Prof. Dr. Carlos Enrique Améndola Cerón Email: amendola (at) math.tu-berlin.de

I am an assistant professor (tenure-track) of Algebraic and Geometric Methods in Data Analysis at the Institute of Mathematics of the Technical University of Berlin.

My main research interest is Algebraic Statistics. I am also very interested in developments concerning Applied Algebraic Geometry and Nonlinear Algebra.

Some research keywords: Gaussian mixtures, likelihood geometry, moment varieties, graphical models, path signatures.

Research Projects

Berlin Mathematics Research Center MATH+ Project AA3-16: Likelihood Geometry of Max-Linear Bayesian Networks (2023-)

DFG Collaborative Research Center Rough Analysis, Stochastic Dynamics and Related Fields CRC / TRR 388:
Project A04: Algebra and Geometry of Path Signatures (2024-)
Project B01: Statistical Learning from Path Observations (2024-)

DFG Priority Program Combinatorial Synergies SPP 2458: Combinatorial Methods for Learning Max-Linear Bayesian Networks (2024-)

Highlights

Since 2022, I am an editor of the Algebraic Statistics (AStat) journal, check it out for the latest research and for submitting your work!

In 2023, Eliana Duarte, Thomas Kahle, and I organized an Algebraic Statistics Seminar between Berlin, Leipzig and Magdeburg.

From July 2021 - 2023, I was the Bernoulli Society Web editor. Before that, I was the e-Briefs editor.

During 2020-2021, I helped organize the Algebraic Statistics Online Seminar (ASOS).

About Me

I graduated as a Mathematics PhD student in Bernd Sturmfels' Einstein Fellowship research group in Nonlinear Algebra at TU Berlin. My co-advisor was Christian Haase.

Previously I was part of the Nonlinear Algebra research group at the Max-Planck Institute for Mathematics in the Sciences. I was a postdoc at TU Munich in the research groups Mathematical Statistics led by Mathias Drton (formerly by Claudia Klüppelberg) and Optimization and Data Analysis led by Felix Krahmer. During the Winter Semester 2020/2021, I was substitute professor at the Institute of Stochastics of Ulm University.

I have a B.S. in Applied Mathematics from Instituto Tecnológico Autónomo de México and an additional B.S. in Mathematics degree from Universidad Nacional Autónoma de México . I also obtained an M.S. in Mathematics from Courant Institute at New York University.

Publications and Preprints

Tropical Combinatorics of Max-Linear Bayesian Networks (with K. Ferry).

Maximum Likelihood Estimation of Log-Affine Models Using Detailed-Balanced Reaction Networks (with O. Henriksson, J. Rodriguez and P. Yu).

A Bridge between Invariant Theory and Maximum Likelihood Estimation (with K. Kohn, P. Reichenbach and A. Seigal), SIAM SIGEST Award Article , Vol. 66, No. 4 (2024) 721-747.

On the Maximum Likelihood Degree of Gaussian Graphical Models (with R. Dinu, M. Michałek and M. Vodička).

Likelihood Geometry of Reflexive Polytopes (with J. Oldekop), Algebraic Statistics 15(1) (2024) 113-143.

Decomposing Tensor Spaces via Path Signatures (with F. Galuppi, A. Ríos, P. Santarsiero and T. Seynnaeve), Journal of Pure and Applied Algebra , 229(1) (2024).

Differential Equations for Gaussian Statistical Models with Rational Maximum Likelihood Estimator (with L. Gustafsson, K. Kohn, O. Marigliano and A. Seigal), SIAM Journal on Applied Algebra and Geometry , 8(3) (2024) 465-492.

Convex Hulls of Curves: Volumes and Signatures (with D. Lee and C. Meroni),Geometric Science of Information (GSI'23) , (2023) 455-464.

Identifiability in Continuous Lyapunov Models (with P. Dettling, R. Homs, M. Drton and N. Hansen), SIAM Journal on Matrix Analysis and Applications, 44(4) (2023).

Third-Order Moment Varieties of Linear Non-Gaussian Graphical Models (with M. Drton, A. Grosdos, R. Homs and E. Robeva), Information and Inference, 12(3) (2023).

Estimating Gaussian Mixtures Using Sparse Polynomial Moment Systems (with J. Lindberg and J. Rodriguez), SIAM Journal on Mathematics of Data Science (SIMODS), to appear.

Markov Equivalence of Max-Linear Bayesian Networks (with B. Hollering, S. Sullivant and N. Tran), Uncertainty in Artificial Intelligence, PLMR 161 (2021) 1746-1755.

An Invitation to Tropical Alexandrov Curvature (with A. Monod), Algebraic Statistics 14(2) (2023) 181-214.

Computing Maximum Likelihood Estimates for Gaussian Graphical Models with Macaulay2 (with L. García Puente, R. Homs, O. Kuznetsova and H. Motwani), Journal of Software for Algebra and Geometry 12(1) (2022) 1-10.

Toric Invariant Theory for Maximum Likelihood Estimation in Log-linear Models (with K. Kohn, P. Reichenbach and A. Seigal), Algebraic Statistics 12(2) (2021) 187-211.

