# Lie-Adaptive Inversion of Signature via Pfeffer-Seigal-Sturmfels Algorithm

Rémi Vaucher

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Rémi Vaucher. Lie-Adaptive Inversion of Signature via Pfeffer-Seigal-Sturmfels Algorithm. Geometric Science of Informations, Oct 2025, Saint Malo, France. hal-05197804

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Submitted on 3 Aug 2025

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Distributed under a Creative Commons Attribution 4.0 International License
# Lie-Adaptive Inversion of Signature via Pfeffer-Seigal-Sturmfels Algorithm

Rémi Vaucher¹,²[0009-0005-4315-7870]

¹ Laboratoire ERIC, Bron 69676, France

² Halias Technologies, Meylan 38240, France, remi.vaucher@halias.fr

**Abstract.** Since rough path signatures were introduced into machine learning by Terry Lyons [5], the practical inversion of the Signature transform remains an open problem. Several approaches have been proposed, ranging from insertion methods [1,3] to optimal transport techniques [7]. Each of these methods is only an approximation of the inversion, based on optimization problems. Our work extends the framework of Pfeffer, Seigal, and Sturmfels [6] to incorporate the Lie group structure of $G^N(\mathbb{R}^d)$, the signature space. The original framework uses an expression of the i-th level of signature $S^{(i)}$ as a sequence of k-mode tensor product between a given functional base and the decomposition matrix corresponding to the target path in this base. In this paper, we aim to go beyond a mean-squared loss by constructing a sequence of tensors that respects the Lie group structure. Additionally, we propose a method to recover a path of wanted length rather than merely the shortest one. Finally, we evaluate our approach on multi-dimensional Brownian paths.

**Keywords:** Signature Inversion · Lie Group · Optimization · Tensors.

## 1 Introduction : Expressing Signature of paths in the piecewise linear dictionary.

Consider $X : [a, b] \to \mathbb{R}^d$ a path of bounded variations:

$$X \in BV([a, b], \mathbb{R}^d) = \left\{ X : [a, b] \to \mathbb{R}^d \in L^1, \sup_{P \in \mathcal{P}} \sum_{i=0}^{n_P-1} \|X(t_{i+1}) - X(t_i)\| < \infty \right\}.$$

where $\mathcal{P}$ is the set of every subdivision of $[a, b]$.

The signature of $X$, noted $S_{[a,b]}(X)$, is the tensor sequence of increasing dimension

$$S_{[a,b]}(X) = 1 \oplus S^{(1)}(X) \oplus S^{(2)}(X) \oplus \dots \oplus S^{(k)}(X) \oplus \dots \in T(\mathbb{R}^d).$$

with $S_{[a,b]}^{(k)}(X) \in \mathbb{R}^{\otimes d}$ and

$$(S^{(k)}(X))_{i_1 \dots i_k} = \int_{a < t_1 < \dots < t_k < b} \dots \int_{a < t_1 < \dots < t_k < b} dX^{i_1}(t_{i_1}) \dots dX^{i_k}(t_{i_k}).$$
Usually, we consider $S^{\leq k}(X) = 1 \oplus S^{(1)}(X) \oplus \cdots \oplus S^{(k)}(X)$ as the **truncated signature** of $X$. This object lives in a **Lie group** $G^k(\mathbb{R}^d)$ called **the signature space**.

Signature has many useful properties:

- Invariance under translation,

- Invariance under reparametrization,

- Signature is characteristic of the distribution of the path $X$.

We invite the reader to discover everything about signature in [2].

This paper addresses the problem of inverting this signature, i.e. recovering the path $X^*$ from its signature $S(X^*)$. Inverting the signature is of fundamental interest in machine learning and stochastic modeling, as it enables the reconstruction or generation of paths from compact tensorial summaries, with potential applications in signal generation, control, and trajectory synthesis.

