Viacheslav Borovitskiy (Hiring PhD Students)
@vabor112.bsky.social
Assistant Professor in Machine Learning @ University of Edinburgh.
PhD in Mathematics. Ex ETH Zurich.
See https://vab.im/vacancies/
PhD in Mathematics. Ex ETH Zurich.
See https://vab.im/vacancies/
Details: vab.im/vacancies/.
Viacheslav Borovitskiy
personal page
vab.im
October 24, 2025 at 11:35 AM
Details: vab.im/vacancies/.
If you are interested in Matérn kernels on more general complexes (cellular/higher dimension), check out arxiv.org/abs/2311.01198 by @miniapeur.bsky.social et al.
Gaussian Processes on Cellular Complexes
In recent years, there has been considerable interest in developing machine learning models on graphs to account for topological inductive biases. In particular, recent attention has been given to Gau...
arxiv.org
January 29, 2025 at 4:57 PM
If you are interested in Matérn kernels on more general complexes (cellular/higher dimension), check out arxiv.org/abs/2311.01198 by @miniapeur.bsky.social et al.
This makes them perform well on tasks ranging from modeling 𝒂𝒓𝒃𝒊𝒕𝒓𝒂𝒈𝒆-𝒇𝒓𝒆𝒆 𝒎𝒂𝒓𝒌𝒆𝒕𝒔 to 𝒑𝒊𝒑𝒆 𝒏𝒆𝒕𝒘𝒐𝒓𝒌𝒔 or 𝒐𝒄𝒆𝒂𝒏 𝒄𝒖𝒓𝒓𝒆𝒏𝒕𝒔 (arxiv.org/abs/2310.19450).
Hodge-Compositional Edge Gaussian Processes
We propose principled Gaussian processes (GPs) for modeling functions defined over the edge set of a simplicial 2-complex, a structure similar to a graph in which edges may form triangular faces. This...
arxiv.org
January 29, 2025 at 4:56 PM
This makes them perform well on tasks ranging from modeling 𝒂𝒓𝒃𝒊𝒕𝒓𝒂𝒈𝒆-𝒇𝒓𝒆𝒆 𝒎𝒂𝒓𝒌𝒆𝒕𝒔 to 𝒑𝒊𝒑𝒆 𝒏𝒆𝒕𝒘𝒐𝒓𝒌𝒔 or 𝒐𝒄𝒆𝒂𝒏 𝒄𝒖𝒓𝒓𝒆𝒏𝒕𝒔 (arxiv.org/abs/2310.19450).
Crucially, the resulting kernels support 𝒂𝒖𝒕𝒐𝒎𝒂𝒕𝒊𝒄 𝒓𝒆𝒍𝒆𝒗𝒂𝒏𝒄𝒆 𝒅𝒆𝒕𝒆𝒓𝒎𝒊𝒏𝒂𝒕𝒊𝒐𝒏 of the relative importance of the three Hodge decomposition parts.
January 29, 2025 at 4:56 PM
Crucially, the resulting kernels support 𝒂𝒖𝒕𝒐𝒎𝒂𝒕𝒊𝒄 𝒓𝒆𝒍𝒆𝒗𝒂𝒏𝒄𝒆 𝒅𝒆𝒕𝒆𝒓𝒎𝒊𝒏𝒂𝒕𝒊𝒐𝒏 of the relative importance of the three Hodge decomposition parts.
On top of these, you can define a 𝑯𝒐𝒅𝒈𝒆-𝒄𝒐𝒎𝒑𝒐𝒔𝒊𝒕𝒊𝒐𝒏𝒂𝒍 𝑴𝒂𝒕𝒆́𝒓𝒏 𝒌𝒆𝒓𝒏𝒆𝒍, which is a linear combination of the pure div, pure curl, and harmonic Matérn kernels, each possible with a different set of hyperparameters.
January 29, 2025 at 4:56 PM
On top of these, you can define a 𝑯𝒐𝒅𝒈𝒆-𝒄𝒐𝒎𝒑𝒐𝒔𝒊𝒕𝒊𝒐𝒏𝒂𝒍 𝑴𝒂𝒕𝒆́𝒓𝒏 𝒌𝒆𝒓𝒏𝒆𝒍, which is a linear combination of the pure div, pure curl, and harmonic Matérn kernels, each possible with a different set of hyperparameters.
This leads to the 𝑯𝒐𝒅𝒈𝒆 𝑫𝒆𝒄𝒐𝒎𝒑𝒐𝒔𝒊𝒕𝒊𝒐𝒏 which splits any edge flow into three parts: pure divergence (curl-free), pure curl (div-free), and harmonic (curl-free & div-free). This allows three different Matérn kernels , one for each part.
January 29, 2025 at 4:56 PM
This leads to the 𝑯𝒐𝒅𝒈𝒆 𝑫𝒆𝒄𝒐𝒎𝒑𝒐𝒔𝒊𝒕𝒊𝒐𝒏 which splits any edge flow into three parts: pure divergence (curl-free), pure curl (div-free), and harmonic (curl-free & div-free). This allows three different Matérn kernels , one for each part.
However, a simplicial 2-complex offers much more. Its structure allows characterizing key properties of edge flows using the discrete concepts of divergence (𝒅𝒊𝒗) and 𝒄𝒖𝒓𝒍, measuring how edge flows diverge at nodes and circulate along faces.
January 29, 2025 at 4:56 PM
However, a simplicial 2-complex offers much more. Its structure allows characterizing key properties of edge flows using the discrete concepts of divergence (𝒅𝒊𝒗) and 𝒄𝒖𝒓𝒍, measuring how edge flows diverge at nodes and circulate along faces.
Any simplicial 2-complex comes with a 𝑯𝒐𝒅𝒈𝒆 𝑳𝒂𝒑𝒍𝒂𝒄𝒊𝒂𝒏 matrix, which can be used to define Matérn kernels on its edge set in exactly as in the Matérn GPs on Graphs paper (arxiv.org/abs/2010.15538).
Matérn Gaussian Processes on Graphs
Gaussian processes are a versatile framework for learning unknown functions in a manner that permits one to utilize prior information about their properties. Although many different Gaussian process m...
arxiv.org
January 29, 2025 at 4:56 PM
Any simplicial 2-complex comes with a 𝑯𝒐𝒅𝒈𝒆 𝑳𝒂𝒑𝒍𝒂𝒄𝒊𝒂𝒏 matrix, which can be used to define Matérn kernels on its edge set in exactly as in the Matérn GPs on Graphs paper (arxiv.org/abs/2010.15538).
However, if you want to keep it simple and think about graphs rather than simplicial 2-complexes, the interface allows it: the library can define a reasonable set of triangles for you, for any graph you provide.
January 29, 2025 at 4:56 PM
However, if you want to keep it simple and think about graphs rather than simplicial 2-complexes, the interface allows it: the library can define a reasonable set of triangles for you, for any graph you provide.
First, a bit about theory. Mathematically, the kernels are defined on the edge set of a 𝒔𝒊𝒎𝒑𝒍𝒊𝒄𝒊𝒂𝒍 2-𝒄𝒐𝒎𝒑𝒍𝒆𝒙, i.e. a graph along with a set of triangular faces formed by some of its edges.
January 29, 2025 at 4:56 PM
First, a bit about theory. Mathematically, the kernels are defined on the edge set of a 𝒔𝒊𝒎𝒑𝒍𝒊𝒄𝒊𝒂𝒍 2-𝒄𝒐𝒎𝒑𝒍𝒆𝒙, i.e. a graph along with a set of triangular faces formed by some of its edges.
