Processing Systems},
 editor = {C.J. Burges and L. Bottou and M. Welling and Z. Ghahramani and K.Q. Weinberger},
 pages = {},
 publisher = {Curran Associates, Inc.},
 title = {Sinkhorn Distances: Lightspeed Computation of Optimal Transport},
 url = {https://proceedings.neurips.cc/paper_files/paper/2013/file/af21d0c97db2e27e13572cbf59eb343d-Paper.pdf},
 volume = {26},
 year = {2013}
}

@inproceedings{
rout2022generative,
title={Generative Modeling with Optimal Transport Maps},
author={Litu Rout and Alexander Korotin and Evgeny Burnaev},
booktitle={International Conference on Learning Representations},
year={2022},
url={https://openreview.net/forum?id=5JdLZg346Lw}
}

@inproceedings{Black_2020, series={FAT* ’20},
   title={FlipTest: fairness testing via optimal transport},
   url={http://dx.doi.org/10.1145/3351095.3372845},
   DOI={10.1145/3351095.3372845},
   booktitle={Proceedings of the 2020 Conference on Fairness, Accountability, and Transparency},
   publisher={ACM},
   author={Black, Emily and Yeom, Samuel and Fredrikson, Matt},
   year={2020},
   month=jan, pages={111–121},
   collection={FAT* ’20} }

@misc{decolnet2021compilationsuccinctnessresultsarithmetic,
      title={A Compilation of Succinctness Results for Arithmetic Circuits}, 
      author={Alexis de Colnet and Stefan Mengel},
      year={2021},
      eprint={2110.13014},
      archivePrefix={arXiv},
      primaryClass={cs.CC},
      url={https://arxiv.org/abs/2110.13014}, 
}

@misc{zhang2024restructuringtractableprobabilisticcircuits,
      title={Restructuring Tractable Probabilistic Circuits}, 
      author={Honghua Zhang and Benjie Wang and Marcelo Arenas and Guy Van den Broeck},
      year={2024},
      eprint={2411.12256},
      archivePrefix={arXiv},
      primaryClass={cs.AI},
      url={https://arxiv.org/abs/2411.12256}, 
}

@inproceedings{Bhattacharyya_2023, series={IJCAI-2023},
   title={On Approximating Total Variation Distance},
   url={http://dx.doi.org/10.24963/ijcai.2023/387},
   DOI={10.24963/ijcai.2023/387},
   booktitle={Proceedings of the Thirty-Second International Joint Conference on Artificial Intelligence},
   publisher={International Joint Conferences on Artificial Intelligence Organization},
   author={Bhattacharyya, Arnab and Gayen, Sutanu and Meel, Kuldeep S. and Myrisiotis, Dimitrios and Pavan, A. and Vinodchandran, N. V.},
   year={2023},
   month=aug, pages={3479–3487},
   collection={IJCAI-2023} }

@article{Feng_2023,
   title={A simple polynomial-time approximation algorithm for the total variation distance between two product distributions},
   volume={Volume 2},
   ISSN={2751-4838},
   url={http://dx.doi.org/10.46298/theoretics.23.7},
   DOI={10.46298/theoretics.23.7},
   journal={TheoretiCS},
   publisher={Centre pour la Communication Scientifique Directe (CCSD)},
   author={Feng, Weiming and Guo, Heng and Jerrum, Mark and Wang, Jiaheng},
   year={2023},
   month=jun }


@InProceedings{strudel,
  title = 	 {Strudel: Learning Structured-Decomposable Probabilistic Circuits},
  author =       {Dang, Meihua and Vergari, Antonio and Van den Broeck, Guy},
  booktitle = 	 {Proceedings of the 10th International Conference on Probabilistic Graphical Models},
  pages = 	 {137--148},
  year = 	 {2020},
  editor = 	 {Jaeger, Manfred and Nielsen, Thomas Dyhre},
  volume = 	 {138},
  series = 	 {Proceedings of Machine Learning Research},
  month = 	 {23--25 Sep},
  publisher =    {PMLR},
  pdf = 	 {http://proceedings.mlr.press/v138/dang20a/dang20a.pdf},
  url = 	 {https://proceedings.mlr.press/v138/dang20a.html}
}