Likelihood Geometry of Correlation Models (with P. Zwiernik), Le Matematiche , 76(2) (2021) 559-583.

The Maximum Likelihood Degree of Linear Spaces of Symmetric Matrices (with L. Gustafsson, K. Kohn, O. Marigliano and A. Seigal), Le Matematiche , 76(2) (2021) 535-557.

Structure Learning for Cyclic Linear Causal Models (with P. Dettling, M. Drton, F. Onori and J. Wu), Uncertainty in Artificial Intelligence, UAI 2020.

Invariant Theory and Scaling Algorithms for Maximum Likelihood Estimation (with K. Kohn, P. Reichenbach and A. Seigal), SIAM Journal on Applied Algebra and Geometry , 5(2) (2021) 304-337.

Conditional Independence in Max-linear Bayesian Networks (with C. Klüppelberg, S. Lauritzen and N. Tran), Annals of Applied Probability, 31(1) (2022) 1-45.

Maximum Likelihood Estimation of Toric Fano Varieties (with D. Kosta and K. Kubjas), Algebraic Statistics 11(1) (2020) 5-30.

Moment Identifiability of Homoscedastic Gaussian Mixtures (with D. Agostini and K. Ranestad), Foundations of Computational Mathematics 21 (2021) 695-724.

Autocovariance Varieties of Moving Average Random Fields (with V.S. Pham), Journal of Symbolic Computation (2020), presented at MEGA 2019.

Tapas of Algebraic Statistics (with M. Casanellas and L. García Puente), Notices of the American Mathematical Society , Vol. 65, No. 8 (2018) 936-938.

Varieties of Signature Tensors (with P. Friz and B. Sturmfels), Forum of Mathematics, Sigma , 7 (2019) e10.

Discrete Gaussian Distributions via Theta Functions (with D. Agostini), SIAM Journal on Applied Algebra and Geometry , 3(1) (2019) 1-30.

Computing Tropical Varieties in Macaulay2 (with K. Kohn, S. Lamboglia, D. Maclagan, B. Smith, J. Sommars, P. Tripoli and M. Zajaczkowska).

Maximum Number of Modes of Gaussian Mixtures (with A. Engström and C. Haase), Information and Inference: A Journal of the IMA , 9(3) (2019) 587-600.

The Maximum Likelihood Degree of Toric Varieties (with N. Bliss, I. Burke, C. Gibbons, M. Helmer, S. Hoşten, E. Nash, J. Rodriguez and D. Smolkin), Journal of Symbolic Computation , 92 (2018) 222-242.

Solving Parameterized Polynomial Systems with Decomposable Projections (with J. Lindberg and J. Rodriguez), presented at MEGA 2017 Code

Algebraic Identifiability of Gaussian Mixtures (with K. Ranestad and B. Sturmfels), International Mathematics Research Notices , 21 (2018) 6556-6580.

Moment Varieties of Gaussian Mixtures (with J-C. Faugère and B. Sturmfels), Algebraic Statistics , 7 (2016) 14-28.

Maximum Likelihood Estimates for Gaussian Mixtures are Transcendental (with M. Drton and B. Sturmfels), Mathematical Aspects of Computer and Information Sciences , (eds. I. Kotsireas, S. Rump and C. Yap), MACIS 2015, Berlin, (2016) 579-590.

PhD Thesis Algebraic Statistics of Gaussian Mixtures (2017). Advisor: Bernd Sturmfels , Christian Haase

Thesis Bases de Gröbner: una variedad de aplicaciones (Gröbner Bases: a variety of applications) (2013). Advisor: Dr. Leon Kushner Schnur

Thesis Una Introducción a la Estadística Algebraica (An Introduction to Algebraic Statistics) (2012). Advisor: Dra. Araceli Reyes Guerrero

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PhD Students

Postdocs

Teaching

Winter Semester 2024-2025: Discrete Geometry II

Summer Semester 2024: Discrete Geometry I

Winter Semester 2023-2024: Algebraic Statistics

Summer Semester 2023: Discrete Geometry III

Winter Semester 2022/2023: Discrete Geometry II

Winter Semester 2021/2022: Fundamentals of Mathematical Statistics [MA5441]
Advanced Student Seminar: Algebraic Statistics

Summer Semester 2021: Statistical Inverse Problems [MA5428]

Winter Semester 2020/2021: Angewandte Stochastik II [MA6902]
Markov-Ketten [MA2019]
Graphical Models [MA2025]

Summer Semester 2020: Statistik: Grundlagen [MA2402]

Winter Semester 2019/2020: Fundamentals of Mathematical Statistics [MA5441]

Summer Semester 2019: Reading Seminar: High-Dimensional Statistics

Winter Semester 2018/2019: Probabilistic Methods and Algorithms in Data Analysis [MA4803]
Advanced Student Seminar: Algebraic Statistics

Summer Semester 2018: Statistics for Business Administration (with Introduction to R) [MA9712]

Activities