Projected Gradient Descent Algorithms for Solving Nonlinear Inverse Problems with Generative Priors

Abstract

Linear temporal logic over finite traces (LTLf) satisfiability checking is a fundamental and hard (PSPACE-complete) problem in the artificial intelligence community. We explore teaching end-to-end neural networks to check satisfiability in polynomial time. It is a challenge to characterize the syntactic and semantic features of LTLf via neural networks. To tackle this challenge, we propose LTLfNet, a recursive neural network that captures syntactic features of LTLf by recursively combining the embeddings of sub-formulae. LTLfNet models permutation invariance and sequentiality in the semantics of LTLf through different aggregation mechanisms of sub-formulae. Experimental results demonstrate that LTLfNet achieves good performance in synthetic datasets and generalizes across large-scale datasets. They also show that LTLfNet is competitive with state-of-the-art symbolic approaches such as nuXmv and CDLSC.

Theoretical Perspectives on Deep Learning Methods in Inverse Problems

“…Non-linear measurement models. While Theorem 1 concerns linear observation models, analogous guarantees have been provided for a variety of non-linear measurement models, including 1-bit observations [85], [105], [69], spiked matrix models [7], [28], phase retrieval [84], [53], principal component analysis [87], and general single-index models [88], [83], [86]. While these each come with their own challenges, the intuition behind their associated results is often similar to that discussed above for the linear model, with the m = O k log Lr δ scaling typically remaining.…”

Theoretical Perspectives on Deep Learning Methods in Inverse Problems

“…Non-linear measurement models. While Theorem 1 concerns linear observation models, analogous guarantees have been provided for a variety of non-linear measurement models, including 1-bit observations [78], [98], [65], spiked matrix models [6], [25], phase retrieval [77], [49], principal component analysis [80], and general single-index models [81], [76], [79]. While these each come with their own challenges, the intuition behind their associated results is often similar to that discussed above for the linear model, with the m = O k log Lr δ scaling typically remaining.…”