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Introduction to optimization on smooth manifolds
first order methods
Nicolas Boumal
Nicolas Boumal
Optimization on manifolds is the result of smooth geometry and optimization merging into one elegant modern framework.
Read more
Optimization on manifolds is the result of smooth geometry and optimization merging into one elegant modern framework.
Read more
Optimization on manifolds is the result of smooth geometry and optimization merging into one elegant modern framework.
We start the course at "What is a manifold?", and give the students a firm understanding of submanifolds embedded in real space. This covers numerous applications in engineering and the sciences.
All definitions and theorems are motivated to build time-tested optimization algorithms. The math is precise, to promote understanding and enable computation.
We build our way up to Riemannian gradient descent: the all-important first-order optimization algorithm on manifolds. This includes analysis and implementation.
The lectures follow (and complement) the textbook "An introduction to optimization on smooth manifolds" written by the instructor, also available on his webpage.
From there, students can explore more with numerical tools (such as the toolbox Manopt, which is the subject of the last week of the course). They will also be in a good position to tackle more advanced theoretical tools necessary for second-order optimization algorithms (e.g., Riemannian Hessians). Those are covered in further video lectures available on the instructor's textbook webpage.
Register for this course and see more details by visiting:
OpenCourser.com/course/ac0mrk/introduction
What's inside
Learning objectives
- Recognize smooth manifolds and do calculus on them.
- Manipulate concepts from differential and riemannian geometry.
- Develop geometric tools to work on new manifolds of interest.
- Recognize and formulate a riemannian optimization problem.
- Analyze and implement first-order riemannian optimization algorithms.
- Use toolboxes to accelerate prototyping.
- By the end of the course, you will be able to:
- Recognize smooth manifolds and do calculus on them.
- Manipulate concepts from differential and riemannian geometry.
- Develop geometric tools to work on new manifolds of interest.
- Recognize and formulate a riemannian optimization problem.
- Analyze and implement first-order riemannian optimization algorithms.
- Use toolboxes to accelerate prototyping.
- By the end of the course, you will be able to:
Syllabus
1. Introduction
2. Manifolds and tangent spaces
3. Functions, differentials, retractions and vector fields
4. Riemannian manifolds and gradients
Read more
Syllabus
1. Introduction
2. Manifolds and tangent spaces
3. Functions, differentials, retractions and vector fields
4. Riemannian manifolds and gradients
Read more
5. Riemannian gradient descent
6. Manopt (toolbox for optimization on manifolds)
Good to know
Good to know
Know what's good ,
what to watch for , and
possible dealbreakers
Examines optimization on smooth manifolds, a technique combining differential and Riemannian geometry to optimize algorithms on manifolds
Develops time-tested optimization algorithms grounded in strong mathematical foundations
Taught by Nicolas Boumal, an expert in optimization on smooth manifolds
Provides hands-on experience through the use of the Manopt toolbox for optimization on manifolds
Suitable for learners with a strong background in differential and Riemannian geometry
Requires access to numerical tools and software, such as Manopt
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Activities
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Introduction to optimization on smooth manifolds: first order methods.
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Career center
Learners who complete Introduction to optimization on smooth manifolds: first order methods will develop knowledge and skills
that may be useful to these careers:
Financial Analyst
Financial Analyst
Financial Analysts use financial data to make investment recommendations. They use a variety of techniques, including optimization on smooth manifolds, to develop models that can predict future market behavior. This course would provide a strong foundation in the mathematical and computational techniques used in financial analysis, including optimization on smooth manifolds. The course's focus on developing geometric tools to work on new manifolds of interest would be particularly relevant to Financial Analysts who work with complex financial data.
Machine Learning Engineer
Machine Learning Engineer
Machine Learning Engineers design and implement machine learning models to solve real-world problems. They use a variety of techniques, including optimization on smooth manifolds, to develop models that can learn from data and make predictions. This course would provide a strong foundation in the mathematical and computational techniques used in machine learning, including optimization on smooth manifolds. The course's focus on developing geometric tools to work on new manifolds of interest would be particularly relevant to Machine Learning Engineers who work with complex data structures.
