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 ℝn 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 (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 ℝn which maps a collection of n-dimensional vectors (boldsymbolxi)i onto a collection of target vectors (boldsymbolyi)i. The key idea is to approximate every target boldsymbolyi by a matrix--vector product of the form boldsymbolwidetildeyi = boldsymbolφ(ti) boldsymbolxi, where the matrix boldsymbolφ(ti) belongs to a one-parameter subgroup of GLn (ℝ). Crucially, the value of the parameter ti ∈ ℝ may change between data pairs (boldsymbolxi, boldsymbolyi) and does not need to be known in advance.