Encoding Invariances in Deep Generative Models

Reliable training of generative adversarial networks (GANs) typically require massive datasets in order to model complicated distributions. However, in several applications, training samples obey invariances that are \textit{a priori} known; for example, in complex physics simulations, the training data obey universal laws encoded as well-defined mathematical equations. In this paper, we propose a new generative modeling approach, InvNet, that can efficiently model data spaces with known invariances. We devise an adversarial training algorithm to encode them into data distribution. We validate our framework in three experimental settings: generating images with fixed motifs; solving nonlinear partial differential equations (PDEs); and reconstructing two-phase microstructures with desired statistical properties. We complement our experiments with several theoretical results

Constrained image generation using binarized neural networks with decision procedures

We consider the problem of binary image generation with given properties. This problem arises in a number of practical applications, including generation of artificial porous medium for an electrode of lithium-ion batteries, for composed materials, etc. A generated image represents a porous medium and, as such, it is subject to two sets of constraints: topological constraints on the structure and process constraints on the physical process over this structure. To perform image generation we need to define a mapping from a porous medium to its physical process parameters. For a given geometry of a porous medium, this mapping can be done by solving a partial differential equation (PDE). However, embedding a PDE solver into the search procedure is computationally expensive. We use a binarized neural network to approximate a PDE solver. This allows us to encode the entire problem as a logical formula. Our main contribution is that, for the first time, we show that this problem can be tackled using decision procedures. Our experiments show that our model is able to produce random constrained images that satisfy both topological and process constraints

Precise quantitative analysis of binarized neural networks: a BDD-based approach

As a new programming paradigm, neural-network-based machine learning has expanded its application to many real-world problems. Due to the black-box nature of neural networks, verifying and explaining their behavior are becoming increasingly important, especially when they are deployed in safety-critical applications. Existing verification work mostly focuses on qualitative verification, which asks whether there exists an input (in a specified region) for a neural network such that a property (e.g., local robustness) is violated. However, in many practical applications, such an (adversarial) input almost surely exists, which makes a qualitative answer less meaningful. In this work, we study a more interesting yet more challenging problem, i.e., quantitative verification of neural networks, which asks how often a property is satisfied or violated. We target binarized neural networks (BNNs), the 1-bit quantization of general neural networks. BNNs have attracted increasing attention in deep learning recently, as they can drastically reduce memory storage and execution time with bit-wise operations, which is crucial in recourse-constrained scenarios, e.g., embedded devices for Internet of Things. Toward quantitative verification of BNNs, we propose a novel algorithmic approach for encoding BNNs as Binary Decision Diagrams (BDDs), a widely studied model in formal verification and knowledge representation. By exploiting the internal structure of the BNNs, our encoding translates the input-output relation of blocks in BNNs to cardinality constraints, which are then encoded by BDDs. Based on the new BDD encoding, we develop a quantitative verification framework for BNNs where precise and comprehensive analysis of BNNs can be performed. To improve the scalability of BDD encoding, we also investigate parallelization strategies at various levels. We demonstrate applications of our framework by providing quantitative robustness verification and interpretability for BNNs. An extensive experimental evaluation confirms the effectiveness and efficiency of our approach