This thesis is a proof of concept for the potential of Variational Auto-Encoder (VAE) on representation learning of real-world Knowledge Graphs (KG). Inspired by successful approaches to the generation of molecular graphs, we evaluate the capabilities of our model, the Relational Graph Variational Auto-Encoder (RGVAE). The impact of the modular hyperparameter choices, encoding through graph convolutions, graph matching and latent space prior, is compared. The RGVAE is first evaluated on link prediction. The mean reciprocal rank (MRR) scores on the two datasets FB15K-237 and WN18RR are compared to the embedding-based model DistMult. A variational DistMult and a RGVAE without latent space prior constraint are implemented as control models. The results show that between different settings, the RGVAE with relaxed latent space, scores highest on both datasets, yet does not outperform the DistMult. Further, we investigate the latent space in a twofold experiment: first, linear interpolation between the latent representation of two triples, then the exploration of each latent dimension in a $95\%$ confidence interval. Both interpolations show that the RGVAE learns to reconstruct the adjacency matrix but fails to disentangle. For the last experiment we introduce a new validation method for the FB15K-237 data set. The relation type-constrains of generated triples are filtered and matched with entity types. The observed rate of valid generated triples is insignificantly higher than the random threshold. All generated and valid triples are unseen. A comparison between different latent space priors, using the $\delta$-VAE method, reveals a decoder collapse. Finally we analyze the limiting factors of our approach compared to molecule generation and propose solutions for the decoder collapse and successful representation learning of multi-relational KGs.

The optimization of submodular functions constitutes a viable way to perform clustering. Strong approximation guarantees and feasible optimization w.r.t. streaming data make this clustering approach favorable. Technically, submodular functions map subsets of data to real values, which indicate how "representative" a specific subset is. Optimal sets might then be used to partition the data space and to infer clusters. Exemplar-based clustering is one of the possible submodular functions, but suffers from high computational complexity. However, for practical applications, the particular real-time or wall-clock run-time is decisive. In this work, we present a novel way to evaluate this particular function on GPUs, which keeps the necessities of optimizers in mind and reduces wall-clock run-time. To discuss our GPU algorithm, we investigated both the impact of different run-time critical problem properties, like data dimensionality and the number of data points in a subset, and the influence of required floating-point precision. In reproducible experiments, our GPU algorithm was able to achieve competitive speedups of up to 72x depending on whether multi-threaded computation on CPUs was used for comparison and the type of floating-point precision required. Half-precision GPU computation led to large speedups of up to 452x compared to single-precision, single-thread CPU computations.

Ensemble learning methods are designed to benefit from multiple learning algorithms for better predictive performance. The tradeoff of this improved performance is slower speed and larger size of ensemble learning systems compared to single learning systems. In this paper, we present a novel approach to deal with this problem in Random Forest (RF) as one of the most powerful ensemble methods. The method is based on crossbreeding of the best tree branches to increase the performance of RF in space and speed while keeping the performance in the classification measures. The proposed approach has been tested on a group of synthetic and real datasets and compared to the standard RF approach. Several evaluations have been conducted to determine the effects of the Crossbred RF (CRF) on the accuracy and the number of trees in a forest. The results show better performance of CRF compared to RF.

Objective: Multi-modal functional magnetic resonance imaging (fMRI) can be used to make predictions about individual behavioral and cognitive traits based on brain connectivity networks. Methods: To take advantage of complementary information from multi-modal fMRI, we propose an interpretable multi-modal graph convolutional network (MGCN) model, incorporating the fMRI time series and the functional connectivity (FC) between each pair of brain regions. Specifically, our model learns a graph embedding from individual brain networks derived from multi-modal data. A manifold-based regularization term is then enforced to consider the relationships of subjects both within and between modalities. Furthermore, we propose the gradient-weighted regression activation mapping (Grad-RAM) and the edge mask learning to interpret the model, which is used to identify significant cognition-related biomarkers. Results: We validate our MGCN model on the Philadelphia Neurodevelopmental Cohort to predict individual wide range achievement test (WRAT) score. Our model obtains superior predictive performance over GCN with a single modality and other competing approaches. The identified biomarkers are cross-validated from different approaches. Conclusion and Significance: This paper develops a new interpretable graph deep learning framework for cognitive ability prediction, with the potential to overcome the limitations of several current data-fusion models. The results demonstrate the power of MGCN in analyzing multi-modal fMRI and discovering significant biomarkers for human brain studies.

