Analogical reasoning is a powerful inductive mechanism, widely used in human cognition and increasingly applied in artificial intelligence. Formal frameworks for analogical inference have been developed for Boolean domains, where inference is provably sound for affine functions and approximately correct when close to affine. These results enabled the design of analogy-based classifiers. However, they do not extend to regression tasks or continuous domains.

In this series of seminars we will revisit analogical inference from a foundational perspective. After a brief motivation, we will first present a recently proposed formalism to model numerical analogies that relies on p-generalized means, and that enables a unifying framework that subsume the classical notions of arithmetic, geometric and harmonic analogies. We will derive several interesting properties such as transitivity of conformity, as well as present algorithmic approaches to detect and compute the parameter p.

In the second part of this series, we will leverage this unified formalism and lift analogical reasoning to real-valued domains and various ML&AI downstream tasks. In particular, we will see that it supports analogical inference over continuous functions, and thus both classification and regression tasks. We characterize the class of analogy-preserving functions in this setting and derive both worst-case and average-case error bounds under smoothness assumptions. If time allows, we will also discuss further applications, e.g., on image reconstruction and NLP downstream tasks.

These two seminars are based on several published and recently submitted by Miguel Couceiro and his collaborators, including Francisco Malaca and Francisco Vincente Cunha, respectively, graduate and undergraduate students at the DM@IST.

Some very recent references

1. Francisco Malaca, Yves Lepage, Miguel Couceiro. Numerical analogies through generalized means:notion, properties and algorithmic approaches. Submitted.
2. Francisco Cunha, Yves Lepage, Zied Bouraoui, Miguel Couceiro. Generalizing Analogical Inference Across Boolean and Continuous Domains. Submitted.
3. Jakub Pillion, Miguel Couceiro, Yves Lepage. Analogical pooling for image reconstruction. Submitted.
4. Fadi Badra, Esteban Marquer, Marie-Jeanne Lesot, Miguel Couceiro, David Leake. EnergyCompress: A General Case Base Learning Strategy. To appear in IJCAI2025.
5. Yves Lepage, Miguel Couceiro. Any four real numbers are on all fours with analogy. CoRR abs/2407.18770 (2024)
6. Miguel Couceiro, Erkko Lehtonen. Galois theory for analogical classifiers. Ann. Math. Artif. Intell. 92(1): 29-47 (2024)
7. Pierre Monnin, Cherif-Hassan Nousradine, Lucas Jarnac, Laurel Zuckerman, Miguel Couceiro. KGPRUNE: A Web Application to Extract Subgraphs of Interest from Wikidata with Analogical Pruning. ECAI 2024: 4495-4498
8. Yves Lepage, Miguel Couceiro. Analogie et moyenne généralisée. JIAF-JFPDA 2024: 114-124
9. Lucas Jarnac, Miguel Couceiro, Pierre Monnin. Relevant Entity Selection: Knowledge Graph Bootstrapping via Zero-Shot Analogical Pruning. CIKM 2023: 934-944
10. N. Kumar, and S. Schockaert. Solving hard analogy questions with relation embedding chains. EMNLP 2023, 6224–6236. ACL

Analogical reasoning is a powerful inductive mechanism, widely used in human cognition and increasingly applied in artificial intelligence. Formal frameworks for analogical inference have been developed for Boolean domains, where inference is provably sound for affine functions and approximately correct when close to affine. These results enabled the design of analogy-based classifiers. However, they do not extend to regression tasks or continuous domains.

In this series of seminars we will revisit analogical inference from a foundational perspective. After a brief motivation, we will first present a recently proposed formalism to model numerical analogies that relies on p-generalized means, and that enables a unifying framework that subsume the classical notions of arithmetic, geometric and harmonic analogies. We will derive several interesting properties such as transitivity of conformity, as well as present algorithmic approaches to detect and compute the parameter p.

In the second part of this series, we will leverage this unified formalism and lift analogical reasoning to real-valued domains and various ML&AI downstream tasks. In particular, we will see that it supports analogical inference over continuous functions, and thus both classification and regression tasks. We characterize the class of analogy-preserving functions in this setting and derive both worst-case and average-case error bounds under smoothness assumptions. If time allows, we will also discuss further applications, e.g., on image reconstruction and NLP downstream tasks.

These two seminars are based on several published and recently submitted by Miguel Couceiro and his collaborators, including Francisco Malaca and Francisco Vincente Cunha, respectively, graduate and undergraduate students at the DM@IST.

This lecture first provides an introduction to classical variational inference (VI), a key technique for approximating complex posterior distributions in Bayesian methods, typically by minimizing the Kullback-Leibler (KL) divergence. We'll discuss its principles and
common uses.

Building on this, the lecture introduces Fenchel-Young variational inference (FYVI), a novel generalization that enhances flexibility. FYVI replaces the KL divergence with broader Fenchel-Young (FY) regularizers, with a special focus on those derived from Tsallis
entropies. This approach enables learning posterior distributions with significantly smaller, or sparser, support than the prior, offering
advantages in model interpretability and performance.

This lecture first provides an introduction to classical variational inference (VI), a key technique for approximating complex posterior
distributions in Bayesian methods, typically by minimizing the Kullback-Leibler (KL) divergence. We'll discuss its principles and
common uses.

Building on this, the lecture introduces Fenchel-Young variational inference (FYVI), a novel generalization that enhances flexibility.
FYVI replaces the KL divergence with broader Fenchel-Young (FY) regularizers, with a special focus on those derived from Tsallis
entropies. This approach enables learning posterior distributions with significantly smaller, or sparser, support than the prior, offering
advantages in model interpretability and performance.

We define a nonlinear Fourier transform which maps sequences of contractive $n \times n$ matrices to $SU(2n)$-valued functions on the circle $\mathbb T$. We characterize the image of compactly supported sequences and square-summable sequences on the half-line, and prove that the inverse map is well-defined on $SU(2n)$-valued functions whose diagonal $n \times n$ blocks are outer matrix functions. As an application, we prove infinite generalized quantum signal processing in the fully coherent regime.

Bibliography:

• L. Lin, Lecture Notes on Quantum Algorithms for Scientific Computation, https://arxiv.org/pdf/2201.08309
• D. Oliveira e Silva, Inequalities in nonlinear Fourier analysis. CIM Bulletin 38-39 (2017), 31-35
• T. Tao and C. Thiele, Nonlinear Fourier Analysis, https://arxiv.org/abs/1201.5129

We define a nonlinear Fourier transform which maps sequences of contractive $n \times n$ matrices to $SU(2n)$-valued functions on the circle $\mathbb T$. We characterize the image of compactly supported sequences and square-summable sequences on the half-line, and prove that the inverse map is well-defined on $SU(2n)$-valued functions whose diagonal $n \times n$ blocks are outer matrix functions. As an application, we prove infinite generalized quantum signal processing in the fully coherent regime.