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We propose a novel Auto-Regressive (AR) image generation approach that models images as hierarchical compositions of interpretable visual layers. While AR models have achieved transformative success in language modeling, replicating this…
This paper presents two enhancements to cylindrical algebraic decomposition (CAD) based quantifier elimination (QE) for cases in which multiple equational constraints are present in the given input formula $\phi^*$. The first enhancement…
We present a novel AI-assisted method for decomposing (segmenting) planar CAD (computer-aided design) models into well shaped rectangular blocks as a proof-of-principle of a general decomposition method applicable to complex 2D and 3D CAD…
This paper provides a theoretical and numerical investigation of a penalty decomposition scheme for the solution of optimization problems with geometric constraints. In particular, we consider some situations where parts of the constraints…
We propose a new numerical algorithm for computing the tensor rank decomposition or canonical polyadic decomposition of higher-order tensors subject to a rank and genericity constraint. Reformulating this computational problem as a system…
The evaluation of robustness and reliability of realistic structures in the presence of uncertainty involves costly numerical simulations with a very high number of evaluations. This motivates model order reduction techniques like the…
Vertical decomposition is a widely used general technique for decomposing the cells of arrangements of semi-algebraic sets in ${{\mathbb R}}^d$ into constant-complexity subcells. In this paper, we settle in the affirmative a few…
Many complex chemical problems encoded in terms of physics-based models become computationally intractable for traditional numerical approaches due to their unfavourable scaling with increasing molecular size. Tensor decomposition…
Decomposing an algebraic variety into irreducible or equidimensional components is a fundamental task in classical algebraic geometry and has various applications in modern geometry engineering. Several researchers studied the problem and…
We develop a novel deep learning technique, termed Deep Orthogonal Decomposition (DOD), for dimensionality reduction and reduced order modeling of parameter dependent partial differential equations. The approach consists in the construction…
When building a cylindrical algebraic decomposition (CAD) savings can be made in the presence of an equational constraint (EC): an equation logically implied by a formula. The present paper is concerned with how to use multiple ECs,…
The Canonical Polyadic decomposition (CPD) is a convenient and intuitive tool for tensor factorization; however, for higher-order tensors, it often exhibits high computational cost and permutation of tensor entries, these undesirable…
Black-box model structures are dominated by large multivariate functions. Usually a generic basis function expansion is used, e.g. a polynomial basis, and the parameters of the function are tuned given the data. This is a pragmatic and…
Canonical polyadic decomposition (CPD) is at the core of fast matrix multiplication, a computational problem with widespread implications across several seemingly unrelated problems in computer science. Much recent progress in this field…
We investigate the relation between Cartan decompositions of the unitary group and discrete quantum symmetries. To every Cartan decomposition there corresponds a quantum symmetry which is the identity when applied twice. As an application,…
McCallum-style Cylindrical Algebra Decomposition (CAD) is a major improvement on the original Collins version, and has had many subsequent advances, notably for total or partial equational constraints. But it suffers from a problem with…
A critical step in developing circuits for quantum simulation is to synthesize a desired unitary operator using the circuit building blocks. Studying unitaries and their generators from the Lie algebraic perspective has given rise to…
We introduce high-order dynamical decoupling strategies for open system adiabatic quantum computation. Our numerical results demonstrate that a judicious choice of high-order dynamical decoupling method, in conjunction with an encoding…
Robust optimization is a framework for modeling optimization problems involving data uncertainty and during the last decades has been an area of active research. If we focus on linear programming (LP) problems with i) uncertain data, ii)…
Singular Value Decomposition (SVD) is one of the most useful techniques for analyzing data in linear algebra. SVD decomposes a rectangular real or complex matrix into two orthogonal matrices and one diagonal matrix. In this work we…