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An overview of distance and similarity functions for structured data

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Abstract

The notions of distance and similarity play a key role in many machine learning approaches, and artificial intelligence in general, since they can serve as an organizing principle by which individuals classify objects, form concepts and make generalizations. While distance functions for propositional representations have been thoroughly studied, work on distance functions for structured representations, such as graphs, frames or logical clauses, has been carried out in different communities and is much less understood. Specifically, a significant amount of work that requires the use of a distance or similarity function for structured representations of data usually employs ad-hoc functions for specific applications. Therefore, the goal of this paper is to provide an overview of this work to identify connections between the work carried out in different areas and point out directions for future work.

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Notes

  1. Notice that in the description logics notation, subsumption is written in the reverse order since it is seen as “set inclusion” of their interpretations. Here, \(x_1 \sqsubseteq x_2\) means that \(x_1\) is more general than \(x_2\), while in description logics it has the opposite meaning.

  2. Interestingly, the Weisfeiler–Lehman test is related to the expressive power of Graph Neural Networks (discussed in Sect. 3.5), as it has been shown that a some classes of GNNs are at least as powerful as the Weisfeiler–Lehman in detecting graph isomorphism (Xu et al. 2018).

References

  • Aamodt A, Plaza E (1994) Case-based reasoning: foundational issues, methodological variations, and system approaches. Artif Intell Commun 7(1):39–59

    Google Scholar 

  • Abu-Khzam FN, Samatova NF, Rizk MA, Langston MA (2007) The maximum common subgraph problem: faster solutions via vertex cover. In: IEEE/ACS international conference on computer systems and applications, 2007. AICCSA’07. IEEE, pp 367–373

  • Agrawal R, Faloutsos C, Swami A (1993) Efficient similarity search in sequence databases. In: International conference on foundations of data organization and algorithms. Springer, pp 69–84

  • Aha DW, Kibler D, Albert MK (1991) Instance-based learning algorithms. Mach Learn 6(1):37–66

    Google Scholar 

  • Almohamad H, Duffuaa SO (1993) A linear programming approach for the weighted graph matching problem. IEEE Trans Pattern Anal Mach Intell 15(5):522–525

    Google Scholar 

  • Armengol E, Plaza E (2001) Similarity assessment for relational cbr. In: International conference on case-based reasoning. Springer, pp 44–58

  • Armengol E, Plaza E (2002) Similarity of structured cases in CBR. In: Proceedings from the CCIA held in Castellon, Spain

  • Assali AA, Lenne D, Debray B (2009) Case retrieval in ontology-based cbr systems. In: Annual conference on artificial intelligence. Springer, pp 564–571

  • Baader F, Calvanese D, McGuinness DL, Nardi D, Patel-Schneider PF (eds) (2003) The description logic handbook: theory, implementation, and applications. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Baader F, Horrocks I, Sattler U (2005) Description logics as ontology languages for the semantic web. In: Hutter D, Stephan W (eds) Mechanizing mathematical reasoning. Springer, pp 228–248

    Google Scholar 

  • Babai L (2018) Groups, graphs, algorithms: the graph isomorphism problem. In: Proceedings of international congress of mathematicians 2018

  • Badea L, Nienhuys-Cheng SH (1999) A refinement operator for description logics. In: Cussens J, Frisch A (eds) Inductive logic programming, no. 1866 in Lecture notes in computer science. Springer, pp 40–59

  • Battaglia PW, Hamrick JB, Bapst V, Sanchez-Gonzalez A, Zambaldi V, Malinowski M, Tacchetti A, Raposo D, Santoro A, Faulkner R et al (2018) Relational inductive biases, deep learning, and graph networks. arXiv:180601261

  • Bellet A, Habrard A, Sebban M (2012) Good edit similarity learning by loss minimization. Mach Learn 89(1–2):5–35

    MathSciNet  MATH  Google Scholar 

  • Bellet A, Habrard A, Sebban M (2013) A survey on metric learning for feature vectors and structured data. arXiv:13066709

  • Bergmann R, Stahl A (1998) Similarity measures for object-oriented case representations. In: Advances in case-based reasoning, pp 25–36

  • Bergmann R, Gil Y (2014) Similarity assessment and efficient retrieval of semantic workflows. Inf Syst 40:115–127

