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.