Learning some Lie theory for fun and profit

⊕The phrase “for fun and profit” seems to be a pretty old expression: according to the answers to this StackExchange question, it might date back to Horace’s Ars Poetica (“prodesse et delectare”). I like the idea that books (and ideas!) should be both instructive and enjoyable…

While exploring quaternions and the theory behind them, I noticed an interesting pattern: in the exposition of Solà (2017), 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!)⊕When you push that idea to its extremes, you get 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 math3ma blog, or Riehl (2017) 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 opensThis is why a lot of programming languages now try to make differentiability a 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. JAX is basically a differentiable Numpy, and Julia has always made differentiable programming a priority, via projects such as JuliaDiff and Zygote.

: optimization becomes easier (because you can use gradient descent), you can have random variables, and so on.

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 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 differentiable manifold is a more complex beast.⊕For a more complete introduction to differential geometry and differentiable manifolds, see Lafontaine (2015). It introduces manifolds, differential topology, Lie groups, and more advanced topics, all with little prerequisites (basics of differential calculus).

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⊕Lie theory is named after Sophus Lie, a Norwegian mathematician. As such, “Lie” is pronounced lee. Lie was inspired by Galois’ work on algebraic equations, and wanted to establish a similar general theory for differential equations.

, 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.

Interesting properties of Lie groups

For a complete overview of Lie theory, there are a lot of interesting material that you can find online.⊕There is also a chapter on Lie theory in the amazing Princeton Companion to Mathematics (Gowers, Barrow-Green, and Leader 2010, sec. II.48).

I especially recommend the tutorial by Solà, Deray, and Atchuthan (2018): just enough maths to understand what is going on, but without losing track of the applications. There is also a video tutorial made for the IROS2020 conferenceMore specifically for the workshop on Bringing geometric methods to robot learning, optimization and control.

. For a more complete treatment, Stillwell (2008) is greatI really like John Stillwell as a textbook author. All his books are extremely clear and a pleasure to read.

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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⊕Skew-symmetric matrices are matrices \(A\) such that \(A^\top = -A\): \[ [\boldsymbol\omega]_\times = \begin{bmatrix} 0 & -\omega_x & \omega_y \\ \omega_x & 0 & -\omega_z \\ -\omega_y & \omega_z & 0 \end{bmatrix}. \]

), 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.

⊕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 Solà, Deray, and Atchuthan (2018).

If all this piqued your interest, you can read a very short (only 14 pages!) overview of Lie theory in Solà, Deray, and Atchuthan (2018). 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).

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