Introduction to signatures Nikolas Tapia NTNU Trondheim Feb. 26, - PowerPoint PPT Presentation

Introduction to signatures Nikolas Tapia NTNU Trondheim Feb. 26, 2019 @ Magic 2019, Ilsetra N. Tapia (NTNU) Introduction to signatures Feb. 26, 2019 @ Magic 2019, Ilsetra 1 / 29

Goals Goals N. Tapia (NTNU) Introduction to signatures Feb. 26, 2019 @ Magic 2019, Ilsetra 2 / 29

Goals 1 Signatures 1 For paths on � d 2 For paths on a Lie group 3 Practical questions 2 Rough paths 1 Geometric rough paths 2 Branched rough paths 3 Problem of existence N. Tapia (NTNU) Introduction to signatures Feb. 26, 2019 @ Magic 2019, Ilsetra 3 / 29

Signatures Signatures N. Tapia (NTNU) Introduction to signatures Feb. 26, 2019 @ Magic 2019, Ilsetra 4 / 29

Signatures Signatures were studied by K.-T. Chen in the 50’s in order to clasify smooth curves on manifolds. They were later generalized by T. Lyons by the end of the 90’s to what he called rough paths . Almost 20 years later, they have found many applications in Machine Learning, Data Analysis and trend recognition in time series. N. Tapia (NTNU) Introduction to signatures Feb. 26, 2019 @ Magic 2019, Ilsetra 5 / 29

Signatures The shuffle algebra Consider a d -dimensional vector space V and define T ( V ) ≔ �1 ⊕ V ⊕ ( V ⊗ V ) ⊕ ( V ⊗ V ⊗ V ) ⊕ · · · . For p ≥ 1 , the degree p component T ( V ) p = V ⊗ p is spanned by the set { e i 1 ··· i p ≔ e i 1 ⊗ · · · ⊗ e i p : i 1 , . . . , i p = 1 , . . . , d } In particular dim T ( V ) = ∞ . For a given ψ ∈ T ( V ) ∗ ≔ T ( ( V ) ) we write � � d � ψ , e i 1 ··· i p � e i 1 ··· i p . ψ = p ≥ 0 i 1 ,..., i p =1 N. Tapia (NTNU) Introduction to signatures Feb. 26, 2019 @ Magic 2019, Ilsetra 6 / 29

Signatures The shuffle algebra There are two products on T ( V ) : 1 the tensor product: e i 1 ··· i p ⊗ e i p +1 ··· i p + q = e i 1 ··· i p + q ∈ T ( V ) p + q and, 2 the shuffle product: � e i 1 ··· i p ✁ e i p +1 ··· i p + q = e i σ ( 1 ) i σ ( 2 ) ··· i σ ( p + q ) ∈ T ( V ) p + q . σ ∈ Sh ( p , q ) Examples: e i ✁ e j = e ij + e ji , e i ✁ e j k = e ij k + e jik + e j ki . On both cases 1 ∈ T ( V ) acts as the unit. N. Tapia (NTNU) Introduction to signatures Feb. 26, 2019 @ Magic 2019, Ilsetra 7 / 29

Signatures The shuffle algebra The shuffle algebra carries a coalgebra structure: define ∆ : T ( V ) → T ( V ) � ⊗ T ( V ) by � p − 1 ∆ e i 1 ··· i p ≔ e i 1 ··· i p � ⊗ 1 + 1 � e i 1 ··· i j � ⊗ e i 1 ··· i p + ⊗ e i j +1 ··· i p . j =1 This structure is dual to the tensor product in the sense that if ϕ , ψ ∈ T ( ) then ( V ) � � d � ϕ � ϕ ⊗ ψ = ⊗ ψ , ∆ e i 1 ··· i p � e i 1 ··· i p . p ≥ 0 i 1 ,..., i p =1 N. Tapia (NTNU) Introduction to signatures Feb. 26, 2019 @ Magic 2019, Ilsetra 8 / 29