    Google Scholar 

  • Bergmann R, Kolodner J, Plaza E (2005) Representation in case-based reasoning. Knowl Eng Rev 20(3):209–213

    Google Scholar 

  • Bille P (2005) A survey on tree edit distance and related problems. Theor Comput Sci 337(1):217–239

    MathSciNet  MATH  Google Scholar 

  • Bisson G (1990) Kbg: a knowledge based generalizer. In: Porter B, Mooney R (eds) Machine learning proceedings 1990. Elsevier, Amsterdam, pp 9–15

    Google Scholar 

  • Bisson G (1992) Learing in FOL with a similarity measure. In: Proceedings of AAAI, vol 1992, pp 82–87

  • Borgida A, Walsh TJ, Hirsh H et al (2005) Towards measuring similarity in description logics. Descr Log 147

  • Bournaud I, Courtine M, Jean-Daniel Z (2002) Propositionalization for clustering symbolic relational descriptions. In: International conference on inductive logic programming. Springer, pp 1–16

  • Bunke H (1997) On a relation between graph edit distance and maximum common subgraph. Pattern Recogn Lett 18(8):689–694

    Google Scholar 

  • Bunke H (1999) Error correcting graph matching: on the influence of the underlying cost function. IEEE Trans Pattern Anal Mach Intell 21(9):917–922

    Google Scholar 

  • Bunke H (2000) Graph matching: theoretical foundations, algorithms, and applications. Proc Vis Interface 2000:82–88

    Google Scholar 

  • Bunke H, Shearer K (1998) A graph distance metric based on the maximal common subgraph. Pattern Recogn Lett 19(3):255–259

    MATH  Google Scholar 

  • Carpenter B (1992) The logic of typed feature structures. Cambridge University Press, New York

    MATH  Google Scholar 

  • Champin PA, Solnon C (2003) Measuring the similarity of labeled graphs. In: International conference on case-based reasoning, ICCBR. Springer

  • Chen PPS (1988) The entity-relationship model—toward a unified view of data. Readings in artificial intelligence and databases. Elsevier, Amsterdam, pp 98–111

    Google Scholar 

  • Church A (1940) A formulation of the simple theory of types. J Symb Log 5(2):56–68

    MathSciNet  MATH  Google Scholar 

  • Cilibrasi R, Vitányi PM (2005) Clustering by compression. IEEE Trans Inf Theory 51(4):1523–1545

    MathSciNet  MATH  Google Scholar 

  • Collins M, Duffy N (2002) Convolution kernels for natural language. In: Becker S, Thrun S, Obermayer K (eds) Advances in neural information processing systems. Vancouver, Canada, pp 625–632

    Google Scholar 

  • Conte D, Foggia P, Sansone C, Vento M (2004) Thirty years of graph matching in pattern recognition. Int J Pattern Recogn Artif Intell 18(03):265–298

    Google Scholar 

  • Cover T, Hart P (1967) Nearest neighbor pattern classification. IEEE Trans Inf Theory 13(1):21–27

    MATH  Google Scholar 

  • d’Amato C, Fanizzi N, Esposito F (2006) A dissimilarity measure for alc concept descriptions. In: Proceedings of the 2006 ACM symposium on applied computing. ACM, pp 1695–1699

  • d’Amato C, Staab S, Fanizzi N (2008) On the influence of description logics ontologies on conceptual similarity. In: Proceedings of the 16th international conference on knowledge engineering. Lecture notes in computer science, vol 5268. Springer, pp 48–63

  • Datar M, Immorlica N, Indyk P, Mirrokni VS (2004) Locality-sensitive hashing scheme based on p-stable distributions. In: Proceedings of the twentieth annual symposium on computational geometry. ACM, pp 253–262

  • de Vries GKD, de Rooij S (2015) Substructure counting graph kernels for machine learning from rdf data. Web Semant Sci Serv Agents World Wide Web 35:71–84

    Google Scholar 

  • Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the EM algorithm. Journal R Stat Soc Ser B (Methodol) 39:1–38

    MathSciNet  MATH  Google Scholar 

  • Dobrushin RL (1970) Prescribing a system of random variables by conditional distributions. Theory Probab Appl 15(3):458–486

