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The paper presents a systematic study and implementation of a reconfigurable combinatorial multi-operand adder for use in Deep Learning systems. The size of carry changes with the number of operands and hence a reliable algorithm to…
The solution of conservation laws with parametrized shock waves presents challenges for both high-order numerical methods and model reduction techniques. We introduce an r-adaptivity scheme based on optimal transport and apply it to develop…
We develop a method for optimization in shape spaces, i.e., sets of surfaces modulo re-parametrization. Unlike previously proposed gradient flows, we achieve superlinear convergence rates through a subtle approximation of the shape Hessian,…
Machine learning assumes a pivotal role in our data-driven world. The increasing scale of models and datasets necessitates quick and reliable algorithms for model training. This dissertation investigates adaptivity in machine learning…
We consider a one-dimensional singularly perturbed 4th order problem with the additional feature of a shift term. An expansion into a smooth term, boundary layers and an inner layer yields a formal solution decomposition, and together with…
We develop an intrinsic geometrical setting for higher order constrained field theories. As a main tool we use an appropriate generalization of the classical Skinner-Rusk formalism. Some examples of application are studied, in particular,…
A unified framework for first-order optimization algorithms fornonconvex unconstrained optimization is proposed that uses adaptivelypreconditioned gradients and includes popular methods such as full anddiagonal AdaGrad, AdaNorm, as well as…
A fully tensorial theoretical framework for hypercomplex-valued neural networks is presented. The proposed approach enables neural network architectures to operate on data defined over arbitrary finite-dimensional algebras. The central…
In this paper, we introduce a graph matching method that can account for constraints of arbitrary order, with arbitrary potential functions. Unlike previous decomposition approaches that rely on the graph structures, we introduce a…
After its introduction (initially within a group field theory framework) in [Tanasa A., J. Phys. A: Math. Theor. 45 (2012), 165401, 19 pages, arXiv:1109.0694], the multi-orientable (MO) tensor model grew over the last years into a solid…
The conformal anomaly and the Virasoro algebra are fundamental aspects of 2D conformal field theory and conformally covariant models in planar random geometry. In this article, we explicitly derive the Virasoro algebra from an…
We address the problem of universal domain adaptation (UDA) in ordinal regression (OR), which attempts to solve classification problems in which labels are not independent, but follow a natural order. We show that the UDA techniques…
We comment on a program designed for the study of local chiral algebras and their representations in 2D conformal field theory. Based on the algebraic approach described by W. Nahm, this program efficiently calculates arbitrary n-point…
An inexact framework for high-order adaptive regularization methods is presented, in which approximations may be used for the $p$th-order tensor, based on lower-order derivatives. Between each recalculation of the $p$th-order derivative…
Despite the advances in the representational capacity of approximate distributions for variational inference, the optimization process can still limit the density that is ultimately learned. We demonstrate the drawbacks of biasing the true…
This paper introduces a nonconforming virtual element method for general second-order elliptic problems with variable coefficients on domains with curved boundaries and curved internal interfaces. We prove arbitrary order optimal…
We develop new adaptive algorithms for variational inequalities with monotone operators, which capture many problems of interest, notably convex optimization and convex-concave saddle point problems. Our algorithms automatically adapt to…
Recent work on dictionary learning with set-atoms has shown benefits in anomaly detection. Instead of viewing an atom as a single vector, these methods allow building sparse representations with atoms taken from a set around a central…
Second-order superintegrable systems in dimensions two and three are essentially classified. With increasing dimension, however, the non-linear partial differential equations employed in current methods become unmanageable. Here we propose…
Recent advances in learning aligned multimodal representations have been primarily driven by training large neural networks on massive, noisy paired-modality datasets. In this work, we ask whether it is possible to achieve similar results…