Solving multi-objective optimization problems is important in various applications where users are interested in obtaining optimal policies subject to multiple (yet often conflicting) objectives. A typical approach to obtain the optimal policies is to first construct a loss function based on the scalarization of individual objectives and then derive optimal policies that minimize the scalarized loss function. Albeit simple and efficient, the typical approach provides no insights/mechanisms on the optimization of multiple objectives due to the lack of ability to quantify the inter-objective relationship. To address the issue, we propose to develop a new efficient gradient-based multi-objective reinforcement learning approach that seeks to iteratively uncover the quantitative inter-objective relationship via finding a minimum-norm point in the convex hull of the set of multiple policy gradients when the impact of one objective on others is unknown a priori. In particular, we first propose a new PAOLS algorithm that integrates pruning and approximate optimistic linear support algorithm to efficiently discover the weight-vector sets of multiple gradients that quantify the inter-objective relationship. Then we construct an actor and a multi-objective critic that can co-learn the policy and the multi-objective vector value function. Finally, the weight discovery process and the policy and vector value function learning process can be iteratively executed to yield stable weight-vector sets and policies. To validate the effectiveness of the proposed approach, we present a quantitative evaluation of the approach based on three case studies.

Polynomial chaos expansions (PCEs) have been used in many real-world engineering applications to quantify how the uncertainty of an output is propagated from inputs. PCEs for models with independent inputs have been extensively explored in the literature. Recently, different approaches have been proposed for models with dependent inputs to expand the use of PCEs to more real-world applications. Typical approaches include building PCEs based on the Gram-Schmidt algorithm or transforming the dependent inputs into independent inputs. However, the two approaches have their limitations regarding computational efficiency and additional assumptions about the input distributions, respectively. In this paper, we propose a data-driven approach to build sparse PCEs for models with dependent inputs. The proposed algorithm recursively constructs orthonormal polynomials using a set of monomials based on their correlations with the output. The proposed algorithm on building sparse PCEs not only reduces the number of minimally required observations but also improves the numerical stability and computational efficiency. Four numerical examples are implemented to validate the proposed algorithm.

We consider the problem of ranking $n$ players from partial pairwise comparison data under the Bradley-Terry-Luce model. For the first time in the literature, the minimax rate of this ranking problem is derived with respect to the Kendall's tau distance that measures the difference between two rank vectors by counting the number of inversions. The minimax rate of ranking exhibits a transition between an exponential rate and a polynomial rate depending on the magnitude of the signal-to-noise ratio of the problem. To the best of our knowledge, this phenomenon is unique to full ranking and has not been seen in any other statistical estimation problem. To achieve the minimax rate, we propose a divide-and-conquer ranking algorithm that first divides the $n$ players into groups of similar skills and then computes local MLE within each group. The optimality of the proposed algorithm is established by a careful approximate independence argument between the two steps.

In this study; in order to diagnose congestive heart failure (CHF) patients, non-linear secondorder difference plot (SODP) obtained from raw 256 Hz sampled frequency and windowed record with different time of ECG records are used. All of the data rows are labelled with their belongings to classify much more realistically. SODPs are divided into different radius of quadrant regions and numbers of the points fall in the quadrants are computed in order to extract feature vectors. Fisher's linear discriminant, Naive Bayes, and artificial neural network are used as classifier. The results are considered in two step validation methods as general kfold cross-validation and patient based cross-validation. As a result, it is shown that using neural network classifier with features obtained from SODP, the constructed system could distinguish normal and CHF patients with 100% accuracy rate.

Electrochemical impedance spectra is a widely used tool for characterization of fuel cells and electrochemical conversion systems in general. When applied to the on-line monitoring in context of in-field applications, the disturbances, drifts and sensor noise may cause severe distortions in the evaluated spectra, especially in the low-frequency part. Failure to account for the random effects can implicate difficulties in interpreting the spectra and misleading diagnostic reasoning. In the literature, this fact has been largely ignored. In this paper, we propose a computationally efficient approach to the quantification of the spectral uncertainty by quantifying the uncertainty of the equivalent circuit model (ECM) parameters by means of the Variational Bayes (VB) approach. To assess the quality of the VB posterior estimates, we compare the results of VB approach with those obtained with the Markov Chain Monte Carlo (MCMC) algorithm. Namely, MCMC algorithm is expected to return accurate posterior distributions, while VB approach provides the approximative distributions. By using simulated and real data we show that VB approach generates approximations, which although slightly over-optimistic, are still pretty close to the more realistic MCMC estimates. A great advantage of the VB method for online monitoring is low computational load, which is several orders of magnitude lighter than that of MCMC. The performance of VB algorithm is demonstrated on a case of ECM parameters estimation in a 6 cell solid-oxide fuel cell stack. The complete numerical implementation for recreating the results can be found at https://repo.ijs.si/lznidaric/variational-bayes-supplementary-material.

Robust estimation is an important problem in statistics which aims at providing a reasonable estimator when the data-generating distribution lies within an appropriately defined ball around an uncontaminated distribution. Although minimax rates of estimation have been established in recent years, many existing robust estimators with provably optimal convergence rates are also computationally intractable. In this paper, we study several estimation problems under a Wasserstein contamination model and present computationally tractable estimators motivated by generative adversarial networks (GANs). Specifically, we analyze properties of Wasserstein GAN-based estimators for location estimation, covariance matrix estimation, and linear regression and show that our proposed estimators are minimax optimal in many scenarios. Finally, we present numerical results which demonstrate the effectiveness of our estimators.