    MathSciNet  MATH  Google Scholar 

  • Doyle PG, Snell JL (1984) Random walks and electric networks, vol 22. American Mathematical Society, Providence

    MATH  Google Scholar 

  • Emde W, Wettschereck D (1996) Relational instance based learning. In: Saitta L (ed) Machine learning—proceedings 13th international conference on machine learning. Morgan Kaufmann Publishers, pp 122–130

  • Emele MC, Zajac R (1990) Typed unification grammars. In: Proceedings of the 13th conference on computational linguistics, vol 3. Association for Computational Linguistics, pp 293–298

  • Emmert-Streib F, Dehmer M, Shi Y (2016) Fifty years of graph matching, network alignment and network comparison. Inf Sci 346:180–197

    MathSciNet  MATH  Google Scholar 

  • Ester M, Kriegel HP, Sander J, Xu X et al (1996) A density-based algorithm for discovering clusters in large spatial databases with noise. Kdd 96:226–231

    Google Scholar 

  • Falkenhainer B, Forbus KD, Gentner D (1989) The structure-mapping engine: algorithm and examples. Artif intell 41(1):1–63

    MATH  Google Scholar 

  • Fanizzi N, d’Amato C (2006) A declarative kernel for $\cal{ALC}$ concept descriptions. In: International symposium on methodologies for intelligent systems. Springer, pp 322–331

  • Fanizzi N, d’Amato C, Esposito F (2008) Learning with kernels in description logics. In: Zelezny F, Lavrac N (eds) Inductive logic programming. Springer, pp 210–225

  • Fernández ML, Valiente G (2001) A graph distance metric combining maximum common subgraph and minimum common supergraph. Pattern Recognition Letters 22(6):753–758

    MATH  Google Scholar 

  • French RM (2002) The computational modeling of analogy-making. Trends Cogn Sci 6(5):200–205

    Google Scholar 

  • Gao X, Xiao B, Tao D, Li X (2010) A survey of graph edit distance. Pattern Anal Appl 13(1):113–129

    MathSciNet  MATH  Google Scholar 

  • Gärtner T (2003) A survey of kernels for structured data. ACM SIGKDD Explor Newsl 5(1):49–58

    Google Scholar 

  • Gärtner T, Lloyd JW, Flach PA (2002) Kernels for structured data. Springer, Berlin

    MATH  Google Scholar 

  • Gärtner T, Flach P, Wrobel S (2003) On graph kernels: Hardness results and efficient alternatives. In: Schölkopf B, Warmuth MK (eds) Learning theory and kernel machines. Springer, Berlin, pp 129–143

    MATH  Google Scholar 

  • Gärtner T, Lloyd JW, Flach PA (2004) Kernels and distances for structured data. Mach Learn 57(3):205–232

    MATH  Google Scholar 

  • Gentner D (1983) Structure-mapping: a theoretical framework for analogy. Cogn Sci 7(2):155–170

    Google Scholar 

  • Getoor L, Taskar B (2007) Introduction to statistical relational learning. MIT Press, Cambridge

    MATH  Google Scholar 

  • Göker MH, Roth-Berghofer T (1999) The development and utilization of the case-based help-desk support system homer. Eng Appl Artif Intell 12(6):665–680

    Google Scholar 

  • Goldstone RL, Medin DL, Gentner D (1991) Relational similarity and the nonindependence of features in similarity judgments. Cogn Psychol 23(2):222–262

    Google Scholar 

  • Gollery M (2005) Bioinformatics: sequence and genome analysis. Clin Chem 51(11):2219–2219

    Google Scholar 

  • Golub GH, Van Loan CF (2012) Matrix computations, vol 3. JHU Press, Baltimore

    MATH  Google Scholar 

  • González-Calero PA, Díaz-Agudo B, Gómez-Albarrán M et al (1999) Applying dls for retrieval in case-based reasoning. In: In Proceedings of the 1999 description logics workshop (Dl’99). Linkopings Universitet, Citeseer

  • Haussler D (1999) Convolution kernels on discrete structures. Technical report, Department of Computer Science, University of California at Santa Cruz

  • Hearst MA, Dumais ST, Osuna E, Platt J, Scholkopf B (1998) Support vector machines. IEEE Intell Syst Their Appl 13(4):18–28

