Shape from Shading through Shape Evolution

In this paper, we address the shape-from-shading problem by training deep networks with synthetic images. Unlike conventional approaches that combine deep learning and synthetic imagery, we propose an approach that does not need any external shape dataset to render synthetic images. Our approach consists of two synergistic processes: the evolution of complex shapes from simple primitives, and the training of a deep network for shape-from-shading. The evolution generates better shapes guided by the network training, while the training improves by using the evolved shapes. We show that our approach achieves state-of-the-art performance on a shape-from-shading benchmark

Learning to Prove Theorems via Interacting with Proof Assistants

Humans prove theorems by relying on substantial high-level reasoning and problem-specific insights. Proof assistants offer a formalism that resembles human mathematical reasoning, representing theorems in higher-order logic and proofs as high-level tactics. However, human experts have to construct proofs manually by entering tactics into the proof assistant. In this paper, we study the problem of using machine learning to automate the interaction with proof assistants. We construct CoqGym, a large-scale dataset and learning environment containing 71K human-written proofs from 123 projects developed with the Coq proof assistant. We develop ASTactic, a deep learning-based model that generates tactics as programs in the form of abstract syntax trees (ASTs). Experiments show that ASTactic trained on CoqGym can generate effective tactics and can be used to prove new theorems not previously provable by automated methods. Code is available at https://github.com/princeton-vl/CoqGym.Comment: Accepted to ICML 201

Vector lattices with a Hausdorff uo-Lebesgue topology

We investigate the construction of a Hausdorff uo-Lebesgue topology on a vector lattice from a Hausdorff (o)-Lebesgue topology on an order dense ideal, and what the properties of the topologies thus obtained are. When the vector lattice has an order dense ideal with a separating order continuous dual, it is always possible to supply it with such a topology in this fashion, and the restriction of this topology to a regular sublattice is then also a Hausdorff uo-Lebesgue topology. A regular vector sublattice of $\mathrm{L}_0(X,\Sigma,\mu)$ for a semi-finite measure $\mu$ falls into this category, and the convergence of nets in its Hausdorff uo-Lebesgue topology is then the convergence in measure on subsets of finite measure. When a vector lattice not only has an order dense ideal with a separating order continuous dual, but also has the countable sup property, we show that every net in a regular vector sublattice that converges in its Hausdorff uo-Lebesgue topology always contains a sequence that is uo-convergent to the same limit. This enables us to give satisfactory answers to various topological questions about uo-convergence in this context.Comment: 37 pages. Minor changes; a few references added. Final version, to appear in J. Math. Anal. App