Geometric and topological methods in data analysis, with applications in biology and medecine

Description

Modeling in biology and medicine raises challenging questions for objects spanning multiple scales (from molecules to populations).

The goal of this course is twofold. First, to develop fundamental geometric, topological and machine learning algorithms, which are well understood in terms of performances and guarantees. Second, to use these methods to get insights on complex questions in biology and medicine, either at the simulation stage, or the post-processing / data analysis stage. Application domains include protein structure and function, gene expression, and the analysis of spatial transcriptomics data.

For any questions or concerns please contact Jean-Daniel Boissonnat at Jean-Daniel.Boissonnat[at]inria.fr, Mathieu Carrière at mathieu.carriere[at]inria.fr, or Frédéric Cazals at frederic.cazals[at]inria.fr

Outline

Class 1

Dimensionality reduction methods and complex non-linear motions

Dimensionality reduction methods aim at embedding high-dimensional data into lower-dimensional spaces, while preserving specific properties (pairwise distance, data spread, etc). This lecture will overview selected DR techniques, and apply them to study molecular motions.

• PCA, MDS, diffusion maps
• Application to collective coordinates and the parameterization of molecular motions

Class 2

High-dimensional volumes and densities of states in statistical physics

Statistical physics commands to compute observables / macroscopic properties of molecules using averages over selected so-called ensembles. This lecture will develop algorithms to compute high-dimensional volumes, and the random walks used therein will be used to sample molecular conformations.

• Monte Carlo Markov Chain methods and polytope volume calculations
• Application to sample molecular conformations

Class 3

ToMATo for colocalizing cell types

Slides, Tutorial, Practical Session, Solution of Practical Session

Class 4

Rips persistence for marker gene correlations

Slides, Practical Session, Solution of Practical Session

Class 6

Statistical tests and application to molecular simulation

Molecular simulations explore complex high-dimensional spaces, and two critical questions are to (i) compare ensembles of conformations discovered, and (ii) assess the convergence of a simulation. This course will develop fundamental techniques for both questions.

• High-dimensional two-sample tests
• Application to the comparison of molecular ensembles and the convergence of simulations

Class 5

Triangulation of point clouds

This course will introduce fundamental models in computational geometry and topology that are relevant to represent complex shapes.

• Simplicial complexes and filtrations
• Delaunay triangulations and Voronoi diagrams, randomized algorithms
• Molecules and alpha-shapes

Slides.

Class 8

Meshing non linear manifolds

Constructing faithful discrete approximations of complex shapes is a preprocessing in many applications such as visualization or numerical simulation. The case of surfaces in 3-space is the most important in medical applications but the study of their higher dimensional analogue is of utmost importance when considering dynamical systems. Crucial aspects are here topological and geometric correctness and complexity issues.

• Grids and triangulations to mesh surfaces
• Coxeter triangulations and meshes of higher dimensional manifolds

Slides.

Class SMS

Sampling and Meshing Submanifolds

Summary

Exercises

Complexity Analysis of the Marching Tetrahedron Algorithm.

Slides

Bibliography

Geometric and Topological Inference, ch. 7-8

Class 7

Multi-persistence for immune cell arrangements

Slides, Tutorial

Class TML:A

Topological Machine Learning (II): Guiding ML models

Location

Classes will take place at Sophia Antipolis, M2 room, Campus des Lucioles, 1645 route des Lucioles, Biot 06410.

Validation

This course is validated with data science projects.