Globally Injective ReLU Networks
This work addresses theoretical foundations for injectivity in neural networks, which is crucial for well-posedness in inverse problems and generative models, but it is incremental as it builds on existing mathematical frameworks.
The paper establishes sharp characterizations of injectivity for ReLU networks, showing that an expansivity factor of two is necessary and sufficient for layerwise injectivity, while global injectivity with Gaussian matrices requires larger expansivity between 3.4 and 10.5, and proves that any Lipschitz map can be approximated by an injective ReLU network.
Injectivity plays an important role in generative models where it enables inference; in inverse problems and compressed sensing with generative priors it is a precursor to well posedness. We establish sharp characterizations of injectivity of fully-connected and convolutional ReLU layers and networks. First, through a layerwise analysis, we show that an expansivity factor of two is necessary and sufficient for injectivity by constructing appropriate weight matrices. We show that global injectivity with iid Gaussian matrices, a commonly used tractable model, requires larger expansivity between 3.4 and 10.5. We also characterize the stability of inverting an injective network via worst-case Lipschitz constants of the inverse. We then use arguments from differential topology to study injectivity of deep networks and prove that any Lipschitz map can be approximated by an injective ReLU network. Finally, using an argument based on random projections, we show that an end-to-end -- rather than layerwise -- doubling of the dimension suffices for injectivity. Our results establish a theoretical basis for the study of nonlinear inverse and inference problems using neural networks.