Invariants of Multidimensional Time Series Based on Their Iterated-Integral Signature

Abstract

We introduce a novel class of features for multidimensional time series that are invariant with respect to transformations of the ambient space. The general linear group, the group of rotations and the group of permutations of the axes are considered. The starting point for their construction is Chen's iterated-integral signature.

Varieties of Signature Tensors

“…The d × d × · · · × d tensor σ (k) (X) can be computed from the m × m × · · · × m tensor σ k (C mono ) in Example 2.2 by multiplying each of its k sides with the d × m matrix X. This is the tensor analogue to the congruence action on matrix space seen in (15). The x i j are homogeneous coordinates on the projective space P dm−1 over an algebraically closed field K that contains R. The matrix-tensor multiplication described above defines a rational map of degree k:…”

“…We can use invariant theory to show that these orbits are different. According to Diehl and Reizenstein [15,Remark 14], the space of SL(d, K)-invariants linear forms on (K 2 ) ⊗4 has dimension 2 and is spanned by…”

Learning Paths from Signature Tensors

“…Its entry q ij is the Lévy area of the projection of ψ onto the plane indexed by i and j, the signed area between the planar path and the segment connecting its endpoints. For background on signature tensors of paths and their applications see [1,8,10,12,21,22,23].…”

“…Signature tensors were introduced by Chen [7], and they play an important role in stochastic analysis [12,21]. We refer to [22,23] for the recovery problem, and to [10,13,17,18] for algorithms and applications. Our point of departure is the approach to signature tensors via algebraic geometry that was proposed in [1].…”

Signatures of paths transformed by polynomial maps

“…Coming from stochastic analysis, the signatures are becoming more relevant in other areas, such as algebraic geometry or combinatorics, and we would like to highlight some recent work. For instance, in [DR18], J. Diehl and J. Reizenstein offer a combinatorial approach to the understanding of invariants of multidimensional time series based on their signature. Another reference is [AFS18], in which C. Améndola, P. Friz, and B. Sturmfels look at the varieties of signatures of tensors for both deterministic and random paths, focusing on piecewise linear paths and polynomials paths, among others.…”