@article{bounding,
 ISSN = {03067734, 17515823},
 URL = {http://www.jstor.org/stable/1403865},
 abstract = {When studying convergence of measures, an important issue is the choice of probability metric. We provide a summary and some new results concerning bounds among some important probability metrics/distances that are used by statisticians and probabilists. Knowledge of other metrics can provide a means of deriving bounds for another one in an applied problem. Considering other metrics can also provide alternate insights. We also give examples that show that rates of convergence can strongly depend on the metric chosen. Careful consideration is necessary when choosing a metric. /// Le choix de métrique de probabilité est une décision très importante lorsqu'on étudie la convergence des mesures. Nous vous fournissons avec un sommaire de plusieurs métriques/distances de probabilité couramment utilisées par des statisticiens(nes) at par des probabilistes, ainsi que certains nouveaux résultats qui se rapportent à leurs bornes. Avoir connaissance d'autres métriques peut vous fournir avec un moyen de dériver des bornes pour une autre métrique dans un problème appliqué. Le fait de prendre en considération plusieurs métriques vous permettra d'approcher des problèmes d'une manière différente. Ainsi, nous vous démontrons que les taux de convergence peuvent dépendre de façon importante sur votre choix de métrique. Il est donc important de tout considérer lorsqu'on doit choisir une métrique.},
 author = {Alison L. Gibbs and Francis Edward Su},
 journal = {International Statistical Review / Revue Internationale de Statistique},
 number = {3},
 pages = {419--435},
 publisher = {[Wiley, International Statistical Institute (ISI)]},
 title = {On Choosing and Bounding Probability Metrics},
 urldate = {2024-10-01},
 volume = {70},
 year = {2002}
}

@INPROCEEDINGS{tcloseness,
  author={Li, Ninghui and Li, Tiancheng and Venkatasubramanian, Suresh},
  booktitle={2007 IEEE 23rd International Conference on Data Engineering}, 
  title={t-Closeness: Privacy Beyond k-Anonymity and l-Diversity}, 
  year={2007},
  volume={},
  number={},
  pages={106-115},
  keywords={Privacy;Earth;Computer science;Publishing;Motion measurement;Databases;Data security;Diseases;Remuneration;Protection},
  doi={10.1109/ICDE.2007.367856}}


@article{darwiche2003differential,
  title={A differential approach to inference in Bayesian networks},
  author={Darwiche, Adnan},
  journal={Journal of the ACM (JACM)},
  volume={50},
  number={3},
  pages={280--305},
  year={2003},
  publisher={ACM New York, NY, USA}
}

@inproceedings{poon2011sum,
  title={Sum-product networks: A new deep architecture},
  author={Poon, Hoifung and Domingos, Pedro},
  booktitle={2011 IEEE International Conference on Computer Vision Workshops (ICCV Workshops)},
  pages={689--690},
  year={2011},
  organization={IEEE}
}

@inproceedings{rahman2014cutset,
  title={Cutset networks: A simple, tractable, and scalable approach for improving the accuracy of chow-liu trees},
  author={Rahman, Tahrima and Kothalkar, Prasanna and Gogate, Vibhav},
  booktitle={Machine Learning and Knowledge Discovery in Databases: European Conference, ECML PKDD 2014, Nancy, France, September 15-19, 2014. Proceedings, Part II 14},
  pages={630--645},
  year={2014},
  organization={Springer}
}

@article{circuitem,
author = {Desana, Mattia and Schnörr, Christoph},
year = {2016},
month = {04},
pages = {},
title = {Expectation Maximization for Sum-Product Networks as Exponential Family Mixture Models},
doi = {10.48550/arXiv.1604.07243}
}

