41 research outputs found
Recognition of mathematical handwriting on whiteboards
Automatic recognition of handwritten mathematics has enjoyed significant improvements in the past decades. In particular, online recognition of mathematical formulae has seen a number of important advancements. However, in reality most mathematics is still taught and developed on regular whiteboards and offline recognition remains an open and challenging task in this area. In this thesis we develop methods to recognise mathematics from static images of handwritten expressions on whiteboards, while leveraging the strength of online recognition systems by transforming offline data into online information. Our approach is based on trajectory recovery techniques, that allow us to reconstruct the actual stroke information necessary for online recognition. To this end we develop a novel recognition process especially designed to deal with whiteboards by prudently extracting information from colour images. To evaluate our methods we use an online recogniser for the recognition task, which is specifically trained for recognition of maths symbols. We present our experiments with varying quality and sources of images. In particular, we have used our approach successfully in a set of experiments using Google Glass for capturing images from whiteboards, in which we achieve highest accuracies of 88.03% and 84.54% for segmentation and recognition of mathematical symbols respectively
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Finding Critical and Gradient-Flat Points of Deep Neural Network Loss Functions
Despite the fact that the loss functions of deep neural networks are highly non-convex, gradient-based optimization algorithms converge to approximately the same performance from many random initial points. This makes neural networks easy to train, which, combined with their high representational capacity and implicit and explicit regularization strategies, leads to machine-learned algorithms of high quality with reasonable computational cost in a wide variety of domains.
One thread of work has focused on explaining this phenomenon by numerically characterizing the local curvature at critical points of the loss function, where gradients are zero. Such studies have reported that the loss functions used to train neural networks have no local minima that are much worse than global minima, backed up by arguments from random matrix theory. More recent theoretical work, however, has suggested that bad local minima do exist.
In this dissertation, we show that one cause of this gap is that the methods used to numerically find critical points of neural network losses suffer, ironically, from a bad local minimum problem of their own. This problem is caused by gradient-flat points, where the gradient vector is in the kernel of the Hessian matrix of second partial derivatives. At these points, the loss function becomes, to second order, linear in the direction of the gradient, which violates the assumptions necessary to guarantee convergence for second order critical point-finding methods. We present evidence that approximately gradient-flat points are a common feature of several prototypical neural network loss functions