Belief propagation is a heuristic approach for solving large-scale statistical inference problems. It is an easy-to-implement heuristic based on message-passing that usually converges quickly. As such, it has become very popular, and it has proved to be successful in a wide range of applications. Its success in practice, however, is at stark contrast to the lack of theoretical understanding of its performance. Belief propagation shares this aspect with many algorithms. The reason for the discrepancy between remarkably good performance in practice and unsatisfactory theoretical understanding is often that performance is analyzed in terms of worst-case analysis; worst-case analysis is often far too pessimistic. Indeed, worst-case instances are often artificially constructed and rarely show up in applications. An adequate theoretical analysis, however, should measure the performance in terms of ``typical' rather than ``pathological' instances. To provide a more realistic analysis of algorithms, the concept of smoothed analysis has been developed: An adversary specifies an instance, and then the expected performance is measured when this instance is slightly randomly perturbed. Smoothed analysis takes into account that practical data is often noisy, e.g., due to measurement errors. Smoothed analysis has already been applied successfully to explain the performance of a variety of algorithms. The aim of this project is a smoothed analysis of belief propagation. We will focus on the application of belief propagation to combinatorial optimization problems. The goal is to get a deeper understanding of its performance and to bridge the gap between theoretical and practical performance of belief propagation.

From recognising faces to diagnosing diseases, convolutional neural networks (CNNs) excel in image recognition. However, their lack of theoretical understanding poses the risk of unexpected behaviour. How do CNNs detect objects? We must take one step back to answer this question. What is the right model for analysing image classification? What are the properties of CNNs? How do they learn? These questions form my research foundation. Through a novel statistical approach, I characterise images as highly structured objects with different geometric deformations. Besides providing theoretical guarantees for CNNs, this Veni redefines our perception of image analysis, bridging theory with practice.

The Dutch Seminar on Optimization is a national online event. It brings together researchers in optimization which are typically spread over different institutes such as Mathematics, Computer Science, Economics and Business. The online seminar is accompanied by a yearly in-person event. This serves primarily three purposes: To (i) help increase the visibility of the Dutch community in optimization, (ii) improve the coherence and smooth functioning of the same community; and (iii) provide a forum for upcoming talent and PhD students to meet their peers, both junior and senior, for networking, exposure, and collaboration.

Deep networks are currently hyped for remarkable successes in various applications and there is wide agreement that a theory for deep learning is missing. Mathematically speaking, neural networks define function classes with a rich mathematical structure that are extremely difficult to analyze because of non-linearity in the parameters. Most existing theoretical results cannot cope with many of the distinctive characteristics of deep networks such as multiple hidden layers or the ReLU activation function. I propose to build statistical theory that lays a mathematical foundation for the analysis of deep neural networks, when the number of parameters exceeds the sample size. Indeed, while optimal estimation rates in high dimensions are slow due to the curse of dimensionality, deep neural networks still perform well. It is thus natural to conjecture that deep neural networks form a flexible class of estimators which can avoid the curse of dimensionality by adapting to various low dimensional structural constraints on the regression function and the design. I will derive the necessary mathematical tools to investigate up to which level this conjecture is true. Apart from combining novel ideas with my recent work on ReLU network, the research program requires techniques from nonparametric/high-dimensional statistics, approximation theory, functional analysis, empirical process theory and inverse problems. My proposal has the potential to open a new direction in mathematical statistics and will lay the basis for an even more general theoretical foundation beyond feedforward and convolutional neural networks. I first plan to develop statistical theory for deep but sparsely connected ReLU networks. Then I suggest to extend these results to convolutional neural networks with image models as input. In both settings, I consider regression and classification. My theory will have several practical applications. It will for instance provide suggestions on how to choose network tuning parameters.