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Employment
2022 — current
Ph.D. Student in Mathematics
Heidelberg University and Heidelberg Institute for Theoretical Studies, Heidelberg, Germany
Topic: Petabyte-scale image analysis for life sciences
Supervisor: Prof. Dr. Vincent Heuveline
2017 — 2022
Ph.D. Student in Mathematics
Heidelberg University, Heidelberg, Germany
Topic: Low-level image analysis on statistical manifolds
Supervisor: Prof. Dr. Christoph Schnörrr, change due to personal interest
2014 — 2016
Working Student
TNG Technology Consulting, Munich, Germany
Topic: Full-stack web development with a focus on frontend architecture
Education
2014 — 2017
Master of Science Mathematics
Heidelberg University, Heidelberg, Germany
Master's Thesis: Perturbed Discrete Functionals for Image Labeling
During my master of science in Mathematics at Heidelberg University I focused on courses in mathematical data analysis and inparticular image analysis.
2013
Erasmus semester
Utrecht University, Utrecht, Netherlands
2010 — 2014
Bachelor of Science Mathematics
TU Munich, Munich, Germany
Bachelor's Thesis: The Dirichlet Prime Number Theorem for arithmetic progressions
We introduce a novel algorithm for estimating optimal parameters of linearized assignment flows for image labeling. An exact formula is derived for the parameter gradient of any loss function that is constrained by the linear system of ODEs determining the linearized assignment flow. We show how to efficiently evaluate this formula using a Krylov subspace and a low-rank approximation. This enables us to perform parameter learning by Riemannian gradient descent in the parameter space, without the need to backpropagate errors or to solve an adjoint equation. Experiments demonstrate that our method performs as good as highly-tuned machine learning software using automatic differentiation. Unlike methods employing automatic differentiation, our approach yields a low-dimensional representation of internal parameters and their dynamics which helps to understand how assignment flows and more generally neural networks work and perform.
@article{Zeilmann2023Learning,
author = {Zeilmann, Alexander and Petra, Stefania and Schn{\" o}rr, Christoph},
journal = {Journal of Mathematical Imaging and Vision},
number = {1},
year = {2023},
month = {jan 17},
pages = {164--184},
publisher = {{Springer Science and Business Media LLC}},
title = {Learning {Linearized} {Assignment} {Flows} for {Image} {Labeling}},
volume = {65},
}
@article{Zeilmann2023Learning,
abstract = {We introduce a novel algorithm for estimating optimal parameters of linearized assignment flows for image labeling. An exact formula is derived for the parameter gradient of any loss function that is constrained by the linear system of ODEs determining the linearized assignment flow. We show how to efficiently evaluate this formula using a Krylov subspace and a low-rank approximation. This enables us to perform parameter learning by Riemannian gradient descent in the parameter space, without the need to backpropagate errors or to solve an adjoint equation. Experiments demonstrate that our method performs as good as highly-tuned machine learning software using automatic differentiation. Unlike methods employing automatic differentiation, our approach yields a low-dimensional representation of internal parameters and their dynamics which helps to understand how assignment flows and more generally neural networks work and perform.},
author = {Zeilmann, Alexander and Petra, Stefania and Schn{\" o}rr, Christoph},
journaltitle = {Journal of Mathematical Imaging and Vision},
shortjournal = {J Math Imaging Vis},
doi = {10.1007/s10851-022-01132-9},
issn = {0924-9907},
number = {1},
date = {2023-01-17},
language = {en},
pages = {164--184},
publisher = {{Springer Science and Business Media LLC}},
title = {Learning {Linearized} {Assignment} {Flows} for {Image} {Labeling}},
url = {http://dx.doi.org/10.1007/s10851-022-01132-9},
volume = {65},
}
Like many other types of cancer, colorectal cancer (CRC) develops through multiple pathways of carcinogenesis. This is also true for colorectal carcinogenesis in Lynch syndrome (LS), the most common inherited CRC syndrome. However, a comprehensive understanding of the distribution of these pathways of carcinogenesis, which allows for tailored clinical treatment and even prevention, is still lacking. We suggest a linear dynamical system modeling the evolution of different pathways of colorectal carcinogenesis based on the involved driver mutations. The model consists of different components accounting for independent and dependent mutational processes. We define the driver gene mutation graphs and combine them using the Cartesian graph product. This leads to matrix components built by the Kronecker sum and product of the adjacency matrices of the gene mutation graphs enabling a thorough mathematical analysis and medical interpretation. Using the Kronecker structure, we developed a mathematical model which we applied exemplarily to the three pathways of colorectal carcinogenesis in LS. Beside a pathogenic germline variant in one of the DNA mismatch repair (MMR) genes, driver mutations in APC, CTNNB1, KRAS and TP53 are considered. We exemplarily incorporate mutational dependencies, such as increased point mutation rates after MMR deficiency, and based on recent experimental data, biallelic somatic CTNNB1 mutations as common drivers of LS-associated CRCs. With the model and parameter choice, we obtained simulation results that are in concordance with clinical observations. These include the evolution of MMR-deficient crypts as early precursors in LS carcinogenesis and the influence of variants in MMR genes thereon. The proportions of MMR-deficient and MMR-proficient APC-inactivated crypts as first measure for the distribution among the pathways in LS-associated colorectal carcinogenesis are compatible with clinical observations. The approach provides a modular framework for modeling multiple pathways of carcinogenesis yielding promising results in concordance with clinical observations in LS CRCs.
