UROP Research Mentor Project Submission Portal: Submission #1303
Submission information
Submission Number: 1303
Submission ID: 21111
Submission UUID: baadfca4-0209-441c-8087-1e2ede6cedb9
Submission URI: /urop-research-mentor-project-submission-portal
Submission Update: /urop-research-mentor-project-submission-portal?token=gVzuRfRW9rfa-AhS-K5X85vttzQyYyiSlSFze_ain-E
Created: Wed, 08/20/2025 - 06:53 PM
Completed: Wed, 08/20/2025 - 10:15 PM
Changed: Tue, 08/26/2025 - 05:25 PM
Remote IP address: 46.110.139.141
Submitted by: Anonymous
Language: English
Is draft: No
Webform: UROP Project Proposal Portal
Submitted to: UROP Research Mentor Project Submission Portal
Research Mentor Information
Rocio Diaz Martin
She/her/hers
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Faculty
Arts and Sciences
Department of Mathematics
Additional Research Mentor(s)
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Overall Project Details
Discovering Matrix Inversion through Transport Theory
Positive Definite Matrices - Measures and Probabilities - Optimal Transport - Wasserestein metric
Yes
2
Mathematics - Computer Science - Electrical Engineering - Physics
On FSU Main Campus
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Partially Remote
5
During business hours
This project aims to design a new way of inverting special types of matrices, which are fundamental objects in mathematics and data science. Instead of relying on traditional formulas, we will use ideas from Optimal Transport, a mathematical framework originally developed to compare probability distributions. Inspired by an algorithm first used in image processing, we will explore a method that updates information step by step along random directions, making it especially useful when the full matrix is too large to store. The final algorithm will be tested on examples, shared as open-source software, and presented through multiple channels, including academic presentations.
- Literature reviews.
- Developing and implementing algorithms through programming.
- Performing benchmark comparisons to evaluate and validate methods or models.
- Formulate and rigorously prove mathematically grounded results.
- Developing and implementing algorithms through programming.
- Performing benchmark comparisons to evaluate and validate methods or models.
- Formulate and rigorously prove mathematically grounded results.
Basic programming skills, preferably in Python (recommended).
Foundation in linear algebra (required).
Some knowledge of probability theory is preferred (recommended).
Foundation in linear algebra (required).
Some knowledge of probability theory is preferred (recommended).
I consider students as junior colleagues, empowering them to grow as collaborators rather than passive learners. I begin by sharing the theoretical foundations -especially intuitive and proof-based thinking- while actively programming side by side with them as a tool for exploration, not just execution.
As a young researcher and new faculty member, I recognize that mentoring presents a challenge for me, but it is one that excites me deeply. I view this process as a two-way learning experience: while students grow as mathematicians and programmers, I grow as a mentor and teacher. Coming from Argentina, I also bring a perspective shaped by my own educational journey, which helps me relate to students navigating diverse paths and backgrounds.
I set clear, mutual goals and maintain open, respectful communication. By inviting students to co-create the learning path, align expectations and build trust. My mentorship is inclusive: I’m attentive to different learning styles and backgrounds, ensuring all students feel valued and encouraged to share their ideas. Being approachable is essential, and I actively cultivate this quality in myself to create a welcoming and supportive environment for everybody.
I guide them to develop mathematical intuition by asking guiding questions and encouraging reflection -helping them formulate logical results and rigorous proofs in their own words. We periodically assess progress, celebrate small victories, and iterate our process to strengthen understanding and confidence.
Ultimately, I aspire for students to become confident, independent thinkers: capable programmers, insightful mathematicians, and critical collaborators who continue learning beyond our time together.
As a young researcher and new faculty member, I recognize that mentoring presents a challenge for me, but it is one that excites me deeply. I view this process as a two-way learning experience: while students grow as mathematicians and programmers, I grow as a mentor and teacher. Coming from Argentina, I also bring a perspective shaped by my own educational journey, which helps me relate to students navigating diverse paths and backgrounds.
I set clear, mutual goals and maintain open, respectful communication. By inviting students to co-create the learning path, align expectations and build trust. My mentorship is inclusive: I’m attentive to different learning styles and backgrounds, ensuring all students feel valued and encouraged to share their ideas. Being approachable is essential, and I actively cultivate this quality in myself to create a welcoming and supportive environment for everybody.
I guide them to develop mathematical intuition by asking guiding questions and encouraging reflection -helping them formulate logical results and rigorous proofs in their own words. We periodically assess progress, celebrate small victories, and iterate our process to strengthen understanding and confidence.