Operations Research Analyst
Operations Research Analyst
Operations Research Analysts use mathematical and computational techniques to solve problems in a variety of industries, including manufacturing, logistics, and healthcare. They use a variety of techniques, including optimization on smooth manifolds, to develop models that can help organizations improve efficiency and profitability. This course would provide a strong foundation in the mathematical and computational techniques used in operations research, including optimization on smooth manifolds. The course's focus on developing geometric tools to work on new manifolds of interest would be particularly relevant to Operations Research Analysts who work with complex systems.
Quantitative Analyst
Quantitative Analyst
Quantitative Analysts use mathematical and statistical models to analyze financial data and make investment decisions. They use a variety of techniques, including optimization on smooth manifolds, to develop models that can predict future market behavior. This course would provide a strong foundation in the mathematical and computational techniques used in quantitative finance, including optimization on smooth manifolds. The course's focus on developing geometric tools to work on new manifolds of interest would be particularly relevant to Quantitative Analysts who work with complex financial data.
Data Scientist
Data Scientist
Data Scientists combine programming skills with knowledge of mathematics and statistics to extract insights from data. They help companies make data-driven decisions, which can lead to improved efficiency, profitability, and customer satisfaction. This course would provide a strong foundation in the mathematical and computational techniques used in data science, including optimization on smooth manifolds. The course's focus on developing geometric tools to work on new manifolds of interest would be particularly relevant to Data Scientists who work with complex data structures.
Data Engineer
Data Engineer
Data Engineers design and build systems to store, manage, and process data. They use a variety of techniques, including optimization on smooth manifolds, to develop systems that are efficient, scalable, and reliable. This course would provide a strong foundation in the mathematical and computational techniques used in data engineering, including optimization on smooth manifolds. The course's focus on developing geometric tools to work on new manifolds of interest would be particularly relevant to Data Engineers who work on complex data systems.
Actuary
Actuary
Actuaries use mathematical and statistical techniques to assess risk and uncertainty. They use a variety of techniques, including optimization on smooth manifolds, to develop models that can predict future events and outcomes. This course would provide a strong foundation in the mathematical and computational techniques used in actuarial science, including optimization on smooth manifolds. The course's focus on developing geometric tools to work on new manifolds of interest would be particularly relevant to Actuaries who work with complex risk models.
Computer Scientist
Computer Scientist
Computer Scientists conduct research in the field of computer science. They develop new theories and algorithms to solve problems in a variety of areas, including artificial intelligence, machine learning, and data science. This course would provide a strong foundation in the mathematical and computational techniques used in computer science, including optimization on smooth manifolds. The course's focus on developing geometric tools to work on new manifolds of interest would be particularly relevant to Computer Scientists who work on complex computational problems.
Risk Manager
Risk Manager
Risk Managers identify and assess risks to organizations and develop strategies to mitigate those risks. They use a variety of techniques, including optimization on smooth manifolds, to develop models that can predict future events and outcomes. This course would provide a strong foundation in the mathematical and computational techniques used in risk management, including optimization on smooth manifolds. The course's focus on developing geometric tools to work on new manifolds of interest would be particularly relevant to Risk Managers who work with complex risk models.
Systems Analyst
Systems Analyst
Systems Analysts design and implement computer systems to meet the needs of organizations. They use a variety of techniques, including optimization on smooth manifolds, to develop systems that are efficient, reliable, and secure. This course would provide a strong foundation in the mathematical and computational techniques used in systems analysis, including optimization on smooth manifolds. The course's focus on developing geometric tools to work on new manifolds of interest would be particularly relevant to Systems Analysts who work on complex systems.
