The threewayin and threewayout framework to treat and exploit ambiguity in data Abstract   In this paper, we address ambiguity, intended as a characteristic of any data expression for which a unique meaning cannot be associated by the computational agent for either lack of information or multiple interpretations of the same configuration. In particular, we will propose and discuss ways in which a decisionsupport classifier can accept ambiguous data and make some (informative) value out of them for the decision maker. Towards this goal we propose a set of learning algorithms within what we call the threewayin and threewayout approach, that is, respectively, learning from partially labeled data and learning classifiers that can abstain. This approach is based on orthopartitions, as a common representation framework, and on threeway decisions and evidence theory, as tools to enable uncertain and approximate reasoning and inference. For both the above learning settings, we provide experimental results and comparisons with standard Machine Learning techniques, and show the advantages and promising performances of the proposed approaches on a collection of benchmarks, including a realworld medical dataset.

Efficient Data Representation by Selecting Prototypes with Importance Weights Prototypical examples that best summarize and compactly represent an underlying complex data distribution, communicate meaningful insights to humans in domains where simple explanations are hard to extract. In this paper, we present algorithms with strong theoretical guarantees to mine these data sets and select prototypes, a.k.a. representatives that optimally describes them. Our work notably generalizes the recent work by Kim et al. (2016) where in addition to selecting prototypes, we also associate nonnegative weights which are indicative of their importance. This extension provides a single coherent framework under which both prototypes and criticisms (i.e. outliers) can be found. Furthermore, our framework works for any symmetric positive definite kernel thus addressing one of the key open questions laid out in Kim et al. (2016). By establishing that our objective function enjoys a key property of that of weak submodularity, we present a fast ProtoDash algorithm and also derive approximation guarantees for the same. We demonstrate the efficacy of our method on diverse domains such as retail, digit recognition (MNIST) and on publicly available 40 health questionnaires obtained from the Center for Disease Control (CDC) website maintained by the US Dept. of Health. We validate the results quantitatively as well as qualitatively based on expert feedback and recently published scientific studies on public health, thus showcasing the power of our technique in providing actionability (for retail), utility (for MNIST), and insight (on CDC datasets), which arguably are the hallmarks of an effective interpretable machine learning method.