publication . Preprint . 2018

The iisignature library: efficient calculation of iterated-integral signatures and log signatures

Reizenstein, Jeremy; Graham, Benjamin;
Open Access English
  • Published: 22 Feb 2018
Abstract
Iterated-integral signatures and log signatures are vectors calculated from a path that characterise its shape. They come from the theory of differential equations driven by rough paths, and also have applications in statistics and machine learning. We present algorithms for efficiently calculating these signatures, and benchmark their performance. We release the methods as a Python package.
Subjects
free text keywords: Computer Science - Data Structures and Algorithms, Computer Science - Mathematical Software, Mathematics - Rings and Algebras
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19 references, page 1 of 2

[1] E. Alpaydin and Fevzi. Alimoglu. Pen-Based Recognition of Handwritten Digits Data Set. 1998. url: https://archive.ics.uci.edu/ml/datasets/Pen-Based+Recognition+of+Handwritten+ Digits (cit. on p. 3).

[2] Johannes Blümlein. “Algebraic relations between harmonic sums and associated quantities”. In: Computer Physics Communications 159.1 (2004), pp. 19-54. url: https://arxiv.org/abs/hepph/0311046 (cit. on p. 9).

[3] Fernando Casas and Ander Murua. “An efficient algorithm for computing the Baker-CampbellHausdorff series and some of its applications”. In: Journal of Mathematical Physics 50.3 (Mar. 2009), p. 033513. eprint: http://arxiv.org/abs/0810.2656 (cit. on pp. 6, 7).

[4] Fernando Casas and Ander Murua. The BCH formula and the symmetric BCH formula up to terms of degree 20. url: http://www.ehu.eus/ccwmuura/bch.html (cit. on p. 6).

[5] Kuo-Tsai Chen. “Integration of Paths - A Faithful Representation of Paths by Noncommutative Formal Power Series”. In: Transactions of the American Mathematical Society 89.2 (1958), pp. 395- 407. url: https://www.jstor.org/stable/1993193 (cit. on pp. 2, 5). [OpenAIRE]

[6] Ilya Chevyrev and Andrey Kormilitzin. “A Primer on the Signature Method in Machine Learning”. 2016. url: http://arxiv.org/abs/1603.03788 (cit. on p. 1).

[7] Joscha Diehl. “Rotation invariants of two dimensional curves based on iterated integrals”. 2013. url: http://arxiv.org/abs/1305.6883 (cit. on pp. 5, 17). [OpenAIRE]

[8] Mike Giles. Private communication via Terry Lyons. 2017 (cit. on p. 8).

[9] Benjamin Graham. “Sparse arrays of signatures for online character recognition”. 2013. url: http: //arxiv.org/abs/1308.0371 (cit. on p. 1).

[10] Marshall Hall. “A basis for free Lie rings and higher commutators in free groups”. In: (1950). url: http://www.ams.org/journals/proc/1950-001-05/S0002-9939-1950-0038336-7/home.html (cit. on p. 6).

[11] Terry Lyons et al. CoRoPa Computational Rough Paths (software library). 2010. url: http:// coropa.sourceforge.net/ (cit. on pp. 1, 6, 11, 12).

[12] Nicholas Nethercote, Robert Walsh, and Jeremy Fitzhardinge. “Building Workload Characterization Tools with Valgrind”. In: IEEE International Symposium on Workload Characterization (2006). url: http://valgrind.org/docs/iiswc2006.pdf (cit. on p. 16).

[13] Jeremy Reizenstein. “Calculation of Iterated-Integral Signatures and Log Signatures”. 2015. url: http://arxiv.org/abs/1712.02757 (cit. on pp. 6, 11).

[14] Christophe Reutenauer. Free Lie Algebras. 1994 (cit. on pp. 5, 6, 9).

[15] Anatoli Illarionovich Shirshov. “Subalgebras of free Lie algebras”. In: Mat. Sbornik N.S. (1953) (cit. on p. 6).

19 references, page 1 of 2
Abstract
Iterated-integral signatures and log signatures are vectors calculated from a path that characterise its shape. They come from the theory of differential equations driven by rough paths, and also have applications in statistics and machine learning. We present algorithms for efficiently calculating these signatures, and benchmark their performance. We release the methods as a Python package.
Subjects
free text keywords: Computer Science - Data Structures and Algorithms, Computer Science - Mathematical Software, Mathematics - Rings and Algebras
Download from
19 references, page 1 of 2

[1] E. Alpaydin and Fevzi. Alimoglu. Pen-Based Recognition of Handwritten Digits Data Set. 1998. url: https://archive.ics.uci.edu/ml/datasets/Pen-Based+Recognition+of+Handwritten+ Digits (cit. on p. 3).

[2] Johannes Blümlein. “Algebraic relations between harmonic sums and associated quantities”. In: Computer Physics Communications 159.1 (2004), pp. 19-54. url: https://arxiv.org/abs/hepph/0311046 (cit. on p. 9).

[3] Fernando Casas and Ander Murua. “An efficient algorithm for computing the Baker-CampbellHausdorff series and some of its applications”. In: Journal of Mathematical Physics 50.3 (Mar. 2009), p. 033513. eprint: http://arxiv.org/abs/0810.2656 (cit. on pp. 6, 7).

[4] Fernando Casas and Ander Murua. The BCH formula and the symmetric BCH formula up to terms of degree 20. url: http://www.ehu.eus/ccwmuura/bch.html (cit. on p. 6).

[5] Kuo-Tsai Chen. “Integration of Paths - A Faithful Representation of Paths by Noncommutative Formal Power Series”. In: Transactions of the American Mathematical Society 89.2 (1958), pp. 395- 407. url: https://www.jstor.org/stable/1993193 (cit. on pp. 2, 5). [OpenAIRE]

[6] Ilya Chevyrev and Andrey Kormilitzin. “A Primer on the Signature Method in Machine Learning”. 2016. url: http://arxiv.org/abs/1603.03788 (cit. on p. 1).

[7] Joscha Diehl. “Rotation invariants of two dimensional curves based on iterated integrals”. 2013. url: http://arxiv.org/abs/1305.6883 (cit. on pp. 5, 17). [OpenAIRE]

[8] Mike Giles. Private communication via Terry Lyons. 2017 (cit. on p. 8).

[9] Benjamin Graham. “Sparse arrays of signatures for online character recognition”. 2013. url: http: //arxiv.org/abs/1308.0371 (cit. on p. 1).

[10] Marshall Hall. “A basis for free Lie rings and higher commutators in free groups”. In: (1950). url: http://www.ams.org/journals/proc/1950-001-05/S0002-9939-1950-0038336-7/home.html (cit. on p. 6).

[11] Terry Lyons et al. CoRoPa Computational Rough Paths (software library). 2010. url: http:// coropa.sourceforge.net/ (cit. on pp. 1, 6, 11, 12).

[12] Nicholas Nethercote, Robert Walsh, and Jeremy Fitzhardinge. “Building Workload Characterization Tools with Valgrind”. In: IEEE International Symposium on Workload Characterization (2006). url: http://valgrind.org/docs/iiswc2006.pdf (cit. on p. 16).

[13] Jeremy Reizenstein. “Calculation of Iterated-Integral Signatures and Log Signatures”. 2015. url: http://arxiv.org/abs/1712.02757 (cit. on pp. 6, 11).

[14] Christophe Reutenauer. Free Lie Algebras. 1994 (cit. on pp. 5, 6, 9).

[15] Anatoli Illarionovich Shirshov. “Subalgebras of free Lie algebras”. In: Mat. Sbornik N.S. (1953) (cit. on p. 6).

19 references, page 1 of 2
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