Related papers: Machine learning modularity
In this paper we explore how machine learning techniques can be applied to the discovery of efficient mathematical identities. We introduce an attribute grammar framework for representing symbolic expressions. Given a set of grammar rules…
Previously, we proved an identity for theta functions of degree eight, and several applications of it were also discussed. This identity is a natural extension of the addition formula for the Weierstrass sigma-function. In this paper we…
Modular exponentiation is crucial to number theory and cryptography, yet remains largely unexplored from a mechanistic interpretability standpoint. We train a 4-layer encoder-decoder Transformer model to perform this operation and…
Complex phenomena are generally modeled with sophisticated simulators that, depending on their accuracy, can be very demanding in terms of computational resources and simulation time. Their time-consuming nature, together with a typically…
An exact transformation, which we call the \emph{master identity}, is obtained for the first time for the series $\sum_{n=1}^{\infty}\sigma_{a}(n)e^{-ny}$ for $a\in\mathbb{C}$ and Re$(y)>0$. New modular-type transformations when $a$ is a…
Machine learning algorithms use error function minimization to fit a large set of parameters in a preexisting model. However, error minimization eventually leads to a memorization of the training dataset, losing the ability to generalize to…
Let L >= 3. Using the moduli interpretation, we define certain elliptic modular forms of level Gamma(L) over any field k where 6L is invertible and k contains the Lth roots of unity. These forms generate a graded algebra R_L, which, over C,…
Large language models (LLMs) have substantially advanced machine learning research, including natural language processing, computer vision, data mining, etc., yet they still exhibit critical limitations in explainability, reliability,…
Transfer learning has recently become the dominant paradigm of machine learning. Pre-trained models fine-tuned for downstream tasks achieve better performance with fewer labelled examples. Nonetheless, it remains unclear how to develop…
Statistics and Optimization are foundational to modern Machine Learning. Here, we propose an alternative foundation based on Abstract Algebra, with mathematics that facilitates the analysis of learning. In this approach, the goal of the…
We propose a generic categorical framework for learning unknown formal languages of various types (e.g. finite or infinite words, weighted and nominal languages). Our approach is parametric in a monad T that represents the given type of…
We show the modular properties of the multiple 'elliptic' gamma functions, which are an extension of those of the theta function and the elliptic gamma function. The modular property of the theta function is known as Jacobi's…
This paper establishes a theoretical framework connecting neural network learning with abstract algebraic structures. We first present a minimal counterexample demonstrating that standard neural networks completely fail on compositional…
Large Language Models demonstrate remarkable mathematical capabilities but at the same time struggle with abstract reasoning and planning. In this study, we explore whether Transformers can learn to abstract and generalize the rules…
Training in machine learning generally consists in finding one model, whose parameters minimize a data-dependent loss. Yet, empirical work shows that ensemble learning, an approach in which multiple models are sampled, can improve…
In this paper, we prove the existence of an efficient algorithm for the computation of $q$-expansions of modular forms of weight $k$ and level $\Gamma$, where $\Gamma \subseteq SL_{2}({\mathbb{Z}})$ is an arbitrary congruence subgroup. We…
In this paper, we propose a novel multi-task learning (MTL) framework, called Self-Paced Multi-Task Learning (SPMTL). Different from previous works treating all tasks and instances equally when training, SPMTL attempts to jointly learn the…
We present the first correct-by-construction learning-based system for step-by-step mathematical integration. The key idea is to learn a policy, represented by a GPT transformer model, which guides the search for the right mathematical…
This paper presents a novel method that allows a machine learning algorithm following the transformation-based learning paradigm \cite{brill95:tagging} to be applied to multiple classification tasks by training jointly and simultaneously on…
In this PhD thesis we study holomorphic and non-holomorphic elliptic analogues of multiple zeta values, namely elliptic multiple zeta values and modular graph functions. Both classes of functions have been discovered very recently, and are…