Signatures Curves on euclidean space Let x : [ 0 , 1 ] → � d be a curve with finite 1 -variation. Its signature over the interval [ s , t ] is the tensor series with coefficients ∫ t x j � S ( x ) s , u , 1 � = 1 , � S ( x ) s , t , e i 1 ··· i p ⊗ e j � ≔ � S ( x ) s , u , e i 1 ··· i p � � u d u . s Example: ∫ t ∫ t ∫ u x j x i u d u = x i t − x i x i � � v � u d v d u . � S ( x ) s , t , e i � = � S ( x ) s , t , e ij � = s , s s s N. Tapia (NTNU) Introduction to signatures Feb. 26, 2019 @ Magic 2019, Ilsetra 9 / 29

Signatures Curves on euclidean space Chen shows that S ( x ) satsifies: 1 the shuffle relation: � S ( x ) s , t , e i 1 ... i p ✁ e i p +1 ··· i p + q � = � S ( x ) s , t , e i 1 ... i p �� S ( x ) s , t , e i p +1 ··· i p + q � . 2 Chen’s rule: if y is another path and x · y is their concatenation, then S ( x · y ) s , t = S ( x ) s , u ⊗ S ( y ) u , t . Moreover, one can show that there exists a constant C > 0 such that |� S ( x ) s , t , e i 1 ··· i p �| ≤ C p p ! | t − s | p . N. Tapia (NTNU) Introduction to signatures Feb. 26, 2019 @ Magic 2019, Ilsetra 10 / 29

Signatures Curves on euclidean space For example ∫ t ∫ u ∫ t ∫ u x j x j x i x i � S ( x ) s , t , e ij + e ji � = � v � u d v d u + � v � u d v d u ∫ t ∫ t s s s s x j ( x j u − x j ( x i u − x i x i s ) � u d u + s ) � u d u = s s s )( x j t − x j = ( x i t − x i s ) = � S ( x ) s , t , e i �� S ( x ) s , t , e j � . N. Tapia (NTNU) Introduction to signatures Feb. 26, 2019 @ Magic 2019, Ilsetra 11 / 29

Signatures Curves on euclidean space Another example: ∫ t x i v d v � S ( x ) s , t , e i � = � ∫ u ∫ t s x i x i � v d v + � v d v = s u = � S ( x ) s , u ⊗ S ( x ) u , t , e i � . N. Tapia (NTNU) Introduction to signatures Feb. 26, 2019 @ Magic 2019, Ilsetra 12 / 29

Signatures Curves on euclidean space Yet another example: ∫ t x j ( x i v − x i v d v � S ( x ) s , t , e ij � = s ) � ∫ u ∫ t s x j x j ( x i v − x i ( x i v − x i s ) � v d v + s ) � v d v = ∫ u ∫ t s u x j x j s )( x j t − x j ( x i v − x i ( x i v − x i v d v + ( x i u − x i v d v + = s ) � u ) � u ) s u = � S ( x ) s , u ⊗ S ( x ) u , t , e ij � N. Tapia (NTNU) Introduction to signatures Feb. 26, 2019 @ Magic 2019, Ilsetra 13 / 29

Signatures Curves on euclidean space Signatures can be easiy computed for certain paths. If x is a straight line, i.e. x t = a + bt with a , b ∈ � d then � p � S ( x ) s , t , e i 1 ··· i p � = ( t − s ) p b i j . p ! j =1 Indeed ∫ t � p ( u − s ) p b i k b j d u � S ( x ) s , t , e i 1 ··· i p ⊗ e j � = p ! s k =1 � p +1 = ( t − s ) p +1 b i k . ( p + 1 ) ! k =1 N. Tapia (NTNU) Introduction to signatures Feb. 26, 2019 @ Magic 2019, Ilsetra 14 / 29