    Google Scholar 

  • Heckerman D, Meek C, Koller D (2007) Probabilistic entity-relationship models, prms, and plate models. In: Getoor L, Taskar B (eds) Introduction to statistical relational learning, MIT Press, pp 201–238

  • Holyoak KJ, Koh K (1987) Surface and structural similarity in analogical transfer. Mem Cogn 15(4):332–340

    Google Scholar 

  • Horváth T, Wrobel S, Bohnebeck U (2001) Relational instance-based learning with lists and terms. Mach Learn 43(1–2):53–80

    MATH  Google Scholar 

  • Horváth T, Gärtner T, Wrobel S (2004) Cyclic pattern kernels for predictive graph mining. In: Proceedings of the tenth ACM SIGKDD international conference on knowledge discovery and data mining. ACM, pp 158–167

  • Hu B, Kalfoglou Y, Alani H, Dupplaw D, Lewis P, Shadbolt N (2006) Semantic metrics. In: International conference on knowledge engineering and knowledge management. Springer, pp 166–181

  • Hutchinson A (1997) Metrics on terms and clauses. In: ECML ’97: proceedings of the 9th European conference on machine learning. Lecture notes in computer science, vol 1224. Springer, pp 138–145

  • Itakura F (1975) Minimum prediction residual principle applied to speech recognition. IEEE Trans Acoust Speech Signal Process 23(1):67–72

    Google Scholar 

  • Jaakkola T, Haussler D (1999) Exploiting generative models in discriminative classifiers. In: Solla SA, Leen TK, Müller K-R (eds) Advances in neural information processing systems. MIT Press, Denver, Colarado, pp 487–493

    Google Scholar 

  • Janowicz K (2006) Sim-dl: towards a semantic similarity measurement theory for the description logic $\cal{ALCNR}$ in geographic information retrieval. In: OTM confederated international conferences “on the move to meaningful internet systems. Springer, pp 1681–1692

  • Jeh G, Widom J (2002) Simrank: a measure of structural-context similarity. In: Proceedings of the eighth ACM SIGKDD international conference on knowledge discovery and data mining. ACM, pp 538–543

  • Jiang JJ, Conrath DW (1997) Semantic similarity based on corpus statistics and lexical taxonomy. arXiv:cmp-lg/9709008

  • Kalfoglou Y, Schorlemmer M (2003) Ontology mapping: the state of the art. Knowl Eng Rev 18(1):1–31

    MATH  Google Scholar 

  • Kashima H, Koyanagi T (2002) Kernels for semi-structured data. ICML 2:291–298

    Google Scholar 

  • Kashima H, Tsuda K, Inokuchi A (2003) Marginalized kernels between labeled graphs. In: Proceedings of the twentieth international conference (ICML 2003). AAAI Press, pp 321–328

  • Katz L (1953) A new status index derived from sociometric analysis. Psychometrika 18(1):39–43

    MATH  Google Scholar 

  • Kaufman L, Rousseeuw P (1987) Clustering by means of medoids. North-Holland, Amsterdam

    Google Scholar 

  • Keogh E, Kasetty S (2003) On the need for time series data mining benchmarks: a survey and empirical demonstration. Data Min Knowl Discov 7(4):349–371

    MathSciNet  Google Scholar 

  • Klein PN (1998) Computing the edit-distance between unrooted ordered trees. In: European symposium on algorithms. Springer, pp 91–102

  • Kok S, Domingos P (2007) Statistical predicate invention. In: Proceedings of the 24th international conference on machine learning. ACM, pp 433–440

  • Kolmogorov AN (1965) Three approaches to the quantitative definition of information. Probl Inf Transm 1(1):1–7

    MathSciNet  Google Scholar 

  • Kramer S, Lavrač N, Flach P (2001) Propositionalization approaches to relational data mining. In: Dzeroski S, Lavrac N (eds) Relational data mining. Springer, pp 262–291

  • Krieger HU, Schäfer U (1995) Efficient parameterizable type expansion for typed feature formalisms

  • Krogel MA, Rawles S, Železnỳ F, Flach PA, Lavrač N, Wrobel S (2003) Comparative evaluation of approaches to propositionalization. In: International conference on inductive logic programming. Springer, pp 197–214