One method [6] is to solve the following least-square regression problem:

Consider that there exists a dictionary $\Psi = (\psi^1, \dots, \psi^m) \in BV([a, b], \mathbb{R}^m)$ and a matrix $A \in \mathcal{M}_{d,m}(\mathbb{R})$ such that $X = A\Psi^T$. Then, $X^*$ is a solution of the following minimization problem

$$ \min_{A \in \mathbb{R}^{n \times m}} \left\| S^{(3)}(X) - [S^{(3)}(\Psi); A, A, A] \right\|_F^2 \quad (1) $$

where $A \in \mathbb{R}^{d \times m}$, and, for $C \in \mathbb{R}^{m \times m \times m}$, the operation $[C; \cdot, \cdot, \cdot]$ is defined by

$$ \begin{aligned} [C; A, A, A]_{\alpha, \beta, \gamma} &= C \times_1 A \times_2 A \times_3 A \\ &= \sum_{i=1}^{m} \sum_{j=1}^{m} \sum_{k=1}^{m} c_{ijk} a_{\alpha,i} a_{\beta,j} a_{\gamma,k}. \end{aligned} \quad (2) $$

In their paper [6], Pfeffer, Seigal and Sturmfels use the characterization (2) for the third tensor of signature - the smallest order to completely define the previous ones - and retrieve $A$ by an optimization on (1). Further details on this characterization can be found in [8]. We used (1) as a starting point for the development of our method.

Since each sampled path is considered as a linear interpolation between successive sampling times, we restrict $\Psi$ to be the piecewise linear basis defined as:

$$ \psi_i(t) = \begin{cases} 0 & \text{if } t \le \frac{i-1}{m} \\ mt - (i-1) & \text{if } \frac{i-1}{m} < t < \frac{i}{m} \\ 1 & \text{if } t \ge \frac{i}{m}. \end{cases} $$

The authors present results on simple theoretical paths. However, this optimization problem is fundamentally a **third-order tensor optimization**, involving sequences in $\mathbb{R}^{d \times d \times d}$. We propose to adapt problem (1) in a Lie adaptive way: The truncated signature space $G^N(\mathbb{R}^d)$ has a Lie group structure,
which reflects the algebraic and geometric properties of path concatenation and reparametrization. Preserving this structure during inversion ensures consistency with the underlying path space and avoids unrealistic artifacts.

As with most signature inversion methods, the Pfeffer-Seigal-Sturmfels algorithm seeks the shortest possible paths. In contrast, we adopt a different strategy by imposing an **exact** path length, introducing an augmentation of the path X. Nevertheless, it is important to recall that the **signature is invariant under reparametrization**. Consequently, our algorithm may recover a reparametrized version of the original path, a feature that we cannot explicitly control.

# 2 Changes in the optimization problem.

## 2.1 Adjust the method to the Lie group structure.

Addition (and then, subtraction) are not defined in the signature space (noted $G(\mathbb{R}^d)$ or $G^N(\mathbb{R}^d)$ for truncated signature space). However, problem (1) is valid in this space, as adding two k-th order tensor gives a valid k-th order tensor. However, this method does not guarantee that the sequence evolves consistently within $G^N(\mathbb{R}^d)$.

As a matter of fact, one could define clearly convergence of a signatures sequence:

**Theorem 1.** Consider $g_n$ and $g$ in $G^N(\mathbb{R}^d)$. Then:

$$ \lim_{n \to +\infty} \|g_n - g\| = 0 \iff \lim_{n \to 0} \|g_n^{-1} \otimes g - \mathbf{1}\| \to 0 $$

where $g^{-1}$ is the signature of the time reversal path, and $\mathbf{1} = (1, 0, \dots, 0, \dots)$.

Returning to our optimization problem, we define $\pi_3(g) : G^N(\mathbb{R}^d) \to (\mathbb{R}^d)^{\otimes 3}$ the canonical projection on the third order tensor, we get

$$ (1) = \min_{A \in \mathbb{R}^{d \times m}} \left\| \pi_3 \left( S^3(X) \otimes S^3(A \overleftarrow{\Psi}) \right) \right\|_F^2 \quad (3) $$

Computing $\pi_3(S^3(X) \otimes S^3(A \overleftarrow{\Psi}))$ require all level of signature up to order 3. One one hand, we can easily compute $S^{(2)}(X)$ and $S^{(1)}(X)$ with the **shuffle products**. With the convention $s_{i_1 \dots i_k} = (S^{(k)}(X))_{i_1, \dots, i_k}$:

$$ \begin{cases} s_i^3 = 3s_{iii} \\ s_i s_{jk} = s_{ijk} + s_{jik} + s_{jki} \end{cases} $$

On the other hand, $S^{(k)}(A\Psi) = [S^{(k)}(X); A, A, \dots, A]_{\underbrace{\dots}_{k \text{ times}}}$.
## 2.2 Constraint on path length.

