Layers, Toposes, and Machine Learning | Category Theory and Machine Learning Series

Introduction to Lesson 7: “Layers, Toposes, and Machine Learning”

In addition to defining specific categories to study concrete machine learning methodologies, one can also generalize the ideas contained in machine learning and connect them with cutting-edge algebraic geometry, higher-order topology, and logic in modern mathematics. This direction explores from a mathematical perspective the foundational architectures (mainly deep neural networks) and ideas in some networks within machine learning, potentially inspiring innovations in fundamental methods.

Representative works include layers and toposes. Layer theory, as an important branch of modern mathematics, provides a systematic method for discussing local algebraic properties on topological spaces, studying an object by examining its relationships with other objects, while aligning with the common idea of learning through network structures found in deep neural networks. Topos theory is a categorical theory that generalizes point-set topology, influencing set theory and mathematical logic, and is part of the general theory development of deep neural networks. This section will primarily introduce the work of Laurent Lafforgue and others in this direction, summarizing some of the inspiring ideas they proposed.Introduction to Laurent LafforgueLaurent Lafforgue is a “genius” mathematician who won a silver medal at the International Mathematical Olympiad at the age of 18. He became a senior researcher at the French National Center for Scientific Research and a mathematics professor at the Institute of Advanced Scientific Studies at the age of 34; at 35, he received the Fields Medal for his outstanding contributions to number theory and algebraic geometry. This award is often referred to as the “Nobel Prize of Mathematics.” At 37, Lafforgue was elected a member of the French Academy of Sciences.Lafforgue made significant contributions to the largest single project in modern mathematical research, the Langlands Program, establishing new connections between number theory and analysis. In 2021, he joined Huawei’s Paris Research Institute, transitioning from academia to industry, where he delves into Grothendieck’s topos theory.In an interview published by Huawei, Lafforgue described the importance of topos theory, stating, “Topos theory is part of the general theoretical development of deep neural networks. For example, it can be used to serve computer architectures for artificial intelligence and can also be used to develop automated systems that assist mathematicians in verifying theorems, discovering theorems, and even promoting the development of new mathematical theories.”Keywords:#Layers #Toposes #Inference #MachineLearning #ModelTheory

References:

1. Laurent Lafforgue: Some sketches for a topos-theoretic AI (2024) https://mat.uab.cat/~rubio/bM2L/Lafforgue-bM2L.pdf

2. Laurent Lafforgue: Some possible roles for AI of Grothendieck topos theory (2022) https://www.laurentlafforgue.org/Expose_Lafforgue_topos_AI_ETH_sept_2022.pdf

3. Model Theory: An Introduction, David Marker, Springer Science & Business Media (2002)

Instructor

Jia Yiyang, Assistant Professor at Japan Women’s University, Former Assistant Professor at Seikei University. His research focuses on computational complexity, algorithms, and category-related theories.

Sharing Information

Sharing Time:April 22, 2024, 20:00-22:00

Sharing Method:

1. Tencent Meeting (Registration for paid course required)

2. Jizhi Academy Video Channel Live Broadcast

“Category Theory and Machine Learning” Series Course

We Look Forward to Your Participation

To help everyone gain an in-depth understanding of the intersection between category theory and machine learning, and to comprehend the categorical significance behind machine learning methods, Jizhi Academy, in collaboration with Assistant Professor Jia Yiyang from Seikei University, has launched the “Category Theory and Machine Learning” series of courses, aimed at researchers in the field of machine learning who wish to delve into theoretical ideas, those in mathematics who want to leverage artificial intelligence to solve problems, and those interested in the application prospects of category theory (such as the potential integration with artificial intelligence and quantum computing). This series of courses will utilize reports, papers, and textbooks related to machine learning and category theory as course materials, introducing important concepts and, more importantly, the hidden ideas behind these concepts. Starting from the categorical perspective, the course will explore specific categorical research methods that establish the background for certain methodologies in machine learning, as well as study certain structures on categories from the perspective of neural network architectures, such as “layers,” “fibers,” and “toposes.” Understanding these complex terms and concepts from a purely mathematical standpoint can take years. The main goal of this course is to guide everyone to rapidly grasp the significance of these concepts and categorical frameworks in machine learning theory and applications without excessive energy expenditure.If you are interested in this topic, you are welcome to join the course and learn and discuss together with the instructor and classmates.1. Course Link: https://campus.swarma.org/course/53052. Detailed Information on the Series Course:Looking at Machine Learning from the Perspective of Category Theory | Launch of the “Category Theory and Machine Learning” Series Course

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Recommended Reading:

1. His Profound Mathematical Ideas Seem to Come from the Void

2. Dialogue with Fields Medalist Lafforgue: Progress and Prospects of Combining Fundamental Mathematics with Engineering Practice in the Era of Large Models

3. From Classical Algebraic Geometry to Modern Algebraic Geometry—Sheaf Theory Part 1: Layers

4. Category Theory for Everyone—Cross-Disciplinary Scientific Methodology | Premium Introductory Course
5. Category Theory for Everyone Season 2—Cross-Disciplinary Scientific Methodology | Premium Introductory Course

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