Related papers: Using Machine Learning to Improve Cylindrical Alge…
Machine Learning (ML) is applicable to scientific problems, i.e. to those which have a well defined answer, only if this answer can be brought to a peculiar form ${\cal G}: X\longrightarrow Z$ with ${\cal G}(\vec x)$ expressed as a…
This paper investigates the foundations of deep learning through insight of geometry, algebra and differential calculus. At is core, artificial intelligence relies on assumption that data and its intrinsic structure can be embedded into…
This paper provides a critical review of the literature on deep learning applications in breast tumor diagnosis using ultrasound and mammography images. It also summarizes recent advances in computer-aided diagnosis (CAD) systems, which…
Cutting planes are essential for solving mixed-integer linear problems (MILPs), because they facilitate bound improvements on the optimal solution value. For selecting cuts, modern solvers rely on manually designed heuristics that are tuned…
A rapidly growing area of research is the use of machine learning approaches such as autoencoders for dimensionality reduction of data and models in scientific applications. We show that the canonical formulation of autoencoders suffers…
Cutting and Packing problems are occurring in different industries with a direct impact on the revenue of businesses. Generally, the goal in Cutting and Packing is to assign a set of smaller objects to a set of larger objects. To solve…
Machine learning has emerged as a transformative tool for solving differential equations (DEs), yet prevailing methodologies remain constrained by dual limitations: data-driven methods demand costly labeled datasets while model-driven…
Derivatives, mostly in the form of gradients and Hessians, are ubiquitous in machine learning. Automatic differentiation (AD), also called algorithmic differentiation or simply "autodiff", is a family of techniques similar to but more…
In this work, we propose a machine learning-based approach to address a specific aspect of the Quantum Marginal Problem: reconstructing a global density matrix compatible with a given set of quantum marginals. Our method integrates a…
We employ techniques of machine-learning, exemplified by support vector machines and neural classifiers, to initiate the study of whether AI can "learn" algebraic structures. Using finite groups and finite rings as a concrete playground, we…
In this article, we briefly describe various tools and approaches that algebraic geometry has to offer to straighten bent objects. Throughout this article we will consider a specific example of a bent or curved piece of paper which in our…
Recent literature has shown that symbolic data, such as text and graphs, is often better represented by points on a curved manifold, rather than in Euclidean space. However, geometrical operations on manifolds are generally more complicated…
In the last two decades Computer Aided Diagnostics (CAD) systems were developed to help radiologists analyze screening mammograms. The benefits of current CAD technologies appear to be contradictory and they should be improved to be…
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,…
Designing functional materials requires a deep search through multidimensional spaces for system parameters that yield desirable material properties. For cases where conventional parameter sweeps or trial-and-error sampling are impractical,…
This paper develops a novel mathematical framework for collaborative learning by means of geometrically inspired kernel machines which includes statements on the bounds of generalisation and approximation errors, and sample complexity. For…
A new algorithm to compute cylindrical algebraic decompositions (CADs) is presented, building on two recent advances. Firstly, the output is truth table invariant (a TTICAD) meaning given formulae have constant truth value on each cell of…
This paper builds and extends on the authors' previous work related to the algorithmic tool, Cylindrical Algebraic Decomposition (CAD), and one of its core applications, Real Quantifier Elimination (QE). These topics are at the heart of…
Machine learning has the potential to become an important tool in quantum error correction as it allows the decoder to adapt to the error distribution of a quantum chip. An additional motivation for using neural networks is the fact that…
Any representation of data involves arbitrary investigator choices. Because those choices are external to the data-generating process, each choice leads to an exact symmetry, corresponding to the group of transformations that takes one…