The problem (1) suffers from a lack of uniqueness. To overcome this, the existing solutions drive the optimization problem by a constraint on path length, using the fact that there exist an unique path of shortest length with the exact $S^N(X)$. However, such a solution may significantly deviate from the original one (see [6] Figure 2).

Our approach differs: we allow for multiple possible solutions in exchange for ensuring an exact path length, with the assumption that any exact length path will be close enough to the original one. To accomplish such task, we try to retrieve an **augmentation** of X.

$$ \tilde{X} = (\mathrm{Length}(X |_{[0,t]}), X_t) \in BV([0; T], \mathbb{R}^{d+1}) $$

where $\mathrm{Length}(X_{[0,t]})$ is the length of X between $X_0$ and $X_t$. Moreover:

$$ \mathrm{Length}(X) = \mathrm{Length}(A\Psi) = \sum_{i=1}^{m} \sqrt{\sum_{j=1}^{d} a_{ij}^2} $$

and, as we defined $\tilde{X}^1 = \mathrm{Length}(X)$

$$ S^{111}(\tilde{X}) = \frac{1}{6} (\mathrm{Length}(X |_{[0,T]}) - \mathrm{Length}(X |_{[0,0]}))^{3} = \frac{1}{6} \mathrm{Length}(X)^{3}. $$

Our optimization problem becomes:

$$ \left\{ \begin{array}{l} \min_{A \in \mathbb{R}^{d \times m}} \left\| \pi_3 \left( S^N (\tilde{X} \star A \overleftarrow{\Psi}) \right) \right\|_F^2 \\ S^{111}(X) = \left( \sum_{i=1}^{m} \sqrt{\sum_{j=2}^{d+1} a_{ij}^2} \right)^3. \end{array} \right. \qquad (4) $$

## 2.3 Boundary constraint:

Now that we have a length constraint, we define a **boundary constraint** to ensure that we reach the exact $X_T$. First, we have, for any $i \in \{1, ..., N\}$

$$ S^{ii...i}(X) = \frac{1}{k!} (X_T^i - X_0^i)^k. $$

It is easy to express these coefficients from A. Defining

$$ f_k : \mathbb{R}^d \to \mathbb{R}^d, (x_1, ..., x_d) \mapsto \frac{1}{k!} (x_1^k, ..., x_d^k), $$

we have:

$$ (S^{ii...i}(X))_{1 \le i \le d} = f_k(A(\Psi(T) - \Psi(0))). $$
Our final problem is then:

$$
\begin{equation}
\left\{
\begin{array}{l}
\displaystyle \min_{A \in \mathbb{R}^{d \times m}} \left\| \pi_3 \left( S^N (\tilde{X} \star A \overleftarrow{\Psi}) \right) \right\|_F^2 \\[1.5ex]
\displaystyle S^{111}(X) = \left( \sum_{i=1}^m \sqrt{\sum_{j=2}^{d+1} a_{ij}^2} \right)^3 \\[1.5ex]
\displaystyle \operatorname{Diag}(S^{(3)}(X)) = f_k \left( A (\Psi(T) - \Psi(0)) \right)
\end{array}
\right.
\end{equation}
$$

3 Experiments

We performed several reconstruction:

- Two bidimensional paths on $[0, 1]$:

$$
X_t = (\mathrm{Length}(f|_{[0,t]}), f(t))
$$

with $f(t) = \cos(5t) + \varepsilon_t$, $f(t) = \exp\left(-\left(\frac{5}{t} - 0.5\right)^2\right) + \varepsilon_t$, and $\varepsilon_t \sim \mathcal{N}(0, \sigma^2)$
for every $t \in [0, 1]$.

