The Topological Field Theory of Data: a program towards a novel strategy for data mining through data language

This paper aims to challenge the current thinking in IT for the 'Big Data' question, proposing - almost verbatim, with no formulas - a program aiming to construct an innovative methodology to perform data analytics in a way that returns an automaton as a recognizer of the data language: a Field Theory of Data. We suggest to build, directly out of probing data space, a theoretical framework enabling us to extract the manifold hidden relations (patterns) that exist among data, as correlations depending on the semantics generated by the mining context. The program, that is grounded in the recent innovative ways of integrating data into a topological setting, proposes the realization of a Topological Field Theory of Data, transferring and generalizing to the space of data notions inspired by physical (topological) field theories and harnesses the theory of formal languages to define the potential semantics necessary to understand the emerging patterns.

[1]  Peter J. Denning,et al.  The science in computer science , 2013, CACM.

[2]  Lee Mosher,et al.  Mapping class groups are automatic , 1995 .

[3]  A. Grothendieck,et al.  Théorie des Topos et Cohomologie Etale des Schémas , 1972 .

[4]  Harry B. Hunt,et al.  Errata for the paper "Predecessor existence problems for finite discrete dynamical systems" [TCS 386 (1-2) (2007) 3-37] , 2008, Theor. Comput. Sci..

[5]  Raoul Bott,et al.  The Yang-Mills equations over Riemann surfaces , 1983, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[6]  V. De Silva,et al.  A Weak Definition of Delaunay Triangulation , 2003 .

[7]  Hermann Haken,et al.  Information and Self-Organization: A Macroscopic Approach to Complex Systems , 2010 .

[8]  G. Ermentrout.  Dynamic patterns: The self-organization of brain and behavior , 1997 .

[9]  Morse theory of the moment map for representations of quivers , 2008, 0807.4734.

[10]  F. Kirwan,et al.  Yang–Mills theory and Tamagawa numbers: the fascination of unexpected links in mathematics , 2008, 0801.4733.

[11]  B. Auchmann,et al.  A geometrically defined discrete hodge operator on simplicial cells , 2006, IEEE Transactions on Magnetics.

[12]  Colin Rourke,et al.  Block bundles: I , 1968 .

[13]  Francesco Vaccarino,et al.  Topological Strata of Weighted Complex Networks , 2013, PloS one.

[14]  Hausdorff Convergence of Riemannian Manifolds and Its Applications , 1990 .

[15]  Gunnar E. Carlsson,et al.  Topology and data , 2009 .

[16]  Mikhael Gromov.  Structures métriques pour les variétés riemanniennes , 1981 .

[17]  M. S. Narasimhan,et al.  On the cohomology groups of moduli spaces of vector bundles on curves , 1975 .

[18]  Ingo Runkel,et al.  Topological and conformal field theory as Frobenius algebras , 2005, math/0512076.

[19]  R. L. Mills,et al.  Conservation of Isotopic Spin and Isotopic Gauge Invariance , 1954 .

[20]  W. Thurston,et al.  Three-Dimensional Geometry and Topology, Volume 1: Volume 1 , 1997 .

[21]  P. D. Val.  The Theory and Applications of Harmonic Integrals , 1941, Nature.

[22]  E. Mark Gold,et al.  Complexity of Automaton Identification from Given Data , 1978, Inf. Control..

[23]  James P. Crutchfield,et al.  Computational Mechanics: Pattern and Prediction, Structure and Simplicity , 1999, ArXiv.

[24]  Edward Witten,et al.  Quantum field theory and the Jones polynomial , 1989 .

[25]  S. Barry Cooper.  Incomputability after Alan Turing , 2013, ArXiv.

[26]  W. Crawley-Boevey.  Geometry of the Moment Map for Representations of Quivers , 2001, Compositio Mathematica.

[27]  Emanuela Merelli,et al.  Topology driven modeling: the IS metaphor , 2014, Natural Computing.

[28]  Vinton G. Cerf.  Where is the science in computer science? , 2012, CACM.

[29]  G. Petri,et al.  Homological scaffolds of brain functional networks , 2014, Journal of The Royal Society Interface.

[30]  Lectures on the Langlands program and conformal field theory , 2005, hep-th/0512172.

[31]  Christos H. Papadimitriou,et al.  Elements of the Theory of Computation , 1997, SIGA.

[32]  M. Bridson,et al.  Formal language theory and the geometry of 3-manifolds , 1996 .

[33]  Jos C. M. Baeten,et al.  A characterization of regular expressions under bisimulation , 2007, JACM.

[34]  W. V. Hodge,et al.  The Theory and Applications of Harmonic Integrals , 1941 .

[35]  Graeme Wilkin.  HOMOTOPY GROUPS OF MODULI SPACES OF STABLE QUIVER REPRESENTATIONS , 2009, 0901.4156.

[36]  Is quantum simulation of turbulence within reach , 2014 .

[37]  Sarah Rees.  Hairdressing in groups: a survey of combings and formal languages , 1998 .

[38]  Harry B. Hunt,et al.  Predecessor existence problems for finite discrete dynamical systems , 2007, Theor. Comput. Sci..

[39]  L. Vietoris.  Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen , 1927 .

[40]  Don Zagier,et al.  Elementary aspects of the Verlinde formula and of the Harder-Narasimhan-Atiyah-Bott formula , 1996 .

[41]  Herbert Edelsbrunner,et al.  Computational Topology - an Introduction , 2009 .

[42]  Oliver Knill,et al.  The Dirac operator of a graph , 2013, ArXiv.

[43]  Demian Battaglia,et al.  Quantum-like diffusion over discrete sets , 2003 .

[44]  Eduard Čech,et al.  Théorie générale de l'homologie dans un espace quelconque , 1932 .

[45]  A. Papadopoulos,et al.  Simplicial actions of mapping class groups , 2012 .

[46]  P. Diaconis,et al.  Gibbs sampling, exponential families and orthogonal polynomials , 2008, 0808.3852.