3.15.3 The Log-Signature, the Free Lie Algebra & Signature Uniqueness

Basic Prerequisites: 02 · The Signature Algebra (Chen’s identity and the shuffle product) and Linear Algebra · 02 Vectors, Spaces & Matrices.

---

1. Intuition & Practical Objective

The signature is a huge object - at level $k$ it has $d^k$ components, and level $k+1$ contains most of the information of level $k$ plus a little more. Two questions naturally follow, and they are the subject of this page:

1. Compression: what is the smallest faithful summary? Because Chen’s identity makes the signature a group-like element, its logarithm - the log-signature - exists and retains only the genuinely independent shape information. At level 2 the log-signature is just the Lévy area (antisymmetric), at level 3 a pair of iterated brackets, and so on. In general it is a Lie element of the free Lie algebra - the “causality-free” core of the path, with the shuffle-redundant part stripped away.
2. Uniqueness: does the signature determine the path? The Lyons signature uniqueness result says yes, up to the genuinely irrelevant information: two paths with the same signature differ only by a reparametrisation (or, for rough paths, by a tree-like path - Boedihardjo et al.). The signature is therefore a faithful coordinate system: nothing about the geometry is lost, and nothing irrelevant is kept.

Two practical consequences drive the page:

3. The log-signature is the compression you actually ship to a model. The full signature carries shuffle-redundant coordinates (products of lower terms); the log-signature is the minimal basis - the same descriptive power with far fewer, mutually independent features. This is the standard pre-processing step in signature-based machine learning.
4. Reparametrisation invariance is a feature, not a bug. The signature (and log-signature) does not care how fast the path is traversed - it cares about the route. For geometric shape features this is ideal; it only becomes a problem when the timing of events is itself the signal, which is exactly what lead-lag reintroduces (§05).

The practical objective: be able to compute the log-signature from the signature by the tensor-algebra power series, verify the free-Lie-algebra (primitive) structure numerically, and state precisely the uniqueness theorem - including its contrapositive for distinct non-reparametrised paths.

---

2. Mathematical Ground Truth & Derivations

2.1 The log-signature as a tensor-algebra series

The signature $S=S(X)\in T(({\mathbb{R}}^d))$ is an element with constant term $1$. Its log-signature is

$$\ell (X)=\log S(X)=\sum _{n\ge 1}\frac{(-1{)}^{n+1}}{n}(S-1{)}^{\otimes n},$$

where $\otimes$ is the concatenation (tensor) product and $\log$ is the formal power-series logarithm of an associative algebra with unit. Truncated at level 3, the components are

$${\ell }^i=S^i,{\ell }^{ij}=S^{ij}-\frac{1}{2}{S}^i{S}^j,{\ell }^{ijk}=S^{ijk}-\frac{1}{2}(S^i{S}^{jk}+S^{ij}{S}^k)+\frac{1}{3}{S}^i{S}^j{S}^k,$$

with all sums over prefix/suffix splits of the word (the tensor-algebra structure - see §3). Level 2 gives the Lévy area matrix, which is antisymmetric:

$${\ell }^{ij}=\frac{1}{2}(S^{ij}-S^{ji}),{\ell }^{ij}=-{\ell }^{ji}.$$

2.2 Why the log-signature is a Lie element

A fundamental structural result (Lyons; Reutenauer) is that $\ell (X)$ is a primitive / Lie element: each level-$k$ piece lies in the degree-$k$ part of the free Lie algebra $\mathrm{Lie}({\mathbb{R}}^d)$. Concretely, for $d=2$ and degree 3, the free Lie algebra has the left-nested bracket basis

$$[1,[1,2]]=e_{112}-2e_{121}+e_{211},[2,[1,2]]=2e_{212}-e_{221}-e_{122},$$

so a degree-3 tensor is a Lie element iff it is in the span of these two - a set of linear coefficient relations that can be checked numerically (and is, in §3). The content of this structure: the log-signature is the minimal non-redundant summary - all shuffle products of lower terms have been removed, so its components are genuinely independent. The number of independent degree-$k$ components of the log-signature (dimension of the free Lie algebra) is, by Witt’s formula, $\frac{1}{k}{\sum }_{d^{\prime }|k}\mu (d^{\prime })d^{k/d^{\prime }}$ - far smaller than $d^k$.

2.3 Signature uniqueness (Lyons; Hambly–Lyons; Boedihardjo et al.)

The Lyons uniqueness theorem states: if two paths $X,Y$ of finite $p$-variation satisfy $S(X)=S(Y)$, then they are tree-equivalent - i.e. $Y$ is obtained from $X$ by a reparametrisation composed with the addition of a tree-like path (a path whose signature is the identity). For bounded-variation paths, Hambly & Lyons (2010) prove the stronger statement that the signature map is injective on the reduced path group: the signature determines the path up to reparametrisation only (no tree-like ambiguity survives in that class). For genuinely rough paths the uniqueness is established by Boedihardjo, Geng, Lyons & Yang (2016).

