Nonlinear constraint satisfaction for compressive autoencoders using instance-specific linear operators

Neural networks and traditional compression techniques can reproduce inputs with high accuracy. However, downstream quantities derived from the reproduced inputs might be distorted because the relationship between reconstruction errors on raw data and errors on downstream quantities is not straightforward. Downstream quantities are crucial in many applications. For example, they play key roles in scientific simulations designed to analyze physics phenomena. These quantities of interest (QoI) are usually obtained via complex post-analysis and are nonlinear on raw data. In this paper, we present a post-processing algorithm for nonlinear constraint satisfaction that can be applied to outputs of neural networks or any other algorithms. We assume that the nonlinear QoI can be approximated by QoI neural networks that have multiple fully connected layers with piecewise linear activation units (PLUs). In this work, the last layer features of the neural networks become nonlinear QoI. Given the true features and network weights, we seek to transform the output of compressive autoencoders such that the reconstructions approximately satisfy the nonlinear QoI. To this end, we introduce a nonlinear-to-linear transformation wherein the entire QoI network is represented as instance-specific linear operators. The use of PLUs (or Leaky ReLUs) is critical to this transformation. The conversion of the entire set of weights and PLU nonlinearities into instance-specific linear operators permits us to recast nonlinear constraint satisfaction as the satisfaction of linear constraints imposed on autoencoder outputs. This transformation also comes at a cost: we have to record the "bitstream"path taken by each instance through the QoI network, specifically, the set of chosen PLU linear segments. Increasing the cardinality of PLU segments usually adversely affects the compression ratio since more bits have to be stored. The results on a standard image dataset (Fashion) and a scientific data application (XGC) clearly demonstrate the ability of the nonlinear-by-linear approach to perform nonlinear constraint satisfaction while compressing the primary data.