- A noisy circle

$$
(\cos(t) + \varepsilon_t, \sin(t) + \nu_t)
$$

with $\varepsilon_t, \nu_t \sim \mathcal{N}(0, \sigma^2)$

- Two brownian motions uni and bidimensional:

$$
dX_t \sim \mathcal{N}(0, \Sigma^2), \quad \forall t \in [0, 1].
$$

We used the piecewise linear basis $\Psi$.

We minimized:

$$
\begin{align*}
\mathcal{L}(A) = & \left\| \pi_3 \left( S^{(3)}(\tilde{X}) \otimes \bigoplus_{i=0}^3 [S^{(3)}(\Psi^{-1}); A, A, A] \right) \right\|_F^2 \\
& + \lambda_1 \left| S^{111}(\tilde{X}) - \sum_{j=1}^m \sqrt{\sum_{i=2}^{d+1} a_{ij}^2} \right| \\
& + \lambda_2 \left\| \operatorname{Diag}(S^{(3)}(\tilde{X})) - \left[ (A(\Psi(1) - \Psi(0)))^{(3)} \right]_{iii} \right\|_{1 \le i \le d+1} \\
& + \lambda_{\text{Ridge}} \|A\|_2.
\end{align*}
$$

We used Adam optimizer for 2 × 10⁵ steps. Figures 1 to 4 shows preliminary
results. These experiments (and others) are available on https://github.com/
RemiVaucher/Thesis/tree/main/Chapter5.

- For a good choice of hyperparameters, our reconstruction gives a good approximation of the paths shape with only order three of signature.
- The reconstructed path appears smoother than the original (see Figure 3). An improvement would be to retrieve the distribution of a stochastic process. This will be addressed in future work.

- We have a correct estimation of the variations between $t_0$ and $t_{final}$.

- The paths lengths appear to be accurately reconstructed.

- Average difference at each time looks like a gaussian (see Figure 5).

<figure><img src="image_2.png" /><figcaption>Fig. 1: Reconstruction of a noisy $\cos(t)$.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Fig. 2: Reconstruction for a noisy $\sin(t)$.</figcaption></figure>

# 4 Conclusion and further works

In this paper, we provide a new way to invert signature transform. Although our objective is not to recover the shortest path corresponding to our signature, we obtained promising results for reconstructing Brownian motion paths, at least
<figure><img src="image_4.png" /><figcaption>Fig. 3</figcaption></figure>

through the absolute error metrics. One next study would be to check if our reconstruction is expressing the probabilistic behavior of a brownian motion.

Moreover, this method opens several avenues for future work:

- Random generation of time series: inverting the signature is a kind of grail for signal generation, as generating third order tensor of signature could be easier as generating all the path, especially with high dimensional path.

- Potential improvements in the generation of diffusion processes: by learning how to generate signatures of diffusion paths, one can produce the reverse diffusion paths from its signature.

- Following [4], our next step would be to optimize jointly $C$ in $[C; A, A, A]$ with $A$. In this case, the optimization problem (1) have global minimizer.

## References

1. Chang, J., Lyons, T.: Insertion algorithm for inverting the signature of a path. arXiv preprint arXiv:1907.08423 (2019)

2. Chevyrev, I., Kormilitzin, A.: A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788 (2016)

3. Fermanian, A., Chang, J., Lyons, T., Biau, G.: The insertion method to invert the signature of a path. arXiv preprint arXiv:2304.01862 (2023)
<figure><img src="image_5.png" /><figcaption>Fig. 4</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 5: Error distribution for 200 brownian motion with 5 dimension.</figcaption></figure>

4. Frandsen, A., Ge, R.: Optimization landscape of tucker decomposition. Mathematical Programming **193**(2), 687–712 (2022)

5. Lyons, T.J.: Differential equations driven by rough signals. *Revista Matemática Iberoamericana* **14**(2), 215–310 (1998)

6. Pfeffer, M., Seigal, A., Sturmfels, B.: Learning paths from signature tensors. *SIAM Journal on Matrix Analysis and Applications* **40**(2), 394–416 (2019)

7. Rauscher, M., Scagliotti, A., Pagginelli Patricio, F.: Shortest-path recovery from signature with an optimal control approach. *Mathematics of Control, Signals, and Systems* pp. 1–33 (2024)
8. Seigal, A.L.: Structured tensors and the geometry of data. University of California, Berkeley (2019)