The Signature Method Nikolas Tapia NTNU Trondheim Apr. 16th, 2019 @ Santiago, Chile N. Tapia (NTNU) The Signature Method Apr. 16th, 2019 @ Santiago, Chile 1 / 34
Goals Goals N. Tapia (NTNU) The Signature Method Apr. 16th, 2019 @ Santiago, Chile 2 / 34
Goals 1 Signatures 1 For paths on � d 2 Some applications 3 Geometric Rough Paths 2 Shape recognition 1 Signatures on Lie groups N. Tapia (NTNU) The Signature Method Apr. 16th, 2019 @ Santiago, Chile 3 / 34
Signatures Signatures N. Tapia (NTNU) The Signature Method Apr. 16th, 2019 @ Santiago, Chile 4 / 34
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) The Signature Method Apr. 16th, 2019 @ Santiago, Chile 5 / 34
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 ) 0 acts as the unit. N. Tapia (NTNU) The Signature Method Apr. 16th, 2019 @ Santiago, Chile 6 / 34
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) The Signature Method Apr. 16th, 2019 @ Santiago, Chile 7 / 34
Signatures Curves on euclidean space Let x : [ 0 , 1 ] → � d be a curve of bounded variation. Definition Its signature over the interval [ s , t ] ⊂ [ 0 , 1 ] is the tensor series with coefficients ∫ t � S ( x ) s , u , e i 1 ··· i p − 1 � d x i p � S ( x ) s , u , 1 � ≔ 1 , � S ( x ) s , t , e i 1 ··· i p � ≔ u . s Example: ∫ t ∫ t ∫ u v d x j d x i u = x i t − x i d x i � S ( x ) s , t , e i � = � S ( x ) s , t , e ij � = s , u . s s s In total: ∫ t ∫ t ∫ u 2 ∭ u 1 d x j u 1 d x j d x i d x i d x i u 2 d x k S ( x ) s , t = 1 + u e i + u 2 e ij + u 3 e ij k + · · · s s s s < u 1 < u 2 < u 3 < t N. Tapia (NTNU) The Signature Method Apr. 16th, 2019 @ Santiago, Chile 8 / 34
Signatures Curves on euclidean space Chen (1954) 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: for any s < u < t , we have S ( x ) s , t = S ( x ) s , u ⊗ S ( x ) u , t . 3 If y is another path and x · y is their concatenation then S ( x · y ) s , t = S ( x ) s , t ⊗ S ( y ) s , t . N. Tapia (NTNU) The Signature Method Apr. 16th, 2019 @ Santiago, Chile 9 / 34
Signatures Curves on euclidean space The shuffle identity generalizes integration by parts. ∫ t ∫ u ∫ t ∫ u v d x j d x j d x i v d x i � S ( x ) s , t , e ij + e ji � = u + u ∫ t ∫ t s s s s s ) d x j ( x j u − x j ( x i u − x i s ) d x i = u + 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) The Signature Method Apr. 16th, 2019 @ Santiago, Chile 10 / 34
Signatures Curves on euclidean space Chen’s rule generalizes the splitting of integrals. ∫ t d x i � S ( x ) s , t , e i � = v ∫ u ∫ t s d x i d x i = v + v s u = � S ( x ) s , u ⊗ S ( x ) u , t , e i � . N. Tapia (NTNU) The Signature Method Apr. 16th, 2019 @ Santiago, Chile 11 / 34
Signatures Curves on euclidean space ∫ t s ) d x j ( x i v − x i � S ( x ) s , t , e ij � = v ∫ u ∫ t s s ) d x j s ) d x j ( x i v − x i ( x i v − x i = v + v ∫ u ∫ t s u s ) d x j u ) d x j s )( x j t − x j ( x i v − x i ( x i v − x i v + ( x i u − x i u ) = v + s u = � S ( x ) s , u ⊗ S ( x ) u , t , e ij � N. Tapia (NTNU) The Signature Method Apr. 16th, 2019 @ Santiago, Chile 12 / 34
Signatures Curves on euclidean space Signatures can be easily 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 − 1 ( u − s ) p − 1 � S ( x ) s , t , e i 1 ··· i p � = b i j b p d u ( p − 1 ) ! s j =1 � p = ( t − s ) p b i j . p ! j =1 N. Tapia (NTNU) The Signature Method Apr. 16th, 2019 @ Santiago, Chile 13 / 34
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) The Signature Method Apr. 16th, 2019 @ Santiago, Chile 14 / 34
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 . The signature takes values on a group G with ⊗ as composition. In practice, a suitable truncation of S ( x ) is considered, and this also belongs to a group G m , m > 1 . N. Tapia (NTNU) The Signature Method Apr. 16th, 2019 @ Santiago, Chile 15 / 34
Signatures Some applications The basic workflow for signatures in applications is to convert data streams into paths. This can be done in several ways: linear interpolation, axis paths, lead-lag transforms, cumulative sums, etc. . . One also has to choose the truncation level. There is some redundancy in the signature due to the shuffle relations. A more efficient approach is to work with the so-called log-signature . N. Tapia (NTNU) The Signature Method Apr. 16th, 2019 @ Santiago, Chile 16 / 34
Signatures Some applications 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) The Signature Method Apr. 16th, 2019 @ Santiago, Chile 17 / 34
Signatures Some applications This framework has been applied to 1 finanical data streams, 2 sound compression (Lyons & Sidorova, 2005), 3 chinese character recognition (Graham, 2013; Lianwen, Weixin & Manfei, 2015), 4 pattern recognition in MEG scans (Gyurkó, Lyons & Oberhauser, 2014) and 5 behavioural patterns of patients with bipolar disorder. N. Tapia (NTNU) The Signature Method Apr. 16th, 2019 @ Santiago, Chile 18 / 34
Rough Paths Rough Paths N. Tapia (NTNU) The Signature Method Apr. 16th, 2019 @ Santiago, Chile 19 / 34
Rough Paths Definition (Lyons (1998)) A rough path of roughness θ > 1 is a map X : [ 0 , 1 ] 2 → T ( ( V ) ) ≤ m such that X s , t = X s , u ⊗ X u , t and |� X s , t , e i 1 ··· i p �| ≤ C p | t − s | p / θ , p < m where m ≔ ⌈ θ ⌉ . The (trucated) signature is the “canonical lift” of a path of bounded variation to a rough path of roughness θ . Theorem (Lyons (1998)) Any path X : [ 0 , 1 ] 2 → T ( ( V ) ) ≤ m satisying Chen’s rule and the analytic bound admits a unique extension ˆ X : [ 0 , 1 ] → T ( ( V ) ) with the same properties. N. Tapia (NTNU) The Signature Method Apr. 16th, 2019 @ Santiago, Chile 20 / 34
Rough Paths Definition (Lyons (1998)) A geometric rough path of roughness θ is the limit of canonical lifts of bounded variation paths in a certain θ -variation metric. Geometric rough paths are G m valued, where again m ≔ ⌈ θ ⌉ . Definition (Friz–Victoir (2006)) A weakly-geometric rough path of roughness θ is a G m -valued path of finite θ -variation. Geometric rough paths provide a “universal” description of flows controlled by x . N. Tapia (NTNU) The Signature Method Apr. 16th, 2019 @ Santiago, Chile 21 / 34
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