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Giovanni Paolini

Explore this author’s publications, collaborators and research topics. This catalog record has not been claimed by its author.

ResearcherUnclaimed source record
Machine LearningInformation Theorymath.ITArtificial Intelligencemath.GRmath.GT

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5 publications

preprint2026arXiv

Mapping Uncharted Symmetries: Machine Discovery in Combinatorics

Inspired by long-standing open problems in algebraic combinatorics, we show that modern machine learning can meaningfully contribute to verifiable mathematical discoveries. In particular, we focus on the construction of simple mathematical functions under exact distributional constraints, a setting we formalize as Simple Learning Under Rigid Proportions (SLURP). We tackle this problem by introducing two methods: MapSeek-Functional, which models the desired function alternating pseudo-labeling and supervised training steps; and MapSeek-Symbolic, designed to directly produce symbolic formulas. We successfully apply both methods to a research problem in algebraic combinatorics, discovering a new combinatorial interpretation of the q,tq,t-Narayana polynomials arising from representation theory. To our knowledge, this is the first such interpretation based on noncrossing partitions. Using one discovered statistic, we find a combinatorial proof of the symmetry of these polynomials in a previously unsolved case. To streamline verification and reproducibility, we release all code, including a formalization of all the mathematical discoveries of this paper in Lean 4.

Open access
preprint2021arXiv

Factoring isometries of quadratic spaces into reflections

Let VV be a vector space endowed with a non-degenerate quadratic form QQ. If the base field F\mathbb{F} is different from F2\mathbb{F}_2, it is known that every isometry can be written as a product of reflections. In this article, we detail the structure of the poset of all minimal length reflection factorizations of an isometry. If F\mathbb{F} is an ordered field, we also study factorizations into positive reflections, i.e., reflections defined by vectors of positive norm. We characterize such factorizations, under the hypothesis that the squares of F\mathbb{F} are dense in the positive elements (this includes Archimedean and Euclidean fields). In particular, we show that an isometry is a product of positive reflections if and only if its spinor norm is positive. As a final application, we explicitly describe the poset of all factorizations of isometries of the hyperbolic space.

Open access
preprint2020arXiv

The Information Complexity of Learning Tasks, their Structure and their Distance

We introduce an asymmetric distance in the space of learning tasks, and a framework to compute their complexity. These concepts are foundational for the practice of transfer learning, whereby a parametric model is pre-trained for a task, and then fine-tuned for another. The framework we develop is non-asymptotic, captures the finite nature of the training dataset, and allows distinguishing learning from memorization. It encompasses, as special cases, classical notions from Kolmogorov complexity, Shannon, and Fisher Information. However, unlike some of those frameworks, it can be applied to large-scale models and real-world datasets. Our framework is the first to measure complexity in a way that accounts for the effect of the optimization scheme, which is critical in Deep Learning.

Open access
preprint2020arXiv

Where is the Information in a Deep Neural Network?

Whatever information a deep neural network has gleaned from training data is encoded in its weights. How this information affects the response of the network to future data remains largely an open question. Indeed, even defining and measuring information entails some subtleties, since a trained network is a deterministic map, so standard information measures can be degenerate. We measure information in a neural network via the optimal trade-off between accuracy of the response and complexity of the weights, measured by their coding length. Depending on the choice of code, the definition can reduce to standard measures such as Shannon Mutual Information and Fisher Information. However, the more general definition allows us to relate information to generalization and invariance, through a novel notion of effective information in the activations of a deep network. We establish a novel relation between the information in the weights and the effective information in the activations, and use this result to show that models with low (information) complexity not only generalize better, but are bound to learn invariant representations of future inputs. These relations hinge not only on the architecture of the model, but also on how it is trained, highlighting the complex inter-dependency between the class of functions implemented by deep neural networks, the loss function used for training them from finite data, and the inductive bias implicit in the optimization.

Open access

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