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The problem of decomposing non-manifold object has already been studied in solid modeling. However, the few proposed solutions are limited to the problem of decomposing solids described through their boundaries. In this thesis we study the…

Various topological techniques and tools have been applied to neural networks in terms of network complexity, explainability, and performance. One fundamental assumption of this line of research is the existence of a global (Euclidean)…

The architecture of a neural network and the selection of its activation function are both fundamental to its performance. Equally vital is ensuring these two elements are well-matched, as their alignment is key to achieving effective…

The use of high-dimensional features has become a normal practice in many computer vision applications. The large dimension of these features is a limiting factor upon the number of data points which may be effectively stored and processed,…

This work provides a quaternioinc reprsentation for real symplectic matrices in dimension four, analogous to the pair of unit quaternions representation for special orthogonal matrices. In the process of finding formulae for this…

Hyperparameter tuning can dramatically impact training stability and final performance of large-scale models. Recent works on neural network parameterisations, such as $\mu$P, have enabled transfer of optimal global hyperparameters across…

The ability to decompose a signal in an orthonormal basis (a set of orthogonal components, each normalized to have unit length) using a fast numerical procedure rests at the heart of many signal processing methods and applications. The…

Despite their successes, deep learning models struggle with tasks requiring complex reasoning and function composition. We present a theoretical and empirical investigation into the limitations of Structured State Space Models (SSMs) and…

The development of deep learning architectures is a resource-demanding process, due to a vast design space, long prototyping times, and high compute costs associated with at-scale model training and evaluation. We set out to simplify this…

We present a practical Newton-based method for computing left eigenvalues of quaternion matrices. It uses only standard real/complex linear-algebra kernels via embeddings and applies to matrices of any size. Extensive tests on literature…

Surface parameterization plays an essential role in numerous computer graphics and geometry processing applications. Traditional parameterization approaches are designed for high-quality meshes laboriously created by specialized 3D…

We undertake to develop a successful framework for commutative-associative hypercomplex numbers with the view to explicate and study associated geometric and generalized-relativistic concepts, basing on an interesting possibility to…

Network embedding is an effective technique to learn the low-dimensional representations of nodes in networks. Real-world networks are usually with multiplex or having multi-view representations from different relations. Recently, there has…

Extracting Hamiltonian parameters from available experimental data is a challenge in quantum materials. In particular, real-space spectroscopy methods such as scanning tunneling spectroscopy allow probing electronic states with atomic…

In this talk we go over several new developments regarding the techniques for a large class of non-hermitian matrix models with unitary randomness (complex random numbers). In particular, we discuss: (a) - A diagrammatic approach based on a…

Transfer learning for feature extraction can be used to exploit deep representations in contexts where there is very few training data, where there are limited computational resources, or when tuning the hyper-parameters needed for training…

Despite their widespread success, the application of deep neural networks to functional data remains scarce today. The infinite dimensionality of functional data means standard learning algorithms can be applied only after appropriate…

Embedding based models have been the state of the art in collaborative filtering for over a decade. Traditionally, the dot product or higher order equivalents have been used to combine two or more embeddings, e.g., most notably in matrix…

Motivated by the geometric advantages of quaternions in representing rotations and postures, we propose a quaternion-valued supervised learning Hopfield-structured neural network (QSHNN) with a fully connected structure inspired by the…

Large Language Models (LLMs) have achieved remarkable breakthroughs. However, the huge number of parameters in LLMs require significant amount of memory storage in inference, which prevents their practical deployment in many applications.…