Unified Forms for Kalman and Finite Impulse Response Filtering and Smoothing

The Kalman filter and smoother are optimal state estimators under certain conditions. The Kalman filter is typically presented in a predictor/corrector format, but the Kalman smoother has never been derived in that format. We derive the Kalman smoother in a predictor/corrector format, thus providing a unified form for the Kalman filter and smoother. We also discuss unbiased finite impulse response (UFIR) filters and smoothers, which can provide a suboptimal but robust alternative to Kalman estimators. We derive two unified forms for UFIR filters and smoothers, and we derive lower and upper bounds for their estimation error covariances

Unified Forms for Kalman and Finite Impulse Response Filtering and Smoothing

The Kalman filter and smoother are optimal state estimators under certain conditions. The Kalman filter is typically presented in a predictor/corrector format, but the Kalman smoother has never been derived in that format. We derive the Kalman smoother in a predictor/corrector format, thus providing a unified form for the Kalman filter and smoother. We also discuss unbiased finite impulse response (UFIR) filters and smoothers, which can provide a suboptimal but robust alternative to Kalman estimators. We derive two unified forms for UFIR filters and smoothers, and we derive lower and upper bounds for their estimation error covariances

Definition and learning of logic-based kernels for categorical data, and application to collaborative filtering

The continuous pursuit of better prediction quality has gradually led to the development of increasingly complex machine learning models, e.g., deep neural networks. Despite the great success in many domains, the black-box nature of these models makes them not suitable for applications in which the model understanding is at least as important as the prediction accuracy, such as medical applications. On the other hand, more interpretable models, as decision trees, are in general much less accurate. In this thesis, we try to merge the positive aspects of these two realities, by injecting interpretable elements inside complex methods. We focus on kernel methods which have an elegant framework that decouples learning algorithms from data representations. In particular, the first main contribution of this thesis is the proposal of a new family of Boolean kernels, i.e., kernels defined on binary data, with the aim of creating interpretable feature spaces. Assuming binary input vectors, the core idea is to build embedding spaces in which the dimensions represent logical formulas (of a specific form) of the input variables. As a result the solution of a kernel machine can be represented as a weighted sum of logical propositions, and this allows to extract from it human-readable rules. Our framework provides a constructive and efficient way to calculate Boolean kernels of different forms (e.g., disjunctive, conjunctive, DNF, CNF). We show that on binary classification tasks over categorical datasets the proposed kernels achieve state-of-the-art performances. We also provide some theoretical properties about the expressiveness of such kernels. The second main contribution consists in the development of a new multiple kernel learning algorithm to automatically learn the best representation (avoiding the validation). We start from a theoretical result which states that, under mild conditions, any dot-product kernel can be seen as a linear non-negative combination of Boolean conjunctive kernels. Then, from this combination, our MKL algorithm learns non-parametrically the best combination of the conjunctive kernels. This algorithm is designed to optimize the radius-margin ratio of the combined kernel, which has been demonstrated of being an upper bound of the Leave-One-Out error. An extensive empirical evaluation, on several binary classification tasks, shows how our MKL technique is able to outperform state-of-the-art MKL approaches. A third contribution is the proposal of another kernel family for binary input data, which aims to overcome the limitations of the Boolean kernels. In this case the focus is not exclusively on the interpretability, but also on the expressivity. With this new framework, that we dubbed propositional kernel framework, is possible to build kernel functions able to create feature spaces containing almost any kind of logical propositions. Finally, the last contribution is the application of the Boolean kernels to Recommender Systems, specifically, on top-N recommendation tasks. First of all, we propose a novel kernel-based collaborative filtering method and we apply on top of it our Boolean kernels. Empirical results on several collaborative filtering datasets show how less expressive kernels can alleviate the sparsity issue, which is peculiar in this kind of applications