blog/posts/lie-theory.org

---
title: "Learning some Lie theory for fun and profit "
date: 2020-11-14
tags: maths, linear algebra, dynamical systems
toc: false
---

[fn::{-} The phrase "for fun and profit" seems to be a pretty old
expression: according to the answers to [[https://english.stackexchange.com/q/25205][this StackExchange question]],
it might date back to Horace's [[https://en.wikipedia.org/wiki/Ars_Poetica_(Horace)][/Ars Poetica/]] ("prodesse et
delectare"). I like the idea that books (and ideas!) should be both
instructive and enjoyable...]

While exploring [[./quaternions.html][quaternions]] and the theory behind them, I noticed an
interesting pattern: in the exposition of
cite:sola2017_quater_kinem_error_state_kalman_filter, quaternions and
rotations matrices had exactly the same properties, and the derivation
of these properties was rigorously identical (bar some minor notation
changes).

This is expected because in this specific case, these are just two
representations of the same underlying object: rotations. However,
from a purely mathematical and abstract point of view, it cannot be a
coincidence that you can imbue two different types of objects with
exactly the same properties.

Indeed, this is not a coincidence: the important structure that is
common to the set of rotation matrices and to the set of quaternions
is that of a /Lie group/.

In this post, I want to explain why I find Lie theory interesting,
both in its theoretical aspects (for fun) and in its potential for
real-world application (for profit). I will also give a minimal set of
references that I used to get started.

* Why would that be interesting?

From a mathematical point of view, seeing a common structure in
different objects, such as quaternions and rotation matrices, should
raise alarm signals in our heads. Is there a deeper concept at play
here? If we can find that two objects are two examples of the same
abstract structure, maybe we'll also be able to identify that
structure elsewhere, maybe where it's less obvious. And then, if we
prove interesting theorems on the abstract structure, we'll
essentially get the same theorems on every example of this structure,
and /for free!/ (i.e. without any additional work!)[fn:structure]

[fn:structure]{-} When you push that idea to its extremes, you get
[[https://en.wikipedia.org/wiki/Category_theory][category theory]], which is just the study of (abstract) structure. This
in a fun rabbit hole to get into, and if you're interested, I
recommend the amazing [[https://www.math3ma.com/][math3ma]] blog, or
cite:riehlCategoryTheoryContext2017 for a complete and approachable
treatment.


We can think of it as a kind of factorization: instead of doing the
same thing over and over, we can basically do it /once/ and recall the
general result whenever it is needed, as one would define a function
and call it later in a piece of software.

In this case, Lie theory provides a general framework for manipulating
objects that we want to /combine/ and on which we'd like to compute
/derivatives/. Differentiability is an essentially linear property, in
the sense that it works best in vector spaces. Indeed, think of what
you do to with a derivative: you want to /add/ it to other stuff to
represent increase rates or uncertainties. (And of course, the
differential operator itself is linear.)

Once you can differentiate, a whole new world
opens[fn:differentiability]: optimization becomes easier (because you
can use gradient descent), you can have random variables, and so on.

[fn:differentiability] This is why a lot of programming languages now
try to make differentiability a [[https://en.wikipedia.org/wiki/Differentiable_programming][first-class concept]]. The ability to
differentiate arbitrary programs is a huge bonus for all kinds of
operations common in scientific computing. Pioneering advances were
made in deep learning libraries, such as TensorFlow and PyTorch; but
recent advances are even more exciting. [[https://github.com/google/jax][JAX]] is basically a
differentiable Numpy, and Julia has always made differentiable
programming a priority, via projects such as [[https://www.juliadiff.org/][JuliaDiff]] and [[https://fluxml.ai/Zygote.jl/][Zygote]].


In the case of quaternions, we can define explicitly a differentiation
operator, and prove that it has all the nice properties that we come
to expect from derivatives. Wouldn't it be nice if we could have all
of this automatically? Lie theory gives us the general framework in
which we can imbue non-"linear" objects with differentiability.

* The structure of a Lie group

Continuing on the example of rotations, what common properties can we
identify?

1. Quaternions and rotation matrices can be multiplied together (to
   compose rotations), have an identity element, along with nice
   properties.
2. Quaternions and rotation matrices can be differentiated, and we can
   map them to and from usual vectors in $\mathbb{R}^m$.

These two group of properties actually correspond to common
mathematical structures: a /group/ and a /differentiable manifold/.

You're probably already familiar with [[https://en.wikipedia.org/wiki/Group_(mathematics)][groups]], but let's recall the
basic properties:
- It's a set $G$ equipped with a binary operation $\cdot$.
- The group is closed under the operation: for any element $x,y$ in G,
  $x \cdot y$ is always in $G$.
- The operation is associative: $x \cdot (y \cdot z) = (x \cdot y)
  \cdot z$.
- There is a special element $e$ of $G$ (called the /identity
  element/), such that $x \cdot e = e \cdot x$ for all $x \in G$.
- For every element $x$ of $G$, there is a unique element of $G$
  denoted $x^{-1}$ such that $x \cdot x^{-1} = x^{-1} \cdot x = e$.

A [[https://en.wikipedia.org/wiki/Differentiable_manifold][differentiable manifold]] is a more complex
beast.[fn:differential_geometry] Although the definition is more
complex, we can loosely imagine it as a surface (in higher dimension)
on which we can compute derivatives at every point. This means that
there is a tangent hyperplane at each point, which is a nice vector
space where our derivatives will live.

You can think of the manifold as a tablecloth that has a weird shape,
all kinds of curvatures, but no edges or spikes. The idea here is that
we can define an /atlas/, i.e. a local approximation of the manifold
as a plane. The name is telling: they're called atlases because they
play the exact same role as geographical maps. The Earth is not flat,
it is a sphere with all kinds of deformations (mountains, canyons,
oceans), but we can have planar maps that represent a small area with
a very good precision. Similarly, atlases are the vector spaces that
provide the best linear approximation of a small region around a point
on the manifold.

