Lies
What are Lie elements
Lie elements live in the free Lie Algebra. Group-like elements have a one-to-one correspondence with a stream. That is, for every group-like element, there exists a stream where the signature of that stream is the group-like element. For more information on Lie Algebras, see Reutenauer [Reu93] and Bourbaki [Bou89].
How do Lies fit into RoughPy
You will most commonly encounter Lies when taking the log signature of a path. We can use the Dynkin map to transfer between Lies and signatures.
How to work with Lies
To create a Lie element, we should start by creating data.
>>> import roughpy
>>> from roughpy import Monomial
>>> lie_data_x = [
1 * roughpy.Monomial("x1"), # channel (1)
1 * roughpy.Monomial("x2"), # channel (2)
1 * roughpy.Monomial("x3"), # channel (3)
1 * roughpy.Monomial("x4"), # channel ([1, 2])
1 * roughpy.Monomial("x5"), # channel ([1, 3])
1 * roughpy.Monomial("x6"), # channel ([2, 3])
]
Here roughpy.Monomial creates a monomial object (in this case, a single indeterminate),
and multiplying this by 1 makes this monomial into a polynomial.
We can then use our data to create a Lie element. Here we are creating two Lies with Width 3, Depth 2. The first is using the object above. The second is using the same object, but with y as the indeterminate name.
>>> lie_x = rp.Lie(lie_data_x, width=3, depth=2, dtype=rp.RationalPoly)
>>> print(f"{lie_x=!s}")
lie_x={ { 1(x1) }(1) { 1(x2) }(2) { 1(x3) }(3) { 1(x4) }([1,2]) { 1(x5) }([1,3]) { 1(x6) }([2,3]) }
>>> lie_y = rp.Lie(lie_data_y, width=3, depth=2, dtype=rp.RationalPoly)
>>> print(f"{lie_y=!s}")
lie_y={ { 1(y1) }(1) { 1(y2) }(2) { 1(y3) }(3) { 1(y4) }([1,2]) { 1(y5) }([1,3]) { 1(y6) }([2,3]) }
We can also multply Lies together
>>> result = lie_x * lie_y
>>> print(f"{result=!s}")
result={ { 1(x1 y2) -1(x2 y1) }([1,2]) { 1(x1 y3) -1(x3 y1) }([1,3]) { 1(x2 y3) -1(x3 y2) }([2,3]) }
If we provide a complementary Contexts, then we can use this to create a Free Tensors, from our Lie element.
>>> context = rp.get_context(3, 2, rp.RationalPoly)
>>> tensor_from_lie = context.lie_to_tensor(lie_x).exp()
>>> print(f"{tensor_from_lie=!s}")
tensor_from_lie={ { 1() }() { 1(x1) }(1) { 1(x2) }(2) { 1(x3) }(3) { 1/2(x1^2) }(1,1) { 1(x4) 1/2(x1 x2) }(1,2) { 1(x5) 1/2(x1 x3) }(1,3) { -1(x4) 1/2(x1 x2) }(2,1) { 1/2(x2^2) }(2,2) { 1(x6) 1/2(x2 x3) }(2,3) { -1(x5) 1/2(x1 x3) }(3,1) { -1(x6) 1/2(x2 x3) }(3,2) { 1/2(x3^2) }(3,3) }
We can also go the other way, creating a Lie from a tensor. Using the signature of the stream shown ealier in Streams, we can create a corresponding Lie element.
>>> lie_from_tensor = context.tensor_to_lie(sig.log())
>>> print(f"{lie_from_tensor=!s}")
lie_from_tensor={ 4.5(1) 9(2) }
Literature references
N. Bourbaki. Algebra I: Chapters 1-3. Springer Science & Business Media, August 1998. ISBN 978-3-540-64243-5. Google-Books-ID: STS9aZ6F204C.
Nicolas Bourbaki. Lie Groups and Lie Algebras: Chapters 1-3. Springer Science & Business Media, 1989. ISBN 978-3-540-64242-8. Google-Books-ID: brSYF_rB2ZcC.
Patrick Kidger, James Morrill, James Foster, and Terry Lyons. Neural Controlled Differential Equations for Irregular Time Series. November 2020. arXiv:2005.08926 [cs, stat]. URL: http://arxiv.org/abs/2005.08926 (visited on 2023-12-08), doi:10.48550/arXiv.2005.08926.
Terry Lyons and Andrew D. McLeod. Signature Methods in Machine Learning. January 2024. arXiv:2206.14674 [cs, math, stat]. URL: http://arxiv.org/abs/2206.14674 (visited on 2024-02-14).
Terry J. Lyons, Michael J. Caruana, and Thierry Lévy. Differential Equations Driven by Rough Paths: Ecole d’Eté de Probabilités de Saint-Flour XXXIV-2004. Springer, April 2007. ISBN 978-3-540-71285-5. Google-Books-ID: hOm5BQAAQBAJ.
Christophe Reutenauer. Free Lie Algebras. Clarendon Press, 1993. ISBN 978-0-19-853679-6. Google-Books-ID: cBvvAAAAMAAJ.