  • Kulis B, et al (2013) Metric learning: a survey. Found Trends® Mach Learn 5(4):287–364

  • Kullback S, Leibler RA (1951) On information and sufficiency. Ann Math Stat 22(1):79–86

    MathSciNet  MATH  Google Scholar 

  • Larson J, Michalski RS (1977) Inductive inference of VL decision rules. SIGART Bull 63(63):38–44. https://doi.org/10.1145/1045343.1045369

    Article  Google Scholar 

  • Lavrac N, Dzeroski S (1994) Inductive logic programming. In: Fuchs NE, Gottlob G (eds) WLP. Springer, Berlion, pp 146–160

  • Lehmann J, Hitzler P (2007) A refinement operator based learning algorithm for the LC description logic. In: Blockeel H, Ramon J, Shavlik JW, Tadepalli P (eds) ILP. Lecture notes in computer science, vol 4894. Springer, Berlin, pp 147–160

  • Lehmann J, Haase C (2009) Ideal downward refinement in the EL description logic. In: Raedt LD (ed) ILP. Lecture notes in computer science, vol 5989. Springer, Berlin pp 73–87

  • Leishman D (1989) Analogy as a constrained partial correspondence over conceptual graphs. In: KR, pp 223–234

  • Levenshtein VI (1966) Binary codes capable of correcting deletions, insertions, and reversals. Sov Phys Dokl 10:707–710

    MathSciNet  Google Scholar 

  • Levi G (1973) A note on the derivation of maximal common subgraphs of two directed or undirected graphs. Calcolo 9(4):341

    MathSciNet  MATH  Google Scholar 

  • Li Y, Gu C, Dullien T, Vinyals O, Kohli P (2019) Graph matching networks for learning the similarity of graph structured objects. arXiv preprint arXiv:190412787

  • Lü L, Zhou T (2011) Link prediction in complex networks: a survey. Phys A Stat Mech Its Appl 390(6):1150–1170

    Google Scholar 

  • Luss R, d’Aspremont A (2008) Support vector machine classification with indefinite kernels. In: Advances in neural information processing systems, pp 953–960

  • Mahé P, Ueda N, Akutsu T, Perret JL, Vert JP (2005) Graph kernels for molecular structure–activity relationship analysis with support vector machines. J Chem Inf Model 45(4):939–951

    Google Scholar 

  • Manago M, Bergmann R, Conruyt N, Traphöner R, Pasley J, Le Renard J, Maurer F, Wess S, Althoff KD, Dumont S (1994) Casuel: a common case representation language. INRECA Consortium Available on the World-Wide Web at http://wwwagr informatik unikl de/bergmann/casuel/CASUEL toc2 4

  • Marteau PF (2009) Time warp edit distance with stiffness adjustment for time series matching. IEEE Trans Pattern Anal Mach Intell 31(2):306–318

    Google Scholar 

  • Miller GA (1995) Wordnet: a lexical database for english. Commun ACM 38(11):39–41

    Google Scholar 

  • Minsky M (1974) A framework for representing knowledge, MIT-AI LAboratory Memo 306

  • Mishne G, De Rijke M (2004) Source code retrieval using conceptual similarity. In: Coupling approaches, coupling media and coupling languages for information retrieval, pp 539–554

  • Mitchell TM (1980) The need for biases in learning generalizations. Department of Computer Science, Laboratory for Computer Science Research, Rutgers Univ, New Jersey

    Google Scholar 

  • Mitchell TM, Keller RM, Kedar-Cabelli ST (1986) Explanation-based generalization: a unifying view. Mach Learn 1(1):47–80

    Google Scholar 

  • Montani S, Leonardi G, Quaglini S, Cavallini A, Micieli G et al (2015) A knowledge-intensive approach to process similarity calculation. Expert Syst Appl 42(9):4207–4215

    Google Scholar 

  • Muggleton S, Lodhi H, Amini A, Sternberg MJ (2005) Support vector inductive logic programming. In: International conference on discovery science. Springer, pp 163–175

  • Munkres J (1957) Algorithms for the assignment and transportation problems. J Soc Ind Appl Math 5(1):32–38

