An Unsupervised Algorithm For Learning Lie Group Transformations

By Jascha Narain Sohl-Dickstein, Jimmy C. Wang, Barbara Olshausen. 2010

52 citations recorded by the source corpus.

We present several theoretical contributions which allow Lie groups to be fit to high dimensional datasets. Transformation operators are represented in their eigen-basis, reducing the computational complexity of parameter estimation to that of training a linear transformation model. A transformation specific "blurring" operator is introduced that allows inference to escape local minima via a smoothing of the transformation space. A penalty on traversed manifold distance is added which encourages the discovery of sparse, minimal distance, transformations between states. Both learning and inference are demonstrated using these methods for the full set of affine transformations on natural image patches. Transformation operators are then trained on natural video sequences. It is shown that the learned video transformations provide a better description of inter-frame differences than the standard motion model based on rigid translation.

Subfields: Computer Vision and Pattern Recognition

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Citations in the corpus

Towards a Definition of Disentangled Representations (2018, background): “Earlier work in this area mostly focused on learning transformations in observation space, such as translation and in-plane rotation of objects [62, 70, 19].”
Learning the Irreducible Representations of Commutative Lie Groups (2014, positive): “This problem was solved by Sohl-Dickstein et al. (2010) using an elegant adaptive smoothing technique, making it possible to learn from large transformations.”
Transformation Properties of Learned Visual Representations (2014, methodology): “Unlike previous work on learning group representations (Rao & Ruderman, 1999; Miao & Rao, 2007; Sohl-Dickstein et al., 2010; Wang et al., 2011; Bruna et al., 2013; Cohen & Welling, 2014), our model does not assume a linear action of the group in the input space, but instead acts linearly on a latent”
Unsupervised Transformation Learning via Convex Relaxations (2017, methodology): “Given data x 1 . . . x n ∈ R d with x i ∈ R d as the nearest neighbor of x i , the nonconvex transformation learning problem objective is This problem is nonconvex, and prior authors have commented on the difﬁculty of optimizing similar objectives [11, 14].”
Towards Nonlinear Disentanglement in Natural Data with Temporal Sparse Coding (2020, background): “But, in reality, all of these factors should be Discrete YT [6] YT [32] YT [32] (1.0, 1.0, 1.0) (square, triangle, star_4, spoke_4) (0,9,...,342,351) Table 7: Natural Sprite Conﬁgs.”