Empirical Bounds on Linear Regions of Deep Rectifier Networks

We can compare the expressiveness of neural networks that use rectified linear units (ReLUs) by the number of linear regions, which reflect the number of pieces of the piecewise linear functions modeled by such networks. However, enumerating these regions is prohibitive and the known analytical bounds are identical for networks with same dimensions. In this work, we approximate the number of linear regions through empirical bounds based on features of the trained network and probabilistic inference. Our first contribution is a method to sample the activation patterns defined by ReLUs using universal hash functions. This method is based on a Mixed-Integer Linear Programming (MILP) formulation of the network and an algorithm for probabilistic lower bounds of MILP solution sets that we call MIPBound, which is considerably faster than exact counting and reaches values in similar orders of magnitude. Our second contribution is a tighter activation-based bound for the maximum number of linear regions, which is particularly stronger in networks with narrow layers. Combined, these bounds yield a fast proxy for the number of linear regions of a deep neural network.Comment: AAAI 202

Personalized Automatic Estimation of Self-reported Pain Intensity from Facial Expressions

Pain is a personal, subjective experience that is commonly evaluated through visual analog scales (VAS). While this is often convenient and useful, automatic pain detection systems can reduce pain score acquisition efforts in large-scale studies by estimating it directly from the participants' facial expressions. In this paper, we propose a novel two-stage learning approach for VAS estimation: first, our algorithm employs Recurrent Neural Networks (RNNs) to automatically estimate Prkachin and Solomon Pain Intensity (PSPI) levels from face images. The estimated scores are then fed into the personalized Hidden Conditional Random Fields (HCRFs), used to estimate the VAS, provided by each person. Personalization of the model is performed using a newly introduced facial expressiveness score, unique for each person. To the best of our knowledge, this is the first approach to automatically estimate VAS from face images. We show the benefits of the proposed personalized over traditional non-personalized approach on a benchmark dataset for pain analysis from face images.Comment: Computer Vision and Pattern Recognition Conference, The 1st International Workshop on Deep Affective Learning and Context Modelin