Real-world data sets often provide multiple types of information about the same set of entities. This data is well represented by multi-view graphs, which consist of several distinct sets of edges over the same nodes. These can be used to analyze how entities interact from different viewpoints. Combining multiple views improves the quality of inferences drawn from the underlying data, which has increased interest in developing efficient multi-view graph embedding methods. We propose an algorithm, C-RSP, that generates a common (C) embedding of a multi-view graph using Randomized Shortest Paths (RSP). This algorithm generates a dissimilarity measure between nodes by minimizing the expected cost of a random walk between any two nodes across all views of a multi-view graph, in doing so encoding both the local and global structure of the graph. We test C-RSP on both real and synthetic data and show that it outperforms benchmark algorithms at embedding and clustering tasks while remaining computationally efficient.

We study the learning capacity of empirical risk minimization with regard to the squared loss and a convex hypothesis class consisting of linear functions. While these types of estimators were originally designed for noisy linear regression problems, it recently turned out that they are in fact capable of handling considerably more complicated situations, involving highly non-linear distortions. This work intends to provide a comprehensive explanation of this somewhat astonishing phenomenon. At the heart of our analysis stands the mismatch principle, which is a simple, yet generic recipe to establish theoretical error bounds for empirical risk minimization. The scope of our results is fairly general, permitting arbitrary sub-Gaussian input-output pairs, possibly with strongly correlated feature variables. Noteworthy, the mismatch principle also generalizes to a certain extent the classical orthogonality principle for ordinary least squares. This adaption allows us to investigate problem setups of recent interest, most importantly, high-dimensional parameter regimes and non-linear observation processes. In particular, our theoretical framework is applied to various scenarios of practical relevance, such as single-index models, variable selection, and strongly correlated designs. We thereby demonstrate the key purpose of the mismatch principle, that is, learning (semi-)parametric output rules under large model uncertainties and misspecifications.

In this dissertation, I derive a new method to estimate the Vapnik-Chervonenkis Dimension (VCD) for the class of linear functions. This method is inspired by the technique developed by Vapnik et al. Vapnik et al. (1994). My contribution rests on the approximation of the expected maximum difference between two empirical Losses (EMDBTEL). In fact, I use a cross-validated form of the error to compute the EMDBTEL, and I make the bound on the EMDBTEL tighter by minimizing a constant in of its right upper bound. I also derive two bounds for the true unknown risk using the additive (ERM1) and the multiplicative (ERM2) Chernoff bounds. These bounds depend on the estimated VCD and the empirical risk. These bounds can be used to perform model selection and to declare with high probability, the chosen model will perform better without making strong assumptions about the data generating process (DG). I measure the accuracy of my technique on simulated datasets and also on three real datasets. The model selection provided by VCD was always as good as if not better than the other methods under reasonable conditions.

The time complexity of support vector machines (SVMs) prohibits training on huge data sets with millions of samples. Recently, multilevel approaches to train SVMs have been developed to allow for time efficient training on huge data sets. While regular SVMs perform the entire training in one - time consuming - optimization step, multilevel SVMs first build a hierarchy of problems decreasing in size that resemble the original problem and then train an SVM model for each hierarchy level benefiting from the solved models of previous levels. We present a faster multilevel support vector machine that uses a label propagation algorithm to construct the problem hierarchy. Extensive experiments show that our new algorithm achieves speed-ups up to two orders of magnitude while having similar or better classification quality over state-of-the-art algorithms.

Multi-output is essential in machine learning that it might suffer from nonconforming residual distributions, i.e., the multi-output residual distributions are not conforming to the expected distribution. In this paper we propose "Wrapped Loss Function" to wrap the original loss function to alleviate the problem. This wrapped loss function acts just like original loss function that its gradient can be used for backpropagation optimization. Empirical evaluations show wrapped loss function has advanced properties of faster convergence, better accuracy and improving imbalanced data.

The notion of belief likelihood function of repeated trials is introduced, whenever the uncertainty for individual trials is encoded by a belief measure (a finite random set). This generalises the traditional likelihood function, and provides a natural setting for belief inference from statistical data. Factorisation results are proven for the case in which conjunctive or disjunctive combination are employed, leading to analytical expressions for the lower and upper likelihoods of `sharp' samples in the case of Bernoulli trials, and to the formulation of a generalised logistic regression framework.

Recently, network lasso has drawn many attentions due to its remarkable performance on simultaneous clustering and optimization. However, it usually suffers from the imperfect data (noise, missing values etc), and yields sub-optimal solutions. The reason is that it finds the similar instances according to their features directly, which is usually impacted by the imperfect data, and thus returns sub-optimal results. In this paper, we propose triangle lasso to avoid its disadvantage. Triangle lasso finds the similar instances according to their neighbours. If two instances have many common neighbours, they tend to become similar. Although some instances are profiled by the imperfect data, it is still able to find the similar counterparts. Furthermore, we develop an efficient algorithm based on Alternating Direction Method of Multipliers (ADMM) to obtain a moderately accurate solution. In addition, we present a dual method to obtain the accurate solution with the low additional time consumption. We demonstrate through extensive numerical experiments that triangle lasso is robust to the imperfect data. It usually yields a better performance than the state-of-the-art method when performing data analysis tasks in practical scenarios.

As bonus, you also learn how to share your analysis results with your collegues friends and others and create visual analysis of your results. You will also have access to some practice questions, which will give you hand on experience.

In order to address this weakness, we take the perspective of random forests as an adaptive kernel method. This interpretation follows work by Athey et al. (2018), Hothorn et al. (2004), and Meinshausen (2006), and complements the traditional view of forests as an ensemble method (i.e., an average of predictions made by individual trees). These types of adjustments are particularly important near boundaries, where neighborhoods are asymmetric by necessity, but with many covariates, the adjustments are also important away from boundaries given that local neighborhoods are often unbalanced due to sampling variation. The goal of this paper is improve the accuracy of forests on smooth signals using regression adjustments, potentially in many dimensions. By using the local regression adjustment, it is possible to adjust for asymmetries and imbalances in the set of nearby points used for prediction, ensuring that the weighted average of the feature vector of neighboring points is approximately equal to the target feature vector, and that predictions are centered.

This course teaches and demonstrates some of the most basic, yet crucial concepts in Data Science and Analytics.