Many scientific fields study data with an underlying structure that is non-Euclidean. Some examples include social networks in computational social sciences, sensor networks in communications, functional networks in brain imaging, regulatory networks in genetics, and meshed surfaces in computer graphics. In many applications, such geometric data are large and complex (in the case of social networks, on the scale of billions) and are natural targets for machine-learning techniques. In particular, we would like to use deep neural networks, which have recently proven to be powerful tools for a broad range of problems from computer vision, natural-language processing, and audio analysis. However, these tools have been most successful on data with an underlying Euclidean or grid-like structure and in cases where the invariances of these structures are built into networks used to model them.

Geometric Deep Learning: Going beyond Euclidean data

We propose a novel point signature based on the properties of the heat diffusion process on a shape. Our signature, called the Heat Kernel Signature (or HKS), is obtained by restricting the well-known heat kernel to the temporal domain. Remarkably we show that under certain mild assumptions, HKS captures all of the information contained in the heat kernel, and characterizes the shape up to isometry. This means that the restriction to the temporal domain, on the one hand, makes HKS much more concise and easily commensurable, while on the other hand, it preserves all of the information about the intrinsic geometry of the shape. In addition, HKS inherits many useful properties from the heat kernel, which means, in particular, that it is stable under perturbations of the shape. Our signature also provides a natural and efficiently computable multi-scale way to capture information about neighborhoods of a given point, which can be extremely useful in many applications. To demonstrate the practical relevance of our signature, we present several methods for non-rigid multi-scale matching based on the HKS and use it to detect repeated structure within the same shape and across a collection of shapes.

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A concise and provably informative multi-scale signature based on heat diffusion

Optimal transport (OT) theory can be informally described using the words of the French mathematician Gaspard Monge (1746-1818): A worker with a shovel in hand has to move a large pile of sand lying on a construction site. The goal of the worker is to erect with all that sand a target pile with a prescribed shape (for example, that of a giant sand castle). Naturally, the worker wishes to minimize her total effort, quantified for instance as the total distance or time spent carrying shovelfuls of sand. Mathematicians interested in OT cast that problem as that of comparing two probability distributions, two different piles of sand of the same volume. They consider all of the many possible ways to morph, transport or reshape the first pile into the second, and associate a "global" cost to every such transport, using the "local" consideration of how much it costs to move a grain of sand from one place to another. Recent years have witnessed the spread of OT in several fields, thanks to the emergence of approximate solvers that can scale to sizes and dimensions that are relevant to data sciences. Thanks to this newfound scalability, OT is being increasingly used to unlock various problems in imaging sciences (such as color or texture processing), computer vision and graphics (for shape manipulation) or machine learning (for regression, classification and density fitting). This short book reviews OT with a bias toward numerical methods and their applications in data sciences, and sheds lights on the theoretical properties of OT that make it particularly useful for some of these applications.

Computational Optimal Transport

Modeling tasks, such as surface deformation and editing, can be analyzed by observing the local behavior of the surface. We argue that defining a modeling operation by asking for rigidity of the local transformations is useful in various settings. Such formulation leads to a non-linear, yet conceptually simple energy formulation, which is to be minimized by the deformed surface under particular modeling constraints. We devise a simple iterative mesh editing scheme based on this principle, that leads to detail-preserving and intuitive deformations. Our algorithm is effective and notably easy to implement, making it attractive for practical modeling applications.

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As-rigid-as-possible surface modeling

The computer vision and pattern recognition communities have recently witnessed a surge of feature-based methods in object recognition and image retrieval applications. These methods allow representing images as collections of “visual words” and treat them using text search approaches following the “bag of features” paradigm. In this article, we explore analogous approaches in the 3D world applied to the problem of nonrigid shape retrieval in large databases. Using multiscale diffusion heat kernels as “geometric words,” we construct compact and informative shape descriptors by means of the “bag of features” approach. We also show that considering pairs of “geometric words” (“geometric expressions”) allows creating spatially sensitive bags of features with better discriminative power. Finally, adopting metric learning approaches, we show that shapes can be efficiently represented as binary codes. Our approach achieves state-of-the-art results on the SHREC 2010 large-scale shape retrieval benchmark.

