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Learning expressive probabilistic models correctly describing the data is aubiquitous problem in machine learning. A popular approach for solving it ismapping the observations into a representation space with a simple jointdistribution, which can typically be written as a product of its marginals --thus drawing a connection with the field of nonlinear independent componentanalysis. Deep density models have been widely used for this task, but theirmaximum likelihood based training requires estimating the log-determinant ofthe Jacobian and is computationally expensive, thus imposing a trade-offbetween computation and expressive power. In this work, we propose a newapproach for exact training of such neural networks. Based on relativegradients, we exploit the matrix structure of neural network parameters tocompute updates efficiently even in high-dimensional spaces; the computationalcost of the training is quadratic in the input size, in contrast with the cubicscaling of naive approaches. This allows fast training with objective functionsinvolving the log-determinant of the Jacobian, without imposing constraints onits structure, in stark contrast to autoregressive normalizing flows.

Relative gradient optimization of the Jacobian term in unsupervised deep learning