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Recently, there has been a growing interest in applying machine learning methods to problems in engineering mechanics. In particular, there has been significant interest in applying deep learning techniques to predicting the mechanical…

We consider the fundamental task of optimising a real-valued function defined in a potentially high-dimensional Euclidean space, such as the loss function in many machine-learning tasks or the logarithm of the probability distribution in…

In this thesis I discuss combinatorial optimization problems, from the statistical physics perspective. The starting point are the motivations which brought physicists together with computer scientists and mathematicians to work on this…

Embedding into hyperbolic space is emerging as an effective representation technique for datasets that exhibit hierarchical structure. This development motivates the need for algorithms that are able to effectively extract knowledge and…

Finding meaningful representations and distances of hierarchical data is important in many fields. This paper presents a new method for hierarchical data embedding and distance. Our method relies on combining diffusion geometry, a central…

Hyperbolic geometry is gaining traction in machine learning for its effectiveness at capturing hierarchical structures in real-world data. Hyperbolic spaces, where neighborhoods grow exponentially, offer substantial advantages and…

We initiate a systematic study of intrinsic dimensional versions of classical functional inequalities which capture refined properties of the underlying objects. We focus on model spaces: Euclidean space, Hamming cube, and manifolds of…

Hyperbolic embeddings have demonstrated their effectiveness in capturing measures of uncertainty and hierarchical relationships across various deep-learning tasks, including image segmentation and active learning. However, their application…

The distortion of six different intrinsic metrics and quasi-metrics under conformal and quasiregular mappings is studied in a few simple domains $G\subsetneq\mathbb{R}^n$. The already known inequalities between the hyperbolic metric and…

Words are not created equal. In fact, they form an aristocratic graph with a latent hierarchical structure that the next generation of unsupervised learned word embeddings should reveal. In this paper, justified by the notion of…

There are massive amounts of textual data residing in databases, valuable for many machine learning (ML) tasks. Since ML techniques depend on numerical input representations, word embeddings are increasingly utilized to convert symbolic…

Hyperbolic manifolds for visual representation learning allow for effective learning of semantic class hierarchies by naturally embedding tree-like structures with low distortion within a low-dimensional representation space. The highly…

Due to its geometric properties, hyperbolic space can support high-fidelity embeddings of tree- and graph-structured data, upon which various hyperbolic networks have been developed. Existing hyperbolic networks encode geometric priors not…

Metric learning involves learning a discriminative representation such that embeddings of similar classes are encouraged to be close, while embeddings of dissimilar classes are pushed far apart. State-of-the-art methods focus mostly on…

Hamiltonian Monte Carlo (HMC) has become routinely used for sampling from posterior distributions. Its extension Riemann manifold HMC (RMHMC) modifies the proposal kernel through distortion of local distances by a Riemannian metric. The…

The integration of deep learning techniques and physics-driven designs is reforming the way we address inverse problems, in which accurate physical properties are extracted from complex data sets. This is particularly relevant for quantum…

Most real-world datasets consist of a natural hierarchy between classes or an inherent label structure that is either already available or can be constructed cheaply. However, most existing representation learning methods ignore this…

The rapid evolution of technology has transformed business operations and customer interactions worldwide, with personalization emerging as a key opportunity for e-commerce companies to engage customers more effectively. The application of…

The problem of embedding a set of objects into a low-dimensional Euclidean space based on a matrix of pairwise dissimilarities is fundamental in data analysis, machine learning, and statistics. However, the assumptions of many standard…

Physical data are representations of the fundamental laws governing the Universe, hiding complex compositional structures often well captured by hierarchical graphs. Hyperbolic spaces are endowed with a non-Euclidean geometry that naturally…