Learning a Lie Algebra from Unlabeled Data Pairs

Deep convolutional networks (convnets) show a remarkable ability to learn disentangled representations. In recent years, the generalization of deep learning to Lie groups beyond rigid motion in $\mathbb{R}^n$ has allowed to build convnets over datasets with non-trivial symmetries, such as patterns over the surface of a sphere. However, one limitation of this approach is the need to explicitly define the Lie group underlying the desired invariance property before training the convnet. Whereas rotations on the sphere have a well-known symmetry group ($\mathrm{SO}(3)$), the same cannot be said of many real-world factors of variability. For example, the disentanglement of pitch, intensity dynamics, and playing technique remains a challenging task in music information retrieval. This article proposes a machine learning method to discover a nonlinear transformation of the space $\mathbb{R}^n$ which maps a collection of $n$-dimensional vectors $(\boldsymbol{x}_i)_i$ onto a collection of target vectors $(\boldsymbol{y}_i)_i$. The key idea is to approximate every target $\boldsymbol{y}_i$ by a matrix--vector product of the form $\boldsymbol{\widetilde{y}}_i = \boldsymbolφ(t_i) \boldsymbol{x}_i$, where the matrix $\boldsymbolφ(t_i)$ belongs to a one-parameter subgroup of $\mathrm{GL}_n (\mathbb{R})$. Crucially, the value of the parameter $t_i \in \mathbb{R}$ may change between data pairs $(\boldsymbol{x}_i, \boldsymbol{y}_i)$ and does not need to be known in advance.