@article{mnist,
  title={Gradient-based learning applied to document recognition},
  author={Yann LeCun and Léon Bottou and Yoshua Bengio and and Patrick Haffner},
  journal={Proceedings of the IEEE},
  year={1998},
  publisher={IEEE}
}

@inproceedings{hclt,
 author = {Liu, Anji and Van den Broeck, Guy},
 booktitle = {Advances in Neural Information Processing Systems},
 editor = {M. Ranzato and A. Beygelzimer and Y. Dauphin and P.S. Liang and J. Wortman Vaughan},
 pages = {3558--3570},
 publisher = {Curran Associates, Inc.},
 title = {Tractable Regularization of Probabilistic Circuits},
 url = {https://proceedings.neurips.cc/paper_files/paper/2021/file/1d0832c4969f6a4cc8e8a8fffe083efb-Paper.pdf},
 volume = {34},
 year = {2021}
}


@misc{liu2024scalingtractableprobabilisticcircuits,
      title={Scaling Tractable Probabilistic Circuits: A Systems Perspective}, 
      author={Anji Liu and Kareem Ahmed and Guy Van den Broeck},
      year={2024},
      eprint={2406.00766},
      archivePrefix={arXiv},
      primaryClass={cs.LG},
      url={https://arxiv.org/abs/2406.00766}, 
}

@misc{gurobi,
  author = {{Gurobi Optimization, LLC}},
  title = {{Gurobi Optimizer Reference Manual}},
  year = 2024,
  url = "https://www.gurobi.com"
}

@article{Rahimian_2022,
   title={Frameworks and Results in Distributionally Robust Optimization},
   volume={3},
   ISSN={2777-5860},
   url={http://dx.doi.org/10.5802/ojmo.15},
   DOI={10.5802/ojmo.15},
   journal={Open Journal of Mathematical Optimization},
   publisher={Cellule MathDoc/Centre Mersenne},
   author={Rahimian, Hamed and Mehrotra, Sanjay},
   year={2022},
   month=jul, pages={1–85} }


@inproceedings{dasgupta2008hard,
  title={The hardness of k-means clustering},
  author={Dasgupta, Sanjoy},
  year={2008},
  publisher={UCSD Technical Report},
  url={https://www.cs.columbia.edu/~verma/classes/uml/ref/clustering_kmeans_NPhard_dasgupta.pdf}
}

@inproceedings{mccormicklinear,
  title={Computability of global solutions to factorable nonconvex programs: Part I — Convex underestimating problems},
  author={McCormick, Garth},
  booktitle={Mathematical Programming},
  year={1976},
  volume={10},
  pages={147-175},
}

@book{otbook,
    author = {Cedric Villani},
    volume={338},
    title={Optimal Transport, old and new},
    year={2008},
    publisher={Springer Science \& Business}
}

@book{ctsknapsack,
    author = {Michael Goodrich, Roberto Tamassia},
    title = {Algorithm Design: Foundations, Analysis, and Internet Examples},
    year = {2002},
    publisher = {John Wiley \& Sons},
    pages={259-260}
}

@book{mongeot,
    author = {Gaspard Monge},
    title={M\'emoire sur la th\'eorie des d\'eblais et des remblais},
    year={1781},
    publisher={De l’Imprimerie Royale}
}

@article{kantorovichot,
    author = {Leonid Kantorovich},
    title={On the Transfer of Masses},
    year={1942},
    volume={37},
pages={227-229},
    journal={Dokl. Akad. Nauk. SSSR}
}