So we know what a group and a differential manifold are. As it turns
out, that's all we need to know! What we have defined so far is a /Lie
group/[fn:lie], i.e. a group that is also a differentiable
manifold. The tangent vector space at the identity element is called
the /Lie algebra/.

To take the example of rotation matrices:
- We can combine them (i.e. by matrix multiplication): they form a
  group.
- If we have a function $R : \mathbb{R} \rightarrow
  \mathrm{GL}_3(\mathbb{R})$ defining a trajectory (e.g. the
  successive attitudes of a object in space), we can find derivatives
  of this trajectory! They would represent instantaneous orientation
  change, or angular velocities.

[fn:differential_geometry] {-} For a more complete introduction to
differential geometry and differentiable manifolds, see
cite:lafontaine2015_introd_differ_manif. It introduces manifolds,
differential topology, Lie groups, and more advanced topics, all with
little prerequisites (basics of differential calculus).

[fn:lie] {-} Lie theory is named after [[https://en.wikipedia.org/wiki/Sophus_Lie][Sophus Lie]], a Norwegian
mathematician. As such, "Lie" is pronounced /lee/. Lie was inspired by
[[https://en.wikipedia.org/wiki/%C3%89variste_Galois][Galois']] work on algebraic equations, and wanted to establish a similar
general theory for differential equations.

* Interesting properties of Lie groups

For a complete overview of Lie theory, there are a lot of interesting
material that you can find online.[fn:princeton_companion] I
especially recommend the tutorial by
cite:sola2018_micro_lie_theor_state_estim_robot: just enough maths to
understand what is going on, but without losing track of the
applications. There is also a [[https://www.youtube.com/watch?v=QR1p0Rabuww][video tutorial]] made for the [[https://www.iros2020.org/][IROS2020]]
conference[fn:workshop]. For a more complete treatment,
cite:stillwell2008_naive_lie_theor is great[fn:stillwell].

Because of the group structure, the manifold is similar at every
point: in particular, all the tangent spaces look alike. This is why
the /Lie algebra/, the tangent space at the identity, is so
important. All tangent spaces are vector spaces isomorphic to the Lie
algebra, therefore studying the Lie algebra is sufficient to derive
all the interesting aspects of the Lie group.

Lie algebras are always vector spaces. Even though their definition
may be quite complex (e.g. skew-symmetric matrices in the case of the
group of rotation matrices[fn:skewsymmetric]), we can always find an
isomorphism of vector spaces between the Lie algebra and
$\mathbb{R}^m$ (in the case of finite-dimensional Lie groups). This is
really nice for many applications: for instance, the usual probability
distributions on $\mathbb{R}^m$ translate directly to the Lie algebra.

The final aspect I'll mention is the existence of /exponential maps/,
allowing transferring elements of the Lie algebra to the Lie
group. The operator $\exp$ will wrap an element of the Lie algebra
(i.e. a tangent vector) to its corresponding element of the Lie group
by wrapping along a geodesic of the manifold. There is also a
logarithmic map providing the inverse operation.

[fn::{-} The Lie group (in blue) with its associated Lie algebra
(red). We can see how each element of the Lie algebra is wrapped on
the manifold via the exponential map. Figure from
cite:sola2018_micro_lie_theor_state_estim_robot.]
#+ATTR_HTML: :width 500px :style background-color: #ffffff;
[[../images/lie_exponential.svg]] 

If all this piqued your interest, you can read a very short (only 14
pages!) overview of Lie theory in
cite:sola2018_micro_lie_theor_state_estim_robot. They also expand on
applications to estimation and robotics (as the title suggests), so
they focus on deriving Jacobians and other essential tools for any Lie
group. They also give very detailed examples of common Lie groups
(complex numbers, rotation matrices, quaternions, translations).

[fn:princeton_companion] {-} There is also a chapter on Lie theory in
the amazing /Princeton Companion to Mathematics/
[[citep:gowersPrincetonCompanionMathematics2010][::, §II.48]].

[fn:workshop] More specifically for the workshop on [[https://sites.google.com/view/iros2020-geometric-methods/][Bringing geometric
methods to robot learning, optimization and control]].

[fn:stillwell] I really like John Stillwell as a textbook author. All
his books are extremely clear and a pleasure to read.

[fn:skewsymmetric] {-} [[https://en.wikipedia.org/wiki/Skew-symmetric_matrix][Skew-symmetric matrices]] are matrices $A$ such
that $A^\top = -A$:
\[ [\boldsymbol\omega]_\times = \begin{bmatrix}
0 & -\omega_x & \omega_y \\
\omega_x & 0 & -\omega_z \\
-\omega_y & \omega_z & 0
\end{bmatrix}. \]

* Conclusion

Lie theory is useful because it gives strong theoretical guarantees
whenever we need to linearize something. If you have a system evolving
on a complex geometric structure (for example, the space of rotations,
which is definitely not linear), but you need to use a linear
operation (if you need uncertainties, or you have differential
equations), you have to approximate somehow. Using the Lie structure
of the underlying space, you immediately get a principled way of
defining derivatives, random variables, and so on.

Therefore, for estimation problems, Lie theory provides a strong
backdrop to define state spaces, in which all the usual manipulations
are possible. It has thus seen a spike of interest in the robotics
literature, with applications to estimation, optimal control, general
optimization, and many other fields.

I hope that this quick introduction has motivated you to learn more
about Lie theory, as it is a fascinating topic with a lot of
potential!

* References