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Introduction to optimization on smooth manifolds

first order methods

Nicolas Boumal

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.

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.

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

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Traffic lights

Read about what's good

what should give you pause

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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Create your own learning path. Save this course to your list so you can find it easily later.

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Reviews summary

Foundational riemannian optimization

According to learners, this course offers a phenomenal and rigorous introduction to optimization on smooth manifolds. Students widely praise the instructor's incredibly clear explanations that make intricate mathematical topics accessible, even when the content is dense. The curriculum builds a strong theoretical foundation, leading effectively to Riemannian gradient descent and practical application through the Manopt toolbox, which is often highlighted as a practical highlight. However, some reviewers note that the course is extremely demanding and requires a strong mathematical background in areas like differential geometry, warning it's not for the faint of heart if prerequisites are weak.

Lectures closely follow and complement the instructor's textbook.

"I particularly appreciated how the lectures followed the textbook closely, making it easy to supplement."

"The provided textbook is a must-have companion and helps solidify the understanding."

Effectively integrates the Manopt toolbox for practical use.

"The Manopt section was very practical. Highly recommend for anyone with a solid linear algebra and calculus background."

"The final part on Manopt was a practical highlight."

"The Manopt section is a fantastic bonus, making the theoretical concepts more tangible."

Provides a rigorous and understandable base for advanced topics.

"This course is a phenomenal introduction to optimization on manifolds. The instructor's explanations are incredibly clear, even for such a complex topic."

"Absolutely brilliant! The instructor makes intricate topics accessible. This course provided me with the tools to confidently approach Riemannian gradient descent..."

"This course perfectly bridges the gap between geometry and optimization. The instructor's passion shines through, and the structured approach... is masterful."

"I found the visualizations in the lectures incredibly helpful for understanding the geometry involved."

Material is rigorous and moves at a demanding speed.

"The math is rigorous and sometimes challenging, but the concepts are explained well. I felt the pace was a bit fast in the middle..."

"The course covers essential topics, but it's extremely demanding."

"I did find myself re-watching lectures multiple times, especially the parts on retractions and vector fields."

Course demands a solid background in advanced mathematics.

"If your background in differential geometry is weak, prepare for a struggle. The lectures are comprehensive but sometimes feel rushed."

"It definitely requires a strong mathematical background."

"I came with a strong background in optimization but lacked differential geometry. The course assumes too much prior knowledge. I was lost after the second week."

"I had to spend a lot of time reviewing external resources to keep up with the pace."

Activities

Be better prepared before your course. Deepen your understanding during and after it. Supplement your coursework and achieve mastery of the topics covered in Introduction to optimization on smooth manifolds: first order methods with these activities:

Review 'An Introduction to Optimization on Smooth Manifolds'

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Review foundational concepts of smooth manifolds and optimization covered in the textbook by the instructor.

View An Introduction to Optimization on Smooth... on Amazon

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Read the preface and first two chapters.
Complete the exercises in the first two chapters.

Attend a Workshop on Optimization on Manifolds

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Gain practical insights and hands-on experience.

Browse courses on Optimization

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Research upcoming workshops related to optimization on manifolds.
Attend a workshop that aligns with your learning goals.

Solve Optimization Problems on Manifolds

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Reinforce understanding of optimization algorithms through practice.

Browse courses on Optimization

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Attempt the practice problems provided at the end of each chapter.
Solve the optimization problems from the textbook's companion website.

Five other activities

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Participate in a Coding Challenge on Optimization on Manifolds

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Test and refine problem-solving skills in a competitive environment.

Browse courses on Optimization

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Identify coding challenges or competitions related to optimization on manifolds.
Register and participate in the challenge.

Organize Course Materials for Review

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Prepare course materials for efficient and effective review.

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Review notes from each lecture.
Organize lecture slides, handouts, and assignments.
Create a study guide using the compiled materials.

Follow Tutorials on Riemannian Optimization Algorithms

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Extend knowledge of first-order optimization algorithms and explore advanced concepts.

Browse courses on Optimization

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Identify reputable resources and tutorials on Riemannian optimization algorithms.
Follow the tutorials and complete the accompanying exercises.

Develop a Visualization of Optimization on Manifolds

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Enhance understanding through visual representation and foster creativity.

Browse courses on Visualization

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Choose a suitable software or tool for creating visualizations.
Select an appropriate dataset or scenario for optimization on a manifold.
Design and develop a visualization that effectively conveys the concept.

Mentor a Junior Student in Optimization

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Solidify understanding by teaching and reinforce problem-solving skills.

Browse courses on Optimization

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Identify a junior student who needs assistance with optimization concepts.
Provide guidance and support on homework assignments or projects.
Facilitate discussion and encourage critical thinking.

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:

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.

See salaries and explore the career path for Data Scientist

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.

See salaries and explore the career path for Machine Learning Engineer

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.

See salaries and explore the career path for Quantitative 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.

See salaries and explore the career path for Operations Research 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.

See salaries and explore the career path for Financial Analyst

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.

See salaries and explore the career path for Software Engineer

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.

See salaries and explore the career path for Computer Scientist

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.

See salaries and explore the career path for Systems Analyst

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.

See salaries and explore the career path for Data Engineer

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.

See salaries and explore the career path for Statistician

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.

See salaries and explore the career path for Actuary

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.

See salaries and explore the career path for Risk Manager

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.

See salaries and explore the career path for Business Analyst

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.

See salaries and explore the career path for Operations 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.

See salaries and explore the career path for Quality Manager

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.

Cover image

Cover image

An Introduction to Optimization on Smooth Manifolds

Save

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.

An Introduction to Optimization on Smooth Manifolds

An Introduction to Optimization on Smooth Manifolds

Cover image

Cover image

Calculus on Manifolds

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Provides a comprehensive treatment of calculus on manifolds, which is essential for understanding the mathematical foundations of optimization on manifolds.

Calculus On Manifolds: A Modern Approach To...

Calculus on manifolds (bilingual edition)

Calculus On Manifolds: A Modern Approach To...

Cover image

Cover image

Introduction to Riemannian Manifolds

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Provides a comprehensive introduction to Riemannian manifolds, which are a type of smooth manifold that arise in many applications, including optimization.

Introduction to Riemannian Manifolds

Introduction to Riemannian Manifolds

Cover image

Cover image

Riemannian Geometry

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Provides a more advanced treatment of Riemannian geometry, which is essential for understanding the Riemannian gradient descent algorithm.

Riemannian Geometry

Riemannian Geometry

Cover image

Cover image

Numerical Optimization

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Provides a comprehensive introduction to numerical optimization, including a chapter on optimization on manifolds.

Numerical Optimization

Numerical Optimization

Cover image

Cover image

Introduction to Applied Linear Algebra

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Provides a comprehensive introduction to convex optimization, which powerful tool for solving a wide range of optimization problems.

Introduction to Applied Linear Algebra: Vectors,...

Introduction to Applied Linear Algebra: Vectors,...

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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

Language

English

Transcript

English

Traffic lights

Read about what's good

what should give you pause

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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