Neural Explicit Representations

• Niloy Mitra (University College London)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal We start by exploring how neural networks can explicitly parametrize surfaces while handling surfaces with different genus mapping 2D domains to complex 3D shapes. We will review how these overfitted representations provide direct access to the first and second funda¬mental forms, ultimately enabling the construction of a continuous Laplace neural/shape operator.

Introduction to Geometric Deep Learning for Surfaces

• Emery Pierson (Ecole Polytechnique)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal The course will be divided in two parts: introduction to surface representations (point cloud, meshes) and discretizations of simple quantities. The second part will introduce deep learn-ing (MLP, CNNs, transformers) and the challenges associated to geometric deep learning on surfaces. It is possible that the course will finish during the second lecture.

Fast Forward Session Room: Endenicher Allee 60/1-016 - Lipschitzsaal

Fast Forward Session Room: Endenicher Allee 60/1-016 - Lipschitzsaal

Recognizing Convex Structure in Geometry Problems

• Justin Solomon (MIT)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal We motivate our discussion by developing intuition for the space of convex optimization problems, motivated by function smoothing, shortest path computation, and other problems in geometry. We will see that a rich class of problems can be understood through the lens of convex optimization; moreover, even when a problem is not obviously convex, a series of strategies can be used to derive a convex relaxation whose solution may approximate the solution to or even solve the original problem.

Neural Implicit Representations

• Niloy Mitra (University College London)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal We shift focus to implicit neural fields and their roots in classical geometry, highlighting why they are so popular for high-fidelity shape generation and 4D animation. We will discuss why extracting a clean, usable surface from these implicit mathematical volumes remains a non-trivial computational challenge, especially for modern learning setups.

Geometric Deep Learning for Point Clouds

• Emery Pierson (Ecole Polytechnique)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal The course will propose to review the most popular approaches for learning on point clouds: PointNet and variants, and recent transformers.

Lecture/Lab

• Niloy Mitra (University College London)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal We will numerically compute the first/second fundamental forms on trained neural surfaces. Participants will be able to try their own neural operators and gain hands-on experience performing differential operations within neural frameworks.

Open Problem Session Room: Endenicher Allee 60/1-016 - Lipschitzsaal

Curves in Probabilty Spaces

• Gabriele Steidl (TU Berlin)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal The course deals with absolutely continuous curves in Wasserstein spaces, their continuity equation and flow equation and special curves induced by couplings from optimal transport.

Solving Surface PDEs Using Neural Representations

• Niloy Mitra (University College London)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal We conclude by bridging the gap between geometry and physics by demonstrating how to solve partial differential equations (PDEs) directly on neural manifolds. We will show early efforts on mesh-free methods that can simulate complex physical phenomena without the traditional challenges of mesh discretization.

Discretization and Convex Optimization on Meshes and Graphs

• Justin Solomon (MIT)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal In this lecture, we will see how convex optimization can be applied to problems in geometry processing, whose unknowns are typically associated to elements of a point cloud, graph, or mesh. We will show that geodesic computation, optimal transport, edge-preserving smooth-ing via total variation, and skinning weights computation reduce to finite-dimensional convex problems that can be tackled with standard solvers.

Flow Matching

• Gabriele Steidl (TU Berlin)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal We introduce the flow matching for generative modelling and show the relation to score based diffusion.

Geometric Deep Learning for Surfaces

• Emery Pierson (Ecole Polytechnique)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal The course will present the main approaches for learning approaches: early failures, Diffu-sionNet, Jacobian fields.

Convex Relaxations for Geometric Reasoning

• Justin Solomon (MIT)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal Next, we will study how convex relaxation techniques lead to tractable formulations of chal-lenging computational geometry problems. We will start with early examples of linear pro-gram relaxations for consistent segmentation and conclude with modern research using semidefinite and sum-of-squares relaxations to tackle particularly challenging problems in computer-aided design and surface matching. We will see that convex relaxations can be unreasonably effective in geometry, motivating open questions regarding the tightness of typical relaxations in this domain.

Generative Modelling

• Gabriele Steidl (TU Berlin)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal In this lab lecture, you can try our generative modelling programs and investigate the role of different latent spaces.

Geometrically Consistent 3D Shape Matching Room: Endenicher Allee 60/1-016 - Lipschitzsaal The aim of 3D shape matching is to establish correspondences between semantically similar regions across given surfaces. A desirable property of resulting matchings is geometric consistency, which means that correspondences preserve the neighbourhood relation between shape elements. Yet, in practice, geometric consistency is often overlooked, or only achieved under severely limiting assumptions (e.g. a good initialisation). This lecture will take on a discrete view and cover graph-based formalisms for geometrically consistent shape matching, including foundations on general assignment problems, product graph formalisms for 1D, 2D and 3D matching, as well as recent developments towards globally optimal 3D shape matching with geometric consistency.

Discrete Flow Matching

• Gabriele Steidl (TU Berlin)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal Finally, we will deal with discrete flow matching which plays a role when sampling from discrete distributions and can be used e.g. in large language models.

Open Challenges

• Emery Pierson (Ecole Polytechnique)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal We conclude by a summary of the course: what are the ”solved challenges”, and what is remaining - ”open challenges”, namely: learning high freqency information (details), large scale learning (linked to transformers, also multimodality), transformers for meshes (how to tokenize? how to adapt attention to surfaces?), generative modeling (how to generate a surface/mesh?), other representations (e.g. CAD).

Infinite-Dimensional Convex Models and Modern Parameterizations

• Justin Solomon (MIT)

Room: Endenicher Allee 60/1-016 - Lipschitzsaal We will conclude our discussion of convex models by considering optimization problems for unknown functions, kernels, measures, currents, and other infinite-dimensional objects common in statistics, calculus of variations, measure-valued optimization, and PDE.