Image reconstruction enhanced by regularizers, e.g., to enforce sparsity, low rank

Many learning problems are modeled as optimization problems. With the explosion in deep learning, many of these problems are modeled as non-convex optimization problems -- either by using non-convex objective functions or by the addition of non-convex constraints. While well-studied algorithms with fast convergence guarantees are available for convex problems, such mathematical tools are more limited for non-convex problems. In fact, the general class of non-convex optimization problems is known to be NP-hard (Jain et al., 2017). Coming up with global certificates of optimality is the major difficulty in solving non-convex problems. In this paper, we take the first steps towards solving a special class of non-convex problems, called invex problems, which attain global minima at every stationary point (Hanson, 1981; Ben-Israel and Mond, 1986). Invex problems are tractable in the sense that we can use local certificates of optimality to establish the global optimality conditions.

Connected decision boundaries are useful in several tasks like image segmentation, clustering, alpha-shape or defining a region in nD-space. However, the machine learning literature lacks methods for generating connected decision boundaries using neural networks. Thresholding an invex function, a generalization of a convex function, generates such decision boundaries. This paper presents two methods for constructing invex functions using neural networks. The first approach is based on constraining a neural network with Gradient Clipped-Gradient Penality (GCGP), where we clip and penalise the gradients. In contrast, the second one is based on the relationship of the invex function to the composition of invertible and convex functions. We employ connectedness as a basic interpretation method and create connected region-based classifiers. We show that multiple connected set based classifiers can approximate any classification function. In the experiments section, we use our methods for classification tasks using an ensemble of 1-vs-all models as well as using a single multiclass model on larger-scale datasets. The experiments show that connected set-based classifiers do not pose any disadvantage over ordinary neural network classifiers, but rather, enhance their interpretability. We also did an extensive study on the properties of invex function and connected sets for interpretability and network morphism with experiments on simulated and real-world data sets. Our study suggests that invex function is fundamental to understanding and applying locality and connectedness of input space which is useful for various downstream tasks.