Machine learning assisted exploration for affine Deligne-Lusztig varieties

There are a variety of choices to be made in both computer algebra systems (CASs) and satisfiability modulo theory (SMT) solvers which can impact performance without affecting mathematical correctness. Such choices are candidates for machine learning (ML) approaches, however, there are difficulties in applying standard ML techniques, such as the efficient identification of ML features from input data which is typically a polynomial system. Our focus is selecting the variable ...

Conversion of raw data into insights and knowledge requires substantial amounts of effort from data scientists. Despite breathtaking advances in Machine Learning (ML) and Artificial Intelligence (AI), data scientists still spend the majority of their effort in understanding and then preparing the raw data for ML/AI. The effort is often manual and ad hoc, and requires some level of domain knowledge. The complexity of the effort increases dramatically when data diversity, both ...

This study explores a new methodology for machine learning classification tasks in 2-dimensional visualization space (2-D ML) using Visual knowledge Discovery in lossless General Line Coordinates. It is shown that this is a full machine learning approach that does not require processing n-dimensional data in an abstract n-dimensional space. It enables discovering n-D patterns in 2-D space without loss of n-D information using graph representations of n-D data in 2-D. Specific...

The study of affine Deligne-Lusztig varieties originally arose from arithmetic geometry, but many problems on affine Deligne-Lusztig varieties are purely Lie-theoretic in nature. This survey deals with recent progress on several important problems on affine Deligne-Lusztig varieties. The emphasis is on the Lie-theoretic aspect, while some connections and applications to arithmetic geometry will also be mentioned.

We consider a generalization of low-rank matrix completion to the case where the data belongs to an algebraic variety, i.e. each data point is a solution to a system of polynomial equations. In this case the original matrix is possibly high-rank, but it becomes low-rank after mapping each column to a higher dimensional space of monomial features. Many well-studied extensions of linear models, including affine subspaces and their union, can be described by a variety model. In ...

Prediction and explanation are key objects in supervised machine learning, where predictive models are known as black boxes and explanatory models are known as glass boxes. Explanation provides the necessary and sufficient information to interpret the model output in terms of the model input. It includes assessments of model output dependence on important input variables and measures of input variable importance to model output. High dimensional model representation (HDMR), a...

We review the recent programme of using machine-learning to explore the landscape of mathematical problems. With this paradigm as a model for human intuition - complementary to and in contrast with the more formalistic approach of automated theorem proving - we highlight some experiments on how AI helps with conjecture formulation, pattern recognition and computation.

Hilbert series are a standard tool in algebraic geometry, and more recently are finding many uses in theoretical physics. This summary reviews work applying machine learning to databases of them; and was prepared for the proceedings of the Nankai Symposium on Mathematical Dialogues, 2021.

This paper proposes a data-driven systematic, consistent and non-exhaustive approach to Model Selection, that is an extension of the classical agnostic PAC learning model. In this approach, learning problems are modeled not only by a hypothesis space $\mathcal{H}$, but also by a Learning Space $\mathbb{L}(\mathcal{H})$, a poset of subspaces of $\mathcal{H}$, which covers $\mathcal{H}$ and satisfies a property regarding the VC dimension of related subspaces, that is a suitable...

We utilize machine learning to study the string landscape. Deep data dives and conjecture generation are proposed as useful frameworks for utilizing machine learning in the landscape, and examples of each are presented. A decision tree accurately predicts the number of weak Fano toric threefolds arising from reflexive polytopes, each of which determines a smooth F-theory compactification, and linear regression generates a previously proven conjecture for the gauge group rank ...