The practical reading is the contrapositive, which §3 verifies: if two paths are not reparametrisations of each other, their signatures differ. And the constructive content of the theorem is reparametrisation invariance - $S(X\circ \phi )=S(X)$ for any monotone $\phi$ - which is also verified directly.

---

3. Computational Implementation - the log-signature, its Lie structure, invariance, and uniqueness

We (i) build the log-signature to level 3 by the tensor-algebra series from the exact iterated integrals, (ii) verify the level-2 antisymmetry and the level-3 free-Lie-algebra coefficient relations, (iii) verify reparametrisation invariance by collinear subdivision, and (iv) verify the uniqueness contrapositive on two distinct non-reparametrised paths. Stdlib only, deterministic.

Reading the output.

• The log-signature is a Lie element, verified at degree 2 and 3. Level 2 is exactly antisymmetric ($L^{12}=-0.5$, $L^{21}=+0.5$, residual $0$) - i.e. the Lévy area, an element of ${\wedge }^2{\mathbb{R}}^2={\mathrm{Lie}}_2({\mathbb{R}}^2)$. Level 3 satisfies all five coefficient relations defining the free Lie algebra basis $\{[1,[1,2]],[2,[1,2]]\}$ to $\sim 10^{-15}$ ($L^{111}=L^{222}=0$, $L^{121}=-2L^{112}$, etc.). The log-signature is provably not an arbitrary tensor - it lives in the minimal non-redundant Lie subspace, with $L_3=a[1,[1,2]]+b[2,[1,2]]$, $a=-37/12$, $b=-8/3$.
• Reparametrisation invariance holds to machine precision. Subdividing each segment into 2, 3, or 5 collinear pieces changes the path only by a reparametrisation (more breakpoints on the same route); the signature is unchanged to $\sim 10^{-14}$ at level 3. This is the flip side of uniqueness: the signature is blind to the speed of traversal.
• Uniqueness’s contrapositive is real. P and Q share their endpoints yet are not reparametrisations of each other (opposite orientations), and their signatures differ - the areas are $+4.50$ and $-4.50$. A signature that agreed would force a reparametrisation/tree-like relation, which these paths do not have.
• The compression is dramatic. $d=2$, degree 3: the signature has $2^1+2^2+2^3=14$ components; the log-signature has $\dim \mathrm{Lie}=2+1+2=5$ (Witt’s formula). The log-signature is the shuffle-redundancy-free core - the reason it, not the full signature, is what an ML pipeline consumes (§06).

---

4. Failure Modes & First-Principles Breakdowns

1. Computing the log-signature with the wrong product. The tensor-algebra log uses prefix/suffix (concatenation) splits of words - only 2 proper splits at level 3 ($i|jk$, $ij|k$), not the 6 interleavings. Using shuffle splits (or worse, the Itô convention) yields a non-Lie, non-primitive object whose “log-signature” is meaningless. This is the single most common implementation error.
2. Reading the signature (not the log-signature) as independent features. The signature carries shuffle-redundant coordinates - $S^{ij}+S^{ji}=S^i{S}^j$ - so its components are highly collinear. Models fed raw signature features silently re-learn the shuffle relations; the log-signature removes that redundancy by construction.
3. Forgetting that uniqueness is up to tree-like equivalence. For rough paths the signature determines the path only up to tree-like ambiguity (Boedihardjo et al. 2016); for bounded-variation paths (Hambly–Lyons) the map is injective on the reduced path group. Claiming “the signature IS the path” overstates the theorem for the genuinely rough case.
4. Treating reparametrisation invariance as costless. It is a feature for geometric shape (§1), but if the timing of moves is the signal (e.g. the intraday structure of order flow, or the lead-lag in vol), the raw signature discards exactly that information. Lead-lag/time augmentation (§05) is the deliberate re-insertion of time.

---

5. References

• Lyons, Terry J. (1998), Differential equations driven by rough signals, Rev. Mat. Iberoamericana 14(2)
• Hambly, B. & Lyons, T. (2010), Uniqueness for the signature of a path of bounded variation and the reduced path group, Annals of Mathematics 171(1), 109–167
• Boedihardjo, H., Geng, X., Lyons, T., Yang, D. (2016), The signature of a rough path: uniqueness, Advances in Mathematics 293, 720–737
• Reutenauer, C. (1993), Free Lie Algebras (Oxford)
• Chevyrev & Kormilitzin (2016), A Primer on the Signature Method in Machine Learning, arXiv:1603.03788
• Friz & Victoir (2010), Multidimensional Stochastic Processes as Rough Paths

---

6. Connected Graph Bridges

• Back: 02 · The Signature Algebra
• Forward: 04 · Rough Path Theory · 05 · Failure Modes & Practice · Index Hub
• Theory: Linear Algebra · 02 Vectors & Matrices (tensor products) · Linear Algebra & Matrices · Statistics & Inference