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It is often challenging to pick suitable data features for learning problems. Sometimes certain regions of the data are harder to learn because they are not well characterized by the selected data features. The challenge is amplified when resources for sensing and computation are limited and time-critical, yet reliable decisions must be made. For example, a robotic system for preventing falls of elderly people needs a real-time fall predictor, with low false positive and false negative rates, using a simple wearable sensor to activate a fall prevention mechanism. Here we present a methodology for assessing the learnability of data based on the Lipschitz quotient.We develop a procedure for determining which regions of the dataset contain adversarial data points, input data that look similar but belong to different target classes. Regardless of the learning model, it will be hard to learn such data. We then present a method for determining which additional feature(s) are most effective in improving the predictability of each of these regions. This is a model-independent data analysis that can be executed before constructing a prediction model through machine learning or other techniques. We demonstrate this method on two synthetic datasets and a dataset of human falls, which uses inertial measurement unit signals. For the fall dataset, we identified two groups of adversarial data points and improved the predictability of each group over the baseline dataset, as assessed by Lipschitz, by using 2 different sets of features. This work offers a valuable tool for assessing data learnability that can be applied to not only fall prediction problems, but also other robotics applications that learn from data. 

This paper presents a computational method, called Bootstrapped Koopman Direct Encoding (B-KDE) that allows us to approximate the Koopman operator with high accuracy by combining Koopman Direct Encoding (KDE) with a deep neural network. Deep learning has been applied to the Koopman operator method for finding an effective set of observable functions. Training the network, however, inevitably faces difficulties such as local minima, unless enormous computational efforts are made. Incorporating KDE can solve or alleviate this problem, producing an order of magnitude more accurate prediction. KDE converts the state transition function of a nonlinear system to a linear model in the lifted space of observables that are generated by deep learning. The combined KDE-deep model achieves higher accuracy than that of the deep learning alone. In B-KDE, the combined model is further trained until it reaches a plateau, and this computation is alternated between the neural network learning and the KDE computation. The result of the MSE loss implies that the neural network may get rid of local minima or at least find a smaller local minimum, and further improve the prediction accuracy. The KDE computation however, entails an effective algorithm for computing the inner products of observables and the nonlinear functions of the governing dynamics. Here, a computational method based on the Quasi-Monte Carlo integration is presented. The method is applied to a three-cable suspension robot, which exhibits complex switched nonlinear dynamics due to slack in each cable. The prediction accuracy is compared against its traditional counterparts.