@InProceedings{arjovsky2017wgans,
  title = 	 {{W}asserstein Generative Adversarial Networks},
  author =       {Martin Arjovsky and Soumith Chintala and L{\'e}on Bottou},
  booktitle = 	 {Proceedings of the 34th International Conference on Machine Learning},
  pages = 	 {214--223},
  year = 	 {2017},
  editor = 	 {Precup, Doina and Teh, Yee Whye},
  volume = 	 {70},
  series = 	 {Proceedings of Machine Learning Research},
  month = 	 {06--11 Aug},
  publisher =    {PMLR},
  pdf = 	 {http://proceedings.mlr.press/v70/arjovsky17a/arjovsky17a.pdf},
  url = 	 {https://proceedings.mlr.press/v70/arjovsky17a.html}
}


@book{otbook2,
    author = {Cedric Villani},
    volume={58},
    title={Topics in Optimal Transportation Theory},
    year={2003},
booktitle={Graduate Studies in Mathematics},
    publisher={American Mathematical Society}
}

@inproceedings{routgenerative,
  title={Generative Modeling with Optimal Transport Maps},
  author={Rout, Litu and Korotin, Alexander and Burnaev, Evgeny},
  booktitle={International Conference on Learning Representations},
  year={2022}
}

@inproceedings{salimans2018improving,
  title={Improving GANs using optimal transport},
  author={Salimans, Tim and Metaxas, Dimitris and Zhang, Han and Radford, Alec},
  booktitle={6th International Conference on Learning Representations, ICLR 2018},
  year={2018}
}

@inproceedings{tolstikhin2018wasserstein,
  title={Wasserstein Auto-Encoders},
  author={Tolstikhin, Ilya and Bousquet, Olivier and Gelly, Sylvain and Schoelkopf, Bernhard},
  booktitle={International Conference on Learning Representations},
  year={2018}
}

@article{delon2020wasserstein,
  title={A Wasserstein-type distance in the space of Gaussian mixture models},
  author={Delon, Julie and Desolneux, Agnes},
  journal={SIAM Journal on Imaging Sciences},
  volume={13},
  number={2},
  pages={936--970},
  year={2020},
  publisher={SIAM}
}

@article{chen2018optimal,
  title={Optimal transport for Gaussian mixture models},
  author={Chen, Yongxin and Georgiou, Tryphon T and Tannenbaum, Allen},
  journal={IEEE Access},
  volume={7},
  pages={6269--6278},
  year={2018},
  publisher={IEEE}
}

@inproceedings{choi2017relaxing,
  title={On relaxing determinism in arithmetic circuits},
  author={Choi, Arthur and Darwiche, Adnan},
  booktitle={International Conference on Machine Learning},
  pages={825--833},
  year={2017},
  organization={PMLR}
}


@article{probcirc,
  author    = {Choi, YooJung and Vergari, Antonio and Van den Broeck, Guy},
  title     = {Probabilistic Circuits: A Unifying Framework for Tractable Probabilistic Models},
  month     = {oct},
  year      = {2020},
  url       = "http://starai.cs.ucla.edu/papers/ProbCirc20.pdf",
}

@inproceedings{liang2017CI,
  title={Towards Compact Interpretable Models : Shrinking of Learned Probabilistic Sentential Decision Diagrams},
  author={Liang, Yitao and Van den Broeck, Guy},
  year={2017},
booktitle = {IJCAI 2017 Workshop on Explainable Artificial Intelligence (XAI)},
url={http://starai.cs.ucla.edu/papers/LiangXAI17.pdf}
}

@inproceedings{vergari2021atlas,
    title = {{A Compositional Atlas of Tractable Circuit Operations for Probabilistic Inference}},
    year = {2021},
    booktitle = {Advances in Neural Information Processing Systems},
    author = {Vergari, Antonio and Choi, YooJung and Liu, Anji and Teso, Stefano and den Broeck, Guy},
    pages = {13189--13201},
    volume = {34}
}

@book{meinel1998algorithms,
  title={Algorithms and Data Structures in VLSI Design: OBDD-foundations and applications},
  author={Meinel, Christoph and Theobald, Thorsten},
  year={1998},
  publisher={Springer Science \& Business Media}
}