    MathSciNet  MATH  Google Scholar 

  • Needleman SB, Wunsch CD (1970) A general method applicable to the search for similarities in the amino acid sequence of two proteins. J Mol Biol 48(3):443–453

    Google Scholar 

  • Neuhaus M, Bunke H (2006a) A convolution edit kernel for error-tolerant graph matching. In: 18th international conference on pattern recognition, 2006. ICPR 2006, vol 4. IEEE, pp 220–223

  • Neuhaus M, Bunke H (2006b) Edit distance-based kernel functions for structural pattern classification. Pattern Recogn 39(10):1852–1863

    MATH  Google Scholar 

  • Neuhaus M, Bunke H (2007) Automatic learning of cost functions for graph edit distance. Inf Sci 177(1):239–247

    MathSciNet  MATH  Google Scholar 

  • Ng AY, Jordan MI, Weiss Y et al (2002) On spectral clustering: analysis and an algorithm. Adv Neural Inf Process Syst 2:849–856

    Google Scholar 

  • Nienhuys-Cheng SH (1997) Distance between Herbrand interpretations: a measure for approximations to a target concept. In: Lavrac N, Dzeroski S (eds) Inductive logic programming. Springer, Berlin, pp 213–226

    MATH  Google Scholar 

  • Nikolentzos G, Meladianos P, Limnios S, Vazirgiannis M (2018) A degeneracy framework for graph similarity. Proc IJCAI 2018:2595–2601

    Google Scholar 

  • Ontañón S, Zhu J (2011) The SAM algorithm for analogy-based story generation. In: Seventh artificial intelligence and interactive digital entertainment conference

  • Ontanón S, Plaza E (2012) Similarity measures over refinement graphs. Mach Learn 87:57–92

    MathSciNet  MATH  Google Scholar 

  • Ontañón S, Shokoufandeh A (2016) Refinement-based similarity measures for directed labeled graphs. In: International conference on case-based reasoning. Springer, pp 311–326

  • Ontañón S, Montaña JL, Gonzalez AJ (2014) A dynamic-bayesian network framework for modeling and evaluating learning from observation. Expert Syst Appl 41(11):5212–5226

    Google Scholar 

  • Page L, Brin S, Motwani R, Winograd T (1999) The pagerank citation ranking: bringing order to the web. Technical report, Stanford InfoLab

  • Plaza E (1995) Cases as terms: a feature term approach to the structured representation of cases. In: International conference on case-based reasoning. Springer, pp 265–276

  • Plaza E, Armengol E, Ontañón S (2005) The explanatory power of symbolic similarity in case-based reasoning. Artif Intell Rev 24(2):145–161

    MATH  Google Scholar 

  • Plotkin GD (1970) A note on inductive generalization. In: Meltzer B, Michie D (eds) Machine intelligence, vol 5. Edinburgh University Press, Edinburgh, pp 153–163

    Google Scholar 

  • Pons P, Latapy M (2005) Computing communities in large networks using random walks. In: International symposium on computer and information sciences. Springer, pp 284–293

  • Poole J, Campbell J (1995) A novel algorithm for matching conceptual and related graphs. In: International conference on conceptual structures. Springer, pp 293–307

  • Rada R, Mili H, Bicknell E, Blettner M (1989) Development and application of a metric on semantic nets. IEEE Trans Syst Man Cybern 19(1):17–30

    Google Scholar 

  • Ralaivola L, Swamidass SJ, Saigo H, Baldi P (2005) Graph kernels for chemical informatics. Neural Netw 18(8):1093–1110

    Google Scholar 

  • Ramon J, Bruynooghe M (1998) A framework for defining distances between first-order logic objects. In: International conference on inductive logic programming. Springer, pp 271–280

  • Ramon J, Gärtner T (2003) Expressivity versus efficiency of graph kernels. In: Proceedings of the first international workshop on mining graphs, trees and sequences, pp 65–74

  • Ramoni M, Sebastiani P, Cohen P (2002) Bayesian clustering by dynamics. Mach Learn 47(1):91–121

    MATH  Google Scholar 

  • Read RC, Corneil DG (1977) The graph isomorphism disease. J Graph Theory 1(4):339–363

    MathSciNet  MATH  Google Scholar 

  • Resnik P (1995) Using information content to evaluate semantic similarity in a taxonomy. arXiv:cmp-lg/9511007