From the Medical Hypothesis Over the Modeling Approach To the Mathematical Structure.
The medical hypothesis of multiple pathways of carcinogenesis is widely known for various types of cancer. (A) We present a model for this phenomenon at the example of Lynch syndrome, the most common inherited CRC syndrome, with specific key driver events in the MMR genes, CTNNB1, APC, KRAS and TP53. (B) This current medical understanding of carcinogenesis is translated into a mathematical model using a specific dynamical system, which can be represented by a graph structure, where each vertex in the graph represents a genotypic state and the edges correspond to the transition probabilities between those states. Starting with all colonic crypts in the state of all genes being wild-type and a single MMR germline variant due to Lynch syndrome, we are interested in the distribution of the crypts among the graph at different ages of the patient in order to obtain estimates for the number of crypts in specific states, e.g., adenomatous or cancerous states. (C) The underlying matrix of the dynamical system makes use of the Kronecker sum and product. It is a sparse upper triangular matrix accounting for the assumption that mutations cannot be reverted. This allows fast numerical solving by using the matrix exponential. Each nonzero entry of the matrix represents a connection between genotypic states in the graph.
@article{Haupt2021Mathematical,
author = {Haupt, Saskia and Zeilmann, Alexander and Ahadova, Aysel and Bl{\" a}ker, Hendrik and von Knebel Doeberitz, Magnus and Kloor, Matthias and Heuveline, Vincent},
journal = {PLOS Computational Biology},
editor = {Chen, Jing},
number = {5},
year = {2021},
month = {may 18},
pages = {e1008970},
publisher = {Public Library of Science (PLoS)},
title = {Mathematical modeling of multiple pathways in colorectal carcinogenesis using dynamical systems with {Kronecker} structure},
volume = {17},
}
@article{Haupt2021Mathematical,
abstract = {Like many other types of cancer, colorectal cancer (CRC) develops through multiple pathways of carcinogenesis. This is also true for colorectal carcinogenesis in Lynch syndrome (LS), the most common inherited CRC syndrome. However, a comprehensive understanding of the distribution of these pathways of carcinogenesis, which allows for tailored clinical treatment and even prevention, is still lacking. We suggest a linear dynamical system modeling the evolution of different pathways of colorectal carcinogenesis based on the involved driver mutations. The model consists of different components accounting for independent and dependent mutational processes. We define the driver gene mutation graphs and combine them using the Cartesian graph product. This leads to matrix components built by the Kronecker sum and product of the adjacency matrices of the gene mutation graphs enabling a thorough mathematical analysis and medical interpretation. Using the Kronecker structure, we developed a mathematical model which we applied exemplarily to the three pathways of colorectal carcinogenesis in LS. Beside a pathogenic germline variant in one of the DNA mismatch repair (MMR) genes, driver mutations in \textit{APC}, \textit{CTNNB1}, \textit{KRAS} and \textit{TP53} are considered. We exemplarily incorporate mutational dependencies, such as increased point mutation rates after MMR deficiency, and based on recent experimental data, biallelic somatic \textit{CTNNB1} mutations as common drivers of LS-associated CRCs. With the model and parameter choice, we obtained simulation results that are in concordance with clinical observations. These include the evolution of MMR-deficient crypts as early precursors in LS carcinogenesis and the influence of variants in MMR genes thereon. The proportions of MMR-deficient and MMR-proficient \textit{APC}-inactivated crypts as first measure for the distribution among the pathways in LS-associated colorectal carcinogenesis are compatible with clinical observations. The approach provides a modular framework for modeling multiple pathways of carcinogenesis yielding promising results in concordance with clinical observations in LS CRCs.},
author = {Haupt, Saskia and Zeilmann, Alexander and Ahadova, Aysel and Bl{\" a}ker, Hendrik and von Knebel Doeberitz, Magnus and Kloor, Matthias and Heuveline, Vincent},
journaltitle = {PLOS Computational Biology},
shortjournal = {PLoS Comput Biol},
doi = {10.1371/journal.pcbi.1008970},
editor = {Chen, Jing},
issn = {1553-7358},
number = {5},
date = {2021-05-18},
language = {en},
pages = {e1008970},
eid = {e1008970},
eprint = {34003820},
eprinttype = {pubmed},
publisher = {Public Library of Science (PLoS)},
title = {Mathematical modeling of multiple pathways in colorectal carcinogenesis using dynamical systems with {Kronecker} structure},
url = {http://dx.doi.org/10.1371/journal.pcbi.1008970},
volume = {17},
}
The assignment flow is a smooth dynamical system that evolves on an elementary statistical manifold and performs contextual data labeling on a graph. We derive and introduce the linear assignment flow that evolves nonlinearly on the manifold, but is governed by a linear ODE on the tangent space. Various numerical schemes adapted to the mathematical structure of these two models are designed and studied, for the geometric numerical integration of both flows: embedded Runge–Kutta–Munthe–Kaas schemes for the nonlinear flow, adaptive Runge–Kutta schemes and exponential integrators for the linear flow. All algorithms are parameter free, except for setting a tolerance value that specifies adaptive step size selection by monitoring the local integration error, or fixing the dimension of the Krylov subspace approximation. These algorithms provide a basis for applying the assignment flow to machine learning scenarios beyond supervised labeling, including unsupervised labeling and learning from controlled assignment flows.
