It is a well-know observation that, given two points $x_1,x_2 \in \mathbb{R}^d$, the path signature associated to their linear interpolation is given by the tensor exponential. Precisely, if $\Delta x$ denotes the linear interpolation $x_1 \rightarrow x_2$, then $$S(\Delta x) = \left(1 , \Delta x, \frac{\Delta x^{\otimes 2}}{2!}, \frac{\Delta x^{\otimes 3}}{3!}, ... \right) =: \exp_\otimes(\Delta x) \in T((\mathbb{R}^d)),$$ where $S(\cdot)$ denotes the signature map and $T((\mathbb{R}^d))$ is the extended tensor algebra. Actually, the signature takes value in a subset $T^1$ of $T((\mathbb{R}^d))$ which can be endowed with an inner product, and thus becomes a Hilbert space if one takes the appropriate completions. I refer to this work for a proper definition, but I assume most readers interested in this post are familiar with this.
Now, instead of considering just a line segment, let us consider a piecewise linear path. Specifically, given a finite number of points $x_1,...,x_n$ in $\mathbb{R}^d$, we consider the path corresponding to successive linear interpolations, i.e. $x_1 \rightarrow x_2$ concatenated with $x_2 \rightarrow x_3$, which, in its turn, we concatenate with $x_3 \rightarrow x_4$, and so on. Let us denote this piecewise linear path by $\textbf{x}$.
Thanks to Chen's identity, we know that the signature of $\textbf{x}$ is given by multiplying in the extended tensor algebra the signatures associated to each interpolation, i.e. $$S(\textbf{x}) = \exp_\otimes(\Delta_1 x) \ \otimes \ \exp_\otimes(\Delta_2 x) \ \otimes \ ... \ \otimes \exp_\otimes(\Delta_{n-1} x),$$ where $\Delta_i x$ denotes the interpolated path $x_i \rightarrow x_{i+1}$, and $\otimes$ is the multiplication defined in $T((\mathbb{R}^d))$.
Lastly, let $\textbf{y}$ denote some other piecewise linear interpolation of $y_1,...,y_n\in \mathbb{R}^d$, and let $\langle \cdot, \cdot \rangle_{T^1}$ denote the aforementioned inner product. My question is:
Can we express $\langle S(\textbf{x}), S(\textbf{y}) \rangle_{T^1}$ via the simpler inner products $\langle \exp_\otimes(\Delta_i x), \exp_\otimes(\Delta_i y) \rangle_{T^1}$ with $i \in \{1,...,n-1\}$?
