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Distance geometry and data science

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  • Leo Liberti

    (LIX CNRS, Ecole Polytechnique, Institut Polytechnique de Paris)

Abstract

Data are often represented as graphs. Many common tasks in data science are based on distances between entities. While some data science methodologies natively take graphs as their input, there are many more that take their input in vectorial form. In this survey, we discuss the fundamental problem of mapping graphs to vectors, and its relation with mathematical programming. We discuss applications, solution methods, dimensional reduction techniques, and some of their limits. We then present an application of some of these ideas to neural networks, showing that distance geometry techniques can give competitive performance with respect to more traditional graph-to-vector mappings.

Suggested Citation

  • Leo Liberti, 2020. "Distance geometry and data science," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(2), pages 271-339, July.
    • Handle: RePEc:spr:topjnl:v:28:y:2020:i:2:d:10.1007_s11750-020-00563-0
    DOI: 10.1007/s11750-020-00563-0
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    References listed on IDEAS

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    1. Vach, Werner & Ro[ss]ner, Reinhard & Schumacher, Martin, 1996. "Neural networks and logistic regression: Part II," Computational Statistics & Data Analysis, Elsevier, vol. 21(6), pages 683-701, June.
    2. S. Lehmann & L. K. Hansen, 2007. "Deterministic modularity optimization," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 60(1), pages 83-88, November.
    3. Edoardo Amaldi & Leo Liberti & Francesco Maffioli & Nelson Maculan, 2009. "Edge-swapping algorithms for the minimum fundamental cycle basis problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 69(2), pages 205-233, May.
    4. Ky Vu & Pierre-Louis Poirion & Leo Liberti, 2018. "Random Projections for Linear Programming," Mathematics of Operations Research, INFORMS, vol. 43(4), pages 1051-1071, November.
    5. Douglas S. Gonçalves & Antonio Mucherino & Carlile Lavor & Leo Liberti, 2017. "Recent advances on the interval distance geometry problem," Journal of Global Optimization, Springer, vol. 69(3), pages 525-545, November.
    6. Antonio Mucherino & Carlile Lavor & Leo Liberti & Nelson Maculan, 2012. "The Discretizable Molecular Distance Geometry Problem," Post-Print hal-00756940, HAL.
    7. Carlile Lavor & Leo Liberti & Antonio Mucherino, 2013. "The interval Branch-and-Prune algorithm for the discretizable molecular distance geometry problem with inexact distances," Journal of Global Optimization, Springer, vol. 56(3), pages 855-871, July.
    8. Antonio Mucherino & Carlile Lavor & Leo Liberti, 2012. "The Discretizable Distance Geometry Problem," Post-Print hal-00756943, HAL.
    9. Carlile Lavor & Leo Liberti & Nelson Maculan & Antonio Mucherino, 2012. "The discretizable molecular distance geometry problem," Computational Optimization and Applications, Springer, vol. 52(1), pages 115-146, May.
    10. Leo Liberti & Fabrizio Marinelli, 2014. "Mathematical programming: Turing completeness and applications to software analysis," Journal of Combinatorial Optimization, Springer, vol. 28(1), pages 82-104, July.
    11. G. Xu & S. Tsoka & L. G. Papageorgiou, 2007. "Finding community structures in complex networks using mixed integer optimisation," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 60(2), pages 231-239, November.
    12. Schumacher, Martin & Ro[ss]ner, Reinhard & Vach, Werner, 1996. "Neural networks and logistic regression: Part I," Computational Statistics & Data Analysis, Elsevier, vol. 21(6), pages 661-682, June.
    13. Brendan O’Donoghue & Eric Chu & Neal Parikh & Stephen Boyd, 2016. "Conic Optimization via Operator Splitting and Homogeneous Self-Dual Embedding," Journal of Optimization Theory and Applications, Springer, vol. 169(3), pages 1042-1068, June.
    14. Luca Mencarelli & Youcef Sahraoui & Leo Liberti, 2017. "A multiplicative weights update algorithm for MINLP," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 5(1), pages 31-86, March.
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    Cited by:

    1. Emilio Carrizosa & Cristina Molero-Río & Dolores Romero Morales, 2021. "Mathematical optimization in classification and regression trees," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 29(1), pages 5-33, April.
    2. Emilio Carrizosa & Dolores Romero Morales, 2024. "Guest editorial to the Special Issue on Machine Learning and Mathematical Optimization in TOP-Transactions in Operations Research," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 32(3), pages 351-353, October.
    3. Maurizio Bruglieri & Roberto Cordone & Leo Liberti, 2022. "Maximum feasible subsystems of distance geometry constraints," Journal of Global Optimization, Springer, vol. 83(1), pages 29-47, May.

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