30/05/2025

'I am at the intersection between the real and the theoretical,' says Adriana Monteiro.

IMPA doctoral student Adriana Laurindo Monteiro, as she defines herself, navigates through “different worlds”. A native of Vila Velha (ES), she assures “I feel at home in Rio”. In school, she even thought she would pursue a career in humanities, but ended up in mathematics. In her undergraduate studies, she wanted to do a teaching degree and ended up with a bachelor's degree. The versatility in life and in academia led to the defense of the thesis “Non-parametric Inference in Optimal Transport Problems”, which mixes concepts from pure and applied mathematics. Under the guidance of the institute's researcher Roberto Imbuzeiro, Adriana defends her thesis this Tuesday (3) in room 232 of IMPA. The live broadcast can be followed on the Youtube channel , at 9 am.

“I define myself as a person at the intersection of the real world and the theoretical world. My education here at IMPA, the courses I took, largely went in that direction. As a pure mathematics student, I took courses in probability, analysis, and statistics, which are extremely important, but I also always sought optimization, machine learning theory, and information theory, because I felt that I had reached a point where pure mathematics wasn't going to exhaust me. So, I needed something more concrete, and I think statistics can build a good bridge between these two worlds,” she says.

Adriana completed her undergraduate and master's degrees at the Federal University of Espírito Santo (UFES), during which time she became acquainted with IMPA. Encouraged by her professors, she participated in the institute's traditional Summer Program and immediately felt a connection with it. Shortly after, the young woman packed her bags and left Espírito Santo for Rio de Janeiro to begin her doctoral studies at the institute.

At IMPA, Adriana works in the area of probability and statistics, specializing in the field of Optimal Transport (OT). She also participated in two projects at the IMPA Center for Projects and Innovation (Centro Pi), one in partnership with Hurb, on sales forecasting, and another in partnership with the Rio de Janeiro City Hall, on rainfall forecasting. The development of her thesis gained momentum during a sandwich program she completed in 2024 at Paul Sabatier University in France, and a visit to the University of Valladolid in Spain. This experience allowed her to gain a deeper understanding of the Optimal Transport community, which led to two of her studies from that period being included in her doctoral thesis.

This research encompasses two applications of Optimal Transport Theory in Statistics and one in Machine Learning. The first application is an inference problem given by the continuity equation. The proposal is a kernel estimator that describes the motion of d-dimensional particles following the continuity equation.

“The statistical problem unfolds as follows: we can know the position of a given particle at the initial moment and at certain instants in time. How, then, is it possible to make a good guess as to the directions governing this movement? My result is in line with proposing a non-parametric estimator for this vector field that satisfies this equation, the continuity equation,” explained Adriana.

The second application of OT theory deals with the problem generated by the massive use of Machine Learning algorithms as black boxes. By analyzing the response of models to variations in the distribution of input variables, we can better understand how the algorithm processes the data. Starting from the line of explainability, the idea is to explain why a given algorithm delivers a specific answer. Adriana sought to identify which properties were important to arrive at a final rule.

The third problem involves estimating the projection of a Gaussian probability using an empirical entropic Gaussian projection. The proposal was to apply a non-parametric statistical method called the Goldenshluger–Lepski Method, which allows the selection of the best parameter to attempt to approximate the entropic projection of a Gaussian distribution.

Adriana highlights that, among the main contributions of her thesis, is the relevance and timeliness of an application in Machine Learning. The subject, as a research topic, is often analyzed based on the performance and accuracy of the models. Adriana's thesis, however, focuses more on understanding how AI works.

“This research goes against the grain of many other studies, and that’s precisely why it’s so important. Many AI models are reproducing various tendencies and biases that are not good, because people generally use them as if they were a black box. Research like mine is trying to avoid this type of use. The idea is to analyze an algorithm and try to understand how the information is being processed there,” he explained.

The next step in her career will be a postdoctoral fellowship at the Getúlio Vargas Foundation (FGV), in the School of Applied Mathematics. “I’m very proud of everything I’ve built. It seems like it went by quickly, but four years can also be quite long. I did a lot, and that was very good. IMPA is incredible in that sense; we are presented with so many possibilities, including in various places around the world. That’s amazing!”