Shape google: Geometric words and expressions for invariant shape retrieval

We present a discrete treatment of adapted framed curves, parallel transport, and holonomy, thus establishing the language for a discrete geometric model of thin flexible rods with arbitrary cross section and undeformed configuration. Our approach differs from existing simulation techniques in the graphics and mechanics literature both in the kinematic description---we represent the material frame by its angular deviation from the natural Bishop frame---as well as in the dynamical treatment---we treat the centerline as dynamic and the material frame as quasistatic. Additionally, we describe a manifold projection method for coupling rods to rigid-bodies and simultaneously enforcing rod inextensibility. The use of quasistatics and constraints provides an efficient treatment for stiff twisting and stretching modes; at the same time, we retain the dynamic bending of the centerline and accurately reproduce the coupling between bending and twisting modes. We validate the discrete rod model via quantitative buckling, stability, and coupled-mode experiments, and via qualitative knot-tying comparisons.

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Discrete elastic rods

We introduce the heat method for computing the geodesic distance to a specified subset (e.g., point or curve) of a given domain. The heat method is robust, efficient, and simple to implement since it is based on solving a pair of standard linear elliptic problems. The resulting systems can be prefactored once and subsequently solved in near-linear time. In practice, distance is updated an order of magnitude faster than with state-of-the-art methods, while maintaining a comparable level of accuracy. The method requires only standard differential operators and can hence be applied on a wide variety of domains (grids, triangle meshes, point clouds, etc.). We provide numerical evidence that the method converges to the exact distance in the limit of refinement; we also explore smoothed approximations of distance suitable for applications where greater regularity is required.

Geodesics in heat: A new approach to computing distance based on heat flow

Discrete Laplace operators are ubiquitous in applications spanning geometric modeling to simulation. For robustness and efficiency, many applications require discrete operators that retain key structural properties inherent to the continuous setting. Building on the smooth setting, we present a set of natural properties for discrete Laplace operators for triangular surface meshes. We prove an important theoretical limitation: discrete Laplacians cannot satisfy all natural properties; retroactively, this explains the diversity of existing discrete Laplace operators. Finally, we present a family of operators that includes and extends well-known and widely-used operators.

Discrete Laplace operators: no free lunch

We present a continuum-based discrete model for thin threads of viscous fluid by drawing upon the Rayleigh analogy to elastic rods, demonstrating canonical coiling, folding, and breakup in dynamic simulations. Our derivation emphasizes space-time symmetry, which sheds light on the role of time-parallel transport in eliminating---without approximation---all but an O(n) band of entries of the physical system's energy Hessian. The result is a fast, unified, implicit treatment of viscous threads and elastic rods that closely reproduces a variety of fascinating physical phenomena, including hysteretic transitions between coiling regimes, competition between surface tension and gravity, and the first numerical fluid-mechanical sewing machine. The novel implicit treatment also yields an order of magnitude speedup in our elastic rod dynamics.

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Discrete viscous threads

We introduce the heat method for computing the shortest geodesic distance to an arbitrary subset of a given domain. The heat method is robust, efficient, and simple to implement since it is based on solving a pair of standard linear elliptic problems. The resulting algorithm represents a significant breakthrough in the practical computation of distance on a wide variety of geometric domains, since these problems can be prefactored once and subsequently solved in linear time. In practice, distance can be updated via the heat method an order of magnitude faster than with state-of-the-art methods while maintaining a comparable level of accuracy. We demonstrate that the method converges to the exact geodesic distance in the limit of refinement; we also explore smoothed approximations of distance suitable for applications where differentiability is required.

Max Wardetzky

Papers

Discrete elastic rods

Geodesics in heat: A new approach to computing distance based on heat flow

Discrete Laplace operators: no free lunch

Discrete viscous threads

Geodesics in Heat