  • Resnik P et al (1999) Semantic similarity in a taxonomy: an information-based measure and its application to problems of ambiguity in natural language. J Artif Intell Res: JAIR 11:95–130

    MATH  Google Scholar 

  • Riesen K, Bunke H (2008) Iam graph database repository for graph based pattern recognition and machine learning. In: da Vitoria Lobo N, Kasparis T, Roli F, Kwok JT, Georgiopoulos M, Anagnostopoulos GC, Loog M (eds) Structural, syntactic, and statistical pattern recognition. Springer, Orlando, pp 287–297

  • Riesen K, Bunke H (2009) Approximate graph edit distance computation by means of bipartite graph matching. Image Vis Comput 27(7):950–959

    Google Scholar 

  • Rubner Y, Tomasi C, Guibas LJ (2000) The earth mover’s distance as a metric for image retrieval. Int J Comput Vis 40(2):99–121

    MATH  Google Scholar 

  • Sánchez-Ruiz AA, Ontañón S, González-Calero PA, Plaza E (2011) Measuring similarity in description logics using refinement operators. In: ICCBR, pp 289–303

  • Sánchez-Ruiz AA, Ontañón S, González-Calero PA, Plaza E (2016) Measuring similarity of individuals in description logics over the refinement space of conjunctive queries. J Intell Inf Syst 47(3):447–467

    Google Scholar 

  • Sanfeliu A, Fu KS (1983) A distance measure between attributed relational graphs for pattern recognition. IEEE Trans Syst Man Cybern 3:353–362

    MATH  Google Scholar 

  • Santini S, Jain R (1999) Similarity measures. IEEE Trans Pattern Anal Mach Intell 21(9):871–883

    Google Scholar 

  • Schaaf JW (1996) Fish and shrink. A next step towards efficient case retrieval in large scaled case bases. In: European workshop on advances in case-based reasoning. Springer, pp 362–376

  • Schädler K, Wysotzki F (1999) Comparing structures using a hopfield-style neural network. Appl Intell 11(1):15–30

    Google Scholar 

  • Sebag M (1997) Distance induction in first order logic. In: International conference on inductive logic programming. Springer, pp 264–272

  • Serra J, Arcos JL (2014) An empirical evaluation of similarity measures for time series classification. Knowl Based Syst 67:305–314

    Google Scholar 

  • Shapiro LG, Haralick RM (1981) Structural descriptions and inexact matching. IEEE Trans Pattern Anal Mach Intell 5:504–519

    Google Scholar 

  • Shervashidze N, Schweitzer P, van Leeuwen EJ, Mehlhorn K, Borgwardt KM (2011) Weisfeiler–Lehman graph kernels. J Mach Learn Res 12(1):2539–2561

    MathSciNet  MATH  Google Scholar 

  • Shieber SM (2003) An introduction to unification-based approaches to grammar. Microtome Publishing, New York

    MATH  Google Scholar 

  • Singhal A (2001) Modern information retrieval: a brief overview. IEEE Data Eng Bull 24(4):35–43

    Google Scholar 

  • Small H (1973) Co-citation in the scientific literature: a new measure of the relationship between two documents. J Assoc Inf Sci Technol 24(4):265–269

    MathSciNet  Google Scholar 

  • Smola AJ, Vishwanathan S (2003) Fast kernels for string and tree matching. In: Thrun S, Saul LK, Schölkopf B (eds) Advances in neural information processing systems. MIT Press, Vancouver, Canada, pp 585–592

    Google Scholar 

  • Sørensen TJ (1948) A method of establishing groups of equal amplitude in plant sociology based on similarity of species content. Kongelige Danske Videnskabernes Selskab 5(1–34):4–7

    Google Scholar 

  • Sowa JF (1979) Semantics of conceptual graphs. In: Proceedings of the 17th annual meeting on Association for Computational Linguistics. Association for Computational Linguistics, pp 39–44

  • Spielman DA (2010) Algorithms, graph theory, and linear equations in laplacian matrices. In: Proceedings of the international congress of mathematicians 2010 (ICM 2010) (In 4 Volumes) vol I: plenary lectures and ceremonies vols. II–IV: invited lectures. World Scientific, pp 2698–2722

