{"title": "Deep Signature Transforms", "book": "Advances in Neural Information Processing Systems", "page_first": 3105, "page_last": 3115, "abstract": "The signature is an infinite graded sequence of statistics known to characterise a stream of data up to a negligible equivalence class. It is a transform which has previously been treated as a fixed feature transformation, on top of which a model may be built. We propose a novel approach which combines the advantages of the signature transform with modern deep learning frameworks. By learning an augmentation of the stream prior to the signature transform, the terms of the signature may be selected in a data-dependent way. More generally, we describe how the signature transform may be used as a layer anywhere within a neural network. In this context it may be interpreted as a pooling operation. We present the results of empirical experiments to back up the theoretical justification. Code available at \\texttt{github.com/patrick-kidger/Deep-Signature-Transforms}.", "full_text": "DeepSignatureTransformsPatricBonnier1,\u2217PatrickKidger1,2,\u2217ImanolPerezArribas1,2,\u2217CristopherSalvi1,2,\u2217TerryLyons1,21MathematicalInstitute,UniversityofOxford2TheAlanTuringInstitute,BritishLibrary{bonnier,kidger,perez,salvi,tlyons}@maths.ox.ac.ukAbstractThesignatureisanin\ufb01nitegradedsequenceofstatisticsknowntocharacteriseastreamofdatauptoanegligibleequivalenceclass.Itisatransformwhichhaspreviouslybeentreatedasa\ufb01xedfeaturetransformation,ontopofwhichamodelmaybebuilt.Weproposeanovelapproachwhichcombinestheadvantagesofthesignaturetransformwithmoderndeeplearningframeworks.Bylearninganaugmentationofthestreampriortothesignaturetransform,thetermsofthesignaturemaybeselectedinadata-dependentway.Moregenerally,wedescribehowthesignaturetransformmaybeusedasalayeranywherewithinaneuralnetwork.Inthiscontextitmaybeinterpretedasapoolingoperation.Wepresenttheresultsofempiricalexperimentstobackupthetheoreticaljusti\ufb01cation.Codeavailableatgithub.com/patrick-kidger/Deep-Signature-Transforms.1Introduction1.1Whatisthesignaturetransform?Whendataisorderedsequentiallythenitcomeswithanaturalpath-likestructure:thedatamaybethoughtofasadiscretisationofapathX:[0,1]\u2192V,whereVissomeBanachspace.InpracticeweshallalwaystakeV=Rdforsomed\u2208N.ForexamplethechangingairpressureataparticularlocationmaybethoughtofasapathinR;themotionofapenonpapermaybethoughtofasapathinR2;thechangeswithin\ufb01nancialmarketsmaybethoughtofasapathinRd,withdpotentiallyverylarge.Givenapath,wemayde\ufb01neitssignature,whichisacollectionofstatisticsofthepath.Themapfromapathtoitssignatureiscalledthesignaturetransform.De\ufb01nition1.1.Letx=(x1,...,xn),wherexi\u2208Rd.Letf=(f1,...,fd):[0,1]\u2192Rdbecontinuous,suchthatf(i\u22121n\u22121)=xi,andlinearontheintervalsinbetween.Thenthesignatureofxisde\ufb01nedasthecollectionofiteratedintegrals2Sig(x)=\uf8eb\uf8ec\uf8ed\uf8eb\uf8edZ\u00b7\u00b7\u00b7Z0