@article{Zeilmann2020Geometric,
author = {Zeilmann, Alexander and Savarino, Fabrizio and Petra, Stefania and Schn{\" o}rr, Christoph},
journal = {Inverse Problems},
number = {3},
year = {2020},
month = {feb 26},
pages = {034003},
publisher = {IOP Publishing},
title = {Geometric numerical integration of the assignment flow},
volume = {36},
}
@article{Zeilmann2020Geometric,
abstract = {The assignment flow is a smooth dynamical system that evolves on an elementary statistical manifold and performs contextual data labeling on a graph. We derive and introduce the linear assignment flow that evolves nonlinearly on the manifold, but is governed by a linear ODE on the tangent space. Various numerical schemes adapted to the mathematical structure of these two models are designed and studied, for the geometric numerical integration of both flows: embedded Runge--Kutta--Munthe--Kaas schemes for the nonlinear flow, adaptive Runge--Kutta schemes and exponential integrators for the linear flow. All algorithms are parameter free, except for setting a tolerance value that specifies adaptive step size selection by monitoring the local integration error, or fixing the dimension of the Krylov subspace approximation. These algorithms provide a basis for applying the assignment flow to machine learning scenarios beyond supervised labeling, including unsupervised labeling and learning from controlled assignment flows.},
author = {Zeilmann, Alexander and Savarino, Fabrizio and Petra, Stefania and Schn{\" o}rr, Christoph},
journaltitle = {Inverse Problems},
shortjournal = {Inverse Problems},
doi = {10.1088/1361-6420/ab2772},
issn = {0266-5611},
number = {3},
date = {2020-02-26},
pages = {034003},
publisher = {IOP Publishing},
title = {Geometric numerical integration of the assignment flow},
url = {http://dx.doi.org/10.1088/1361-6420/ab2772},
volume = {36},
}
We present an end-to-end learned algorithm for seeded segmentation. Our method is based on the Random Walker algorithm, where we predict the edge weights of the underlying graph using a convolutional neural network. This can be interpreted as learning context-dependent diffusivities for a linear diffusion process. Besides calculating the exact gradient for optimizing these diffusivities, we also propose simplifications that sparsely sample the gradient and still yield competitive results. The proposed method achieves the currently best results on a seeded version of the CREMI neuron segmentation challenge.
@inproceedings{Cerrone2019Walker,
address = {Long Beach, California},
author = {Cerrone, Lorenzo and Zeilmann, Alexander and Hamprecht, Fred A.},
year = {2019},
month = {jun 16},
pages = {12559--12568},
organization = {IEEE},
title = {End-to-{End} {Learned} {Random} {Walker} for {Seeded} {Image} {Segmentation}},
}
@inproceedings{Cerrone2019Walker,
abstract = {We present an end-to-end learned algorithm for seeded segmentation. Our method is based on the Random Walker algorithm, where we predict the edge weights of the underlying graph using a convolutional neural network. This can be interpreted as learning context-dependent diffusivities for a linear diffusion process. Besides calculating the exact gradient for optimizing these diffusivities, we also propose simplifications that sparsely sample the gradient and still yield competitive results. The proposed method achieves the currently best results on a seeded version of the CREMI neuron segmentation challenge.},
author = {Cerrone, Lorenzo and Zeilmann, Alexander and Hamprecht, Fred A.},
doi = {10.1109/CVPR.2019.01284},
venue = {Long Beach, CA, USA},
isbn = {978-1-72813-293-8},
date = {2019-06-16/2019-06-20},
keywords = {Computer Science - Computer Vision and Pattern Recognition,Computer Science - Machine Learning,ownPublication},
pages = {12559--12568},
location = {Long Beach, California},
publisher = {IEEE},
title = {End-to-{End} {Learned} {Random} {Walker} for {Seeded} {Image} {Segmentation}},
url = {https://ieeexplore.ieee.org/document/8954100},
}