Signatures Curves on euclidean space Therefore S ( x ) s , t = 1 + ( t − s ) b + ( t − s ) 2 b ⊗ b + ( t − s ) 3 b ⊗ b ⊗ b + · · · = exp ⊗ (( t − s ) b ) . 2 6 By Chen’s rule, if x is a general piecewise linear path with slopes b 1 , . . . , b m ∈ � d between times s < t 1 < · · · < t m − 1 < t then S ( x ) s , t = exp ⊗ (( t 1 − s ) b 1 ) ⊗ · · · ⊗ exp ⊗ (( t − t m − 1 ) b m ) . N. Tapia (NTNU) Introduction to signatures Feb. 26, 2019 @ Magic 2019, Ilsetra 15 / 29

Signatures Curves on euclidean space Some further properties: 1 Invariant under reparametrization: if ϕ is an increasing diffeomorphism on [ 0 , 1 ] then S ( x ◦ ϕ ) s , t = S ( x ) s , t . 2 Characterizes the path up-to irreducibility . If S ( x ) = S ( y ) for two irreducible paths then y is a translation of x . N. Tapia (NTNU) Introduction to signatures Feb. 26, 2019 @ Magic 2019, Ilsetra 16 / 29

Signatures Practical questions For some applications one has a set of points sampled in time. One may construct the signature by linear interpolation. How to deal with infinite series? Truncation is one possibility, but to which level? N. Tapia (NTNU) Introduction to signatures Feb. 26, 2019 @ Magic 2019, Ilsetra 17 / 29

Signatures Practical questions Applying some transformations one can read off some information from the signature: 1 Mean 2 Quadratic variation, i.e. variance For Machine Learning applications, levels of the signature are selected as explanatory variables for the features of a path. An example of objective function (taken from Gyurkó, Lyons, Kontkowski & Field; 2014)   � � �   2 � β w � S ( x k ) 0 , 1 , w � − y k � L   � �   min + α | β w |   �  �  β k =1 | w |≤ M | w |≤ M N. Tapia (NTNU) Introduction to signatures Feb. 26, 2019 @ Magic 2019, Ilsetra 18 / 29

Signatures Paths on a manifold Let G be a d -dimensional Lie group with Lie algebra g . The Maurer–Cartan form on G is the pushforward of left translation: ω g ( v ) = ( L g − 1 ) ∗ v , v ∈ T g G . It is a g -valued 1-form, i.e. a smooth section of ( M × g ) ⊗ T ∗ G . In other words, ω g maps T g G into g . In particular, it can be written as ω = X 1 ⊗ ω 1 + · · · + X d ⊗ ω d where ω 1 , . . . , ω d are suitable 1-forms on G and X 1 , . . . , X d is a basis of g . N. Tapia (NTNU) Introduction to signatures Feb. 26, 2019 @ Magic 2019, Ilsetra 19 / 29

Signatures Paths on a manifold Chen defines the signature over the interval [ s , t ] of a smooth curve α : [ 0 , 1 ] → G as the tensor series S ( α ) s , t with coefficients ∫ t � S ( α ) s , u , e i 1 ··· i p � ω j � S ( α ) s , t , 1 � ≔ 1 , � S ( α ) s , t , e i 1 ... i p ⊗ e j � ≔ α u ( � α u ) d u . s When G = � d this definition coincides with the previous one by observing that ω i = d x i , i.e. α 1 α d ω α t ( � α t ) = � t e 1 + · · · + � t e d with e 1 , . . . , e d the canonical basis of � d . N. Tapia (NTNU) Introduction to signatures Feb. 26, 2019 @ Magic 2019, Ilsetra 20 / 29

Signatures Paths on a manifold An example: let G = H 3 be the Heisenberg group, that is,         � � 1 x z � �   : x , y , z ∈ � H 3 ≔ 0 1 y   . � �   0 0 1 Its Lie algebra h 3 is spanned by the matrices X ≔ � � , Y ≔ � � , Z ≔ � � 0 1 0 0 0 0 0 0 1 � � � � � � 0 0 0 0 0 1 0 0 0 � � � � � � 0 0 0 0 0 0 0 0 0 with [ X , Y ] = Z , [ X , Z ] = [ Y , Z ] = 0 . N. Tapia (NTNU) Introduction to signatures Feb. 26, 2019 @ Magic 2019, Ilsetra 21 / 29