Ultimately, I aspire for students to become confident, independent thinkers: capable programmers, insightful mathematicians, and critical collaborators who continue learning beyond our time together.
https://rociodm.github.io/
The project in more detail:
The goal of this project is to develop a new iterative method for approximating the inverse of a matrix using techniques from Optimal Transport (OT) theory. OT is a central area of my current research, and as a new Assistant Professor in the Department of Mathematics, I am thrilled to continue studying it and to share my background. We will begin by exploring both its fundamental theory and applications. We will begin by studying both its fundamental theory and applications. For example, the Wasserstein metric (also known as Earth Mover's Distance) is a key object in OT, widely used in Machine Learning and inspiring developments such as Wasserstein Generative Adversarial Networks (Wasserstein-GANs) by M. Arjovsky et al. in 2017. This theoretical study will also provide an opportunity to explore other important concepts in applied mathematics.
We will focus on symmetric positive definite (SPD) matrices, as they can be interpreted as the covariance matrix of a multivariate Gaussian distribution. This allows us to associate a Gaussian measure to each matrix and apply OT techniques to the corresponding probability distributions, rather than relying solely on traditional linear algebra methods.
As part of the initial steps, the project will include a thorough literature review to understand the foundations and developments of a specific algorithm within the OT framework: the Iterative Distribution Transfer (IDT) algorithm introduced by F. Pitié et al. in 2007. Originally, this method was developed for color grading (adjusting the colors in an image or video to achieve a desired aesthetic or mood) and color transfer (transferring the color style of one image to another), which are important tasks in image processing and computer vision.
Next, we will investigate how to invert SPD matrices using ideas inspired by the IDT algorithm. In particular, we will explore how to leverage one-dimensional updates along random directions, analogous to the IDT updates for probability measures. This approach has strong potential in matrix-free and streaming settings, where memory is limited and direct or incomplete factorizations are impractical -for instance, in cases where only matrix-vector products are accessible, while the matrix itself is not explicitly available.
The project objectives include developing an iterative algorithm for inverting SPD matrices, analyzing its intermediate approximations, studying convergence conditions and rates, and evaluating its efficiency relative to standard inversion techniques. Proof-of-concept examples, demonstrating guaranteed convergence for matrices of different sizes, will be used to build intuition and validate the approach.
The resulting algorithm will preferably be implemented in Python and released as open-source software. In addition, we aim to disseminate our findings through multiple channels, including academic presentations and publications.
The goal of this project is to develop a new iterative method for approximating the inverse of a matrix using techniques from Optimal Transport (OT) theory. OT is a central area of my current research, and as a new Assistant Professor in the Department of Mathematics, I am thrilled to continue studying it and to share my background. We will begin by exploring both its fundamental theory and applications. We will begin by studying both its fundamental theory and applications. For example, the Wasserstein metric (also known as Earth Mover's Distance) is a key object in OT, widely used in Machine Learning and inspiring developments such as Wasserstein Generative Adversarial Networks (Wasserstein-GANs) by M. Arjovsky et al. in 2017. This theoretical study will also provide an opportunity to explore other important concepts in applied mathematics.
We will focus on symmetric positive definite (SPD) matrices, as they can be interpreted as the covariance matrix of a multivariate Gaussian distribution. This allows us to associate a Gaussian measure to each matrix and apply OT techniques to the corresponding probability distributions, rather than relying solely on traditional linear algebra methods.
As part of the initial steps, the project will include a thorough literature review to understand the foundations and developments of a specific algorithm within the OT framework: the Iterative Distribution Transfer (IDT) algorithm introduced by F. Pitié et al. in 2007. Originally, this method was developed for color grading (adjusting the colors in an image or video to achieve a desired aesthetic or mood) and color transfer (transferring the color style of one image to another), which are important tasks in image processing and computer vision.
Next, we will investigate how to invert SPD matrices using ideas inspired by the IDT algorithm. In particular, we will explore how to leverage one-dimensional updates along random directions, analogous to the IDT updates for probability measures. This approach has strong potential in matrix-free and streaming settings, where memory is limited and direct or incomplete factorizations are impractical -for instance, in cases where only matrix-vector products are accessible, while the matrix itself is not explicitly available.
The project objectives include developing an iterative algorithm for inverting SPD matrices, analyzing its intermediate approximations, studying convergence conditions and rates, and evaluating its efficiency relative to standard inversion techniques. Proof-of-concept examples, demonstrating guaranteed convergence for matrices of different sizes, will be used to build intuition and validate the approach.
The resulting algorithm will preferably be implemented in Python and released as open-source software. In addition, we aim to disseminate our findings through multiple channels, including academic presentations and publications.
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UROP Program Elements
Yes
Yes
Yes
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2025
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