The max-product {belief propagation} (BP) is a popular message-passing heuristic for approximating a maximum-a-posteriori (MAP) assignment in a joint distribution represented by a graphical model (GM). In the past years, it has been shown that BP can solve a few classes of linear programming (LP) formulations to combinatorial optimization problems including maximum weight matching, shortest path and network flow, i.e., BP can be used as a message-passing solver for certain combinatorial optimizations. However, those LPs and corresponding BP analysis are very sensitive to underlying problem setups, and it has been not clear what extent these results can be generalized to. In this paper, we obtain a generic criteria that BP converges to the optimal solution of given LP, and show that it is satisfied in LP formulations associated to many classical combinatorial optimization problems including maximum weight perfect matching, shortest path, traveling salesman, cycle packing, vertex/edge cover and network flow.

Perception is fundamental to many robot application areas especially in service robotics. Our aim is to perceive and model an unprepared kitchen scenario with many objects. We start with the perception of a single target object. The modeling relies especially on fusing and merging of weak information from the sensors of the robot in order to localize objects. This requires the representation of various probability distributions of pose in $S_3 \times \mathbb{R}^3$ as orientation and position have to be localized. In this thesis I present a framework for probabilistic modeling of poses in $S_3 \times \mathbb{R}^3$ that represents a large class of probability distributions and provides among others the operations of the fusion and the merge of estimates. Further it offers the propagation of uncertain information data. I work out why we choose to represent the orientation part of a pose by a unit quaternion. The translation part is described either by a 3-dimensional vector or by a purely imaginary quaternion. This depends on whether we define the probability density function or whether we want to represent a transformation which consists of a rotation and a translation by a dual quaternion. A basic probability den- sity function over the poses is defined by a tangent point on the hypersphere and a 6-dimensional Gaussian distribution. The hypersphere is embedded to the R4 which is representing a unit quaternions whereas the Gaussian is defined over the product of the tangent space of the sphere and of the space of translations. The projection of this Gaussian to the hypersphere induces a distribution over poses in $S_3 \times \mathbb{R}^3$. The set of mixtures of projected Gaussians can approximate the probability density functions that arise in our application. Moreover it is closed under the operations introduced in this framework and allows for an efficient implementation.

Diffusion maps are a nonlinear manifold learning technique based on harmonic analysis of a diffusion process over the data. Out-of-sample extensions with computational complexity $\mathcal{O}(N)$, where $N$ is the number of points comprising the manifold, frustrate applications to online learning applications requiring rapid embedding of high-dimensional data streams. We propose landmark diffusion maps (L-dMaps) to reduce the complexity to $\mathcal{O}(M)$, where $M \ll N$ is the number of landmark points selected using pruned spanning trees or k-medoids. Offering $(N/M)$ speedups in out-of-sample extension, L-dMaps enables the application of diffusion maps to high-volume and/or high-velocity streaming data. We illustrate our approach on three datasets: the Swiss roll, molecular simulations of a C$_{24}$H$_{50}$ polymer chain, and biomolecular simulations of alanine dipeptide. We demonstrate up to 50-fold speedups in out-of-sample extension for the molecular systems with less than 4% errors in manifold reconstruction fidelity relative to calculations over the full dataset.

In this paper, we study the missing sample recovery problem using methods based on sparse approximation. In this regard, we investigate the algorithms used for solving the inverse problem associated with the restoration of missed samples of image signal. This problem is also known as inpainting in the context of image processing and for this purpose, we suggest an iterative sparse recovery algorithm based on constrained $l_1$-norm minimization with a new fidelity metric. The proposed metric called Convex SIMilarity (CSIM) index, is a simplified version of the Structural SIMilarity (SSIM) index, which is convex and error-sensitive. The optimization problem incorporating this criterion, is then solved via Alternating Direction Method of Multipliers (ADMM). Simulation results show the efficiency of the proposed method for missing sample recovery of 1D patch vectors and inpainting of 2D image signals.

Machine Learning (ML) has been successfully applied to a wide range of domains and applications. One of the techniques behind most of these successful applications is Ensemble Learning (EL), the field of ML that gave birth to methods such as Random Forests or Boosting. The complexity of applying these techniques together with the market scarcity on ML experts, has created the need for systems that enable a fast and easy drop-in replacement for ML libraries. Automated machine learning (autoML) is the field of ML that attempts to answers these needs. Typically, these systems rely on optimization techniques such as bayesian optimization to lead the search for the best model. Our approach differs from these systems by making use of the most recent advances on metalearning and a learning to rank approach to learn from metadata. We propose autoBagging, an autoML system that automatically ranks 63 bagging workflows by exploiting past performance and dataset characterization. Results on 140 classification datasets from the OpenML platform show that autoBagging can yield better performance than the Average Rank method and achieve results that are not statistically different from an ideal model that systematically selects the best workflow for each dataset. For the purpose of reproducibility and generalizability, autoBagging is publicly available as an R package on CRAN.

