Inverse problems on low-dimensional manifolds

Abstract

We consider abstract inverse problems between infinite-dimensional Banach spaces. These inverse problems are typically nonlinear and ill-posed, making the inversion with limited and noisy measurements a delicate process. In this work, we assume that the unknown belongs to a finitedimensional manifold: this assumption arises in many real-world scenarios where natural objects have a low intrinsic dimension and belong to a certain submanifold of a much larger ambient space. We prove uniqueness and Hölder and Lipschitz stability and derive a globally-convergent reconstruction algorithm from a finite number of measurements. Several examples are discussed.

Continuous Generative Neural Networks

“…More precisely, we show that an injective CGNN yields a Lipschitz stability result for the inverse problem (13); in other words, the inverse map is Lipschitz continuous, and noise in the data is not amplified in the reconstruction. For simplicity, we consider the 1D case with stride 1 2 and non-expansive convolutional layers, but the result can be extended to the 2D case and arbitrary stride as done in Appendix A.4 for Theorem 1.…”

“…The training is done with the Adam optimizer using a learning rate of 0.005 and the loss function, commonly used for VAEs, given by the weighted sum of two terms: the Binary Cross Entropy (BCE) between the original and the generated images and the Kullback-Leibler Divergence (KLD) between the standard Gaussian distribution and the one generated by the encoder in the latent space 1 .…”

“…As a result, the output belongs to the same higher resolution space. For instance, the filter belongs to a space that is twice the resolution of the input space when the stride is 1 2 . This notion of resolution is coherent with the scale parameter used in the continuous setting.…”

“…However the illustration is useful to model the fractional-strided convolution in the continuous setting. A more precise illustration of the 1 2 -strided convolution of equation ( 19) is presented in Figure 5.…”

“…Indeed, another useful property of CGNNs is dimensionality reduction. For ill-posed inverse problems, it is well known that imposing finite-dimensional priors improves the stability and the quality of the reconstruction [6,14,13,5,15], also working with finitely-many measurements [3,31,2,4,1]. In practice, these priors are unknown or cannot be analytically described: yet, they can be approximated by a (trained) CGNN.…”