  • Sussenguth EH (1964) Structure matching in information processing. Harvard University, Cambridge

    MATH  Google Scholar 

  • Tai KC (1979) The tree-to-tree correction problem. J ACM: JACM 26(3):422–433

    MathSciNet  MATH  Google Scholar 

  • Tsai WH, Fu KS (1979) Error-correcting isomorphisms of attributed relational graphs for pattern analysis. IEEE Trans Syst Man Cybern 9(12):757–768

    MATH  Google Scholar 

  • Tsuda K, Kin T, Asai K (2002) Marginalized kernels for biological sequences. Bioinformatics 18(Suppl 1):S268–S275

    Google Scholar 

  • Tversky A (1977) Features of similarity. Psychol Rev 84:327–352

    Google Scholar 

  • Umeyama S (1988) An eigendecomposition approach to weighted graph matching problems. IEEE Trans Pattern Anal Machine Intell 10(5):695–703

    MATH  Google Scholar 

  • Valls-Vargas J, Ontanón S, Zhu J (2014) Toward automatic character identification in unannotated narrative text. In: Seventh intelligent narrative technologies workshop

  • van der Laag PRJ, Nienhuys-Cheng SH (1998) Completeness and properness of refinement operators in inductive logic programming. J Log Program 34(3):201–225

    MathSciNet  MATH  Google Scholar 

  • Vert JP, Tsuda K, Schölkopf B (2004) A primer on kernel methods. Kernel Methods Comput Biol 47:35–70

    Google Scholar 

  • Wallis WD, Shoubridge P, Kraetz M, Ray D (2001) Graph distances using graph union. Pattern Recogn Lett 22(6):701–704

    MATH  Google Scholar 

  • Wang Y, Ishii N (1997) A method of similarity metrics for structured representations. Expert Syst Appl 12(1):89–100

    Google Scholar 

  • Weisfeiler B, Lehman AA (1968) A reduction of a graph to a canonical form and an algebra arising during this reduction. Nauchno-Technicheskaya Informatsia 2(9):12–16

    Google Scholar 

  • Welch TA (1984) A technique for high-performance data compression. Computer 6(17):8–19

    Google Scholar 

  • Wess S (1995) Fallbasiertes Problemlösen in wissensbasierten systemen zur entscheidungsunterst ützung und diagnostik

  • Wettschereck D, Aha DW, Mohri T (1997) A review and empirical evaluation of feature weighting methods for a class of lazy learning algorithms. Artif Intell Rev 11(1–5):273–314

    Google Scholar 

  • Wolpert DH (1996) The lack of a priori distinctions between learning algorithms. Neural Comput 8(7):1341–1390

    Google Scholar 

  • Wu Z, Palmer M (1994) Verbs semantics and lexical selection. In: Proceedings of the 32nd annual meeting on Association for Computational Linguistics. Association for Computational Linguistics, pp 133–138

  • Xu L, King I (2001) A pca approach for fast retrieval of structural patterns in attributed graphs. IEEE Trans Syst Man Cybern Part B (Cybern) 31(5):812–817

    Google Scholar 

  • Xu K, Hu W, Leskovec J, Jegelka S (2018) How powerful are graph neural networks? arXiv:181000826

  • Yang L, Jin R (2006) Distance metric learning: a comprehensive survey. Mich State Univ 2(2):4

    Google Scholar 

  • Zhang K (1989) The editing distance between trees: algorithms and applications. PhD thesis from the New York University

  • Zhang Z, Wang M, Xiang Y, Huang Y, Nehorai A (2018) Retgk: graph kernels based on return probabilities of random walks. In: Bengio S, Wallach HM, Larochelle H, Grauman K, Cesa-Bianchi N, Garnett R, Neural Information Processing Systems Conference (eds) Advances in neural information processing systems, Vancouver, Canada, pp 3964–3974

    Google Scholar 

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  1. Google Research, Mountain View, CA, 94043, USA

    Santiago Ontañón

  2. Drexel University, Philadelphia, PA, 19104, USA

    Santiago Ontañón

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Ontañón, S. An overview of distance and similarity functions for structured data. Artif Intell Rev 53, 5309–5351 (2020). https://doi.org/10.1007/s10462-020-09821-w

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