Quality Manager
Quality Manager
Quality Managers plan and oversee the quality of products and services. They use a variety of techniques, including optimization on smooth manifolds, to develop systems that ensure that products and services meet customer requirements. This course would provide a strong foundation in the mathematical and computational techniques used in quality management, including optimization on smooth manifolds. The course's focus on developing geometric tools to work on new manifolds of interest would be particularly relevant to Quality Managers who work with complex products and services.
Business Analyst
Business Analyst
Business Analysts use mathematical and statistical techniques to analyze business data and make recommendations to improve business performance. They use a variety of techniques, including optimization on smooth manifolds, to develop models that can predict future trends and outcomes. This course would provide a strong foundation in the mathematical and computational techniques used in business analysis, including optimization on smooth manifolds. The course's focus on developing geometric tools to work on new manifolds of interest would be particularly relevant to Business Analysts who work with complex business data.
Statistician
Statistician
Statisticians collect, analyze, and interpret data to help businesses and organizations make informed decisions. They use a variety of techniques, including optimization on smooth manifolds, to develop models that can predict future trends and outcomes. This course would provide a strong foundation in the mathematical and computational techniques used in statistics, including optimization on smooth manifolds. The course's focus on developing geometric tools to work on new manifolds of interest would be particularly relevant to Statisticians who work with complex data structures.
Software Engineer
Software Engineer
Software Engineers design, develop, and maintain software applications. They use a variety of programming languages and tools to create software that meets the needs of users. This course would provide a strong foundation in the mathematical and computational techniques used in software engineering, including optimization on smooth manifolds. The course's focus on developing geometric tools to work on new manifolds of interest would be particularly relevant to Software Engineers who work on complex software systems.
Operations Manager
Operations Manager
Operations Managers plan and oversee the operations of organizations. They use a variety of techniques, including optimization on smooth manifolds, to develop systems that are efficient, effective, and profitable. This course would provide a strong foundation in the mathematical and computational techniques used in operations management, including optimization on smooth manifolds. The course's focus on developing geometric tools to work on new manifolds of interest would be particularly relevant to Operations Managers who work with complex operations.
For more career information including salaries, visit:
OpenCourser.com/course/ac0mrk/introduction
Reading list
We've selected six books
that we think will supplement your
learning. Use these to
develop background knowledge, enrich your coursework, and gain a
deeper understanding of the topics covered in
Introduction to optimization on smooth manifolds: first order methods.
Is written by the instructor of the course and covers the same topics. It valuable resource for both understanding the concepts and implementing the algorithms.
Provides a comprehensive treatment of calculus on manifolds, which is essential for understanding the mathematical foundations of optimization on manifolds.
Provides a comprehensive introduction to Riemannian manifolds, which are a type of smooth manifold that arise in many applications, including optimization.
Save
Provides a more advanced treatment of Riemannian geometry, which is essential for understanding the Riemannian gradient descent algorithm.
Provides a comprehensive introduction to numerical optimization, including a chapter on optimization on manifolds.
Provides a comprehensive introduction to convex optimization, which powerful tool for solving a wide range of optimization problems.
For more information about how these books relate to this course, visit:
OpenCourser.com/course/ac0mrk/introduction
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Effort
5 to 6 hours per week for 6 weeks
Level
Advanced
Via
edX
Institution
École Polytechnique Fédérale de Lausanne
Instructor
Nicolas Boumal
Session
January 1, 2024
- January 1, 2050
Language
English
Transcript
English
Good to know
Examines optimization on smooth manifolds, a technique combining differential and Riemannian geometry to optimize algorithms on manifolds
Develops time-tested optimization algorithms grounded in strong mathematical foundations
Taught by Nicolas Boumal, an expert in optimization on smooth manifolds
Provides hands-on experience through the use of the Manopt toolbox for optimization on manifolds
Suitable for learners with a strong background in differential and Riemannian geometry
Requires access to numerical tools and software, such as Manopt
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