Inverting Deep Generative models, One layer at a time

TL;DR

This paper investigates the theoretical and practical aspects of inverting deep ReLU generative models, providing polynomial-time algorithms for single-layer inversion, NP-hardness results for multiple layers, and high-probability exact recovery conditions for deep models.

Contribution

The paper introduces novel polynomial-time inversion algorithms for single-layer models, proves NP-hardness for multi-layer inversion, and establishes conditions for exact recovery in deep generative models with random weights.

Findings

Single-layer inversion can be done exactly via linear programming.

Multi-layer inversion is NP-hard and pre-image sets can be non-convex.

Exact recovery is possible in polynomial time for deep models with expanding layers and random weights.

Abstract

We study the problem of inverting a deep generative model with ReLU activations. Inversion corresponds to finding a latent code vector that explains observed measurements as much as possible. In most prior works this is performed by attempting to solve a non-convex optimization problem involving the generator. In this paper we obtain several novel theoretical results for the inversion problem. We show that for the realizable case, single layer inversion can be performed exactly in polynomial time, by solving a linear program. Further, we show that for multiple layers, inversion is NP-hard and the pre-image set can be non-convex. For generative models of arbitrary depth, we show that exact recovery is possible in polynomial time with high probability, if the layers are expanding and the weights are randomly selected. Very recent work analyzed the same problem for gradient descent inversion. Their analysis requires significantly higher expansion (logarithmic in the latent dimension) while our proposed algorithm can provably reconstruct even with constant factor expansion. We also provide provable error bounds for different norms for reconstructing noisy observations. Our empirical validation demonstrates that we obtain better reconstructions when the latent dimension is large.