In the setting of high-dimensional linear regression models, we propose two frameworks for constructing pointwise and group confidence sets for penalized estimators which incorporate prior knowledge about the organization of the non-zero coefficients. This is done by desparsifying the estimator as in van de Geer et al. [18] and van de Geer and Stucky [17], then using an appropriate estimator for the precision matrix $\Theta$. In order to estimate the precision matrix a corresponding structured matrix norm penalty has to be introduced. After normalization the result is an asymptotic pivot. The asymptotic behavior is studied and simulations are added to study the differences between the two schemes.

Recommender systems aim to find an accurate and efficient mapping from historic data of user-preferred items to a new item that is to be liked by a user. Towards this goal, energy-based sequence generative adversarial nets (EB-SeqGANs) are adopted for recommendation by learning a generative model for the time series of user-preferred items. By recasting the energy function as the feature function, the proposed EB-SeqGANs is interpreted as an instance of maximum-entropy imitation learning.

Mini-batch stochastic gradient descent and variants thereof have become standard for large-scale empirical risk minimization like the training of neural networks. These methods are usually used with a constant batch size chosen by simple empirical inspection. The batch size significantly influences the behavior of the stochastic optimization algorithm, though, since it determines the variance of the gradient estimates. This variance also changes over the optimization process; when using a constant batch size, stability and convergence is thus often enforced by means of a (manually tuned) decreasing learning rate schedule. We propose a practical method for dynamic batch size adaptation. It estimates the variance of the stochastic gradients and adapts the batch size to decrease the variance proportionally to the value of the objective function, removing the need for the aforementioned learning rate decrease. In contrast to recent related work, our algorithm couples the batch size to the learning rate, directly reflecting the known relationship between the two. On popular image classification benchmarks, our batch size adaptation yields faster optimization convergence, while simultaneously simplifying learning rate tuning. A TensorFlow implementation is available.

EARNING a representation for nodes in a graph, also known as node embedding, has been an important tool for extracting features that can be used in machine learning problems involving graph-structured data [1]-[4]. Perhaps the most widely adopted node embedding is the one based on the eigendecomposition of the adjacency matrix or the graph Laplacian [2], [5], [6]. Recent advances in word embeddings for natural language processing such as [7] has inspired the development of analogous embeddings for nodes in graphs [3], [8]. These so-called "neural" node embeddings have been applied to a number of supervised learning problems such us link prediction and node classification and demonstrated stateof-the-art performance [3], [4], [8]. In contrast to applications to supervised learning problems in graphs, in this work we leverage the neural embedding framework to develop an algorithm for the unsupervised community discovery problem in graphs [9]-[12]. The key idea is straightforward: learn node embeddings such that vectors of similar nodes are close to each other in the latent embedding space. Then, the problem of discovering communities in a graph can be solved by finding clusters in the embedding space. We focus on non-overlapping communities and validate the performance of the new approach through a comprehensive set of experiments on both synthetic and real-world data.

Estimating the log-likelihood gradient with respect to the parameters of a Restricted Boltzmann Machine (RBM) typically requires sampling using Markov Chain Monte Carlo (MCMC) techniques. To save computation time, the Markov chains are only run for a small number of steps, which leads to a biased estimate. This bias can cause RBM training algorithms such as Contrastive Divergence (CD) learning to deteriorate. We adopt the idea behind Population Monte Carlo (PMC) methods to devise a new RBM training algorithm termed Population-Contrastive-Divergence (pop-CD). Compared to CD, it leads to a consistent estimate and may have a significantly lower bias. Its computational overhead is negligible compared to CD. However, the variance of the gradient estimate increases. We experimentally show that pop-CD can significantly outperform CD. In many cases, we observed a smaller bias and achieved higher log-likelihood values. However, when the RBM distribution has many hidden neurons, the consistent estimate of pop-CD may still have a considerable bias and the variance of the gradient estimate requires a smaller learning rate. Thus, despite its superior theoretical properties, it is not advisable to use pop-CD in its current form on large problems.