Going back in time...

Going back in time...

Euclid~300 B.C.

Going back in time...

EuclidLobachevsky

BolyaiRiemann~300 B.C.

~1829

~1830

~1856

Going back in time...

What is the “one true geometry”?

EuclidLobachevsky

BolyaiRiemann~300 B.C.

~1829

~1830

~1856

Felix Klein and the “Erlangen Program” (1872)

Felix Klein and the “Erlangen Program” (1872)

Blueprint for unifying geometries
Lens of invariances and symmetries
Formalised in the language of group theory!

Impact of the Erlangen Program●

Strong impact on geometry; no longer hunting for the “one true geometry”○ Formalised by Élie Cartan in the 1920s●

Physics: Noether’s Theorem (all conservation laws derivable from symmetry!)○ Even enabled the classiﬁcation of elementary particles (irreducible representations)●Category Theory○ “can be regarded as a continuation of the Klein Erlangen Program, in the sense that ageometrical space with its group of transformations is generalized to a category with
its algebra of mappings”; – Eilenber, Lane (creators of CT)

Deep learning, circa 2020

Deep learning, circa 2020

What is the “one true architecture”?

Could GNNs be the answer?
●“If we squint hard enough”, (m)any NNs can be seen as message passing over a graph○ Further, most data we receive from nature is inherently graph-structured○ So, GNNs likely play a part in the “one true architecture” (motivating this course)●

But to formalise this, we need to understand GNNs beyond permutation equivariance!

...now it’s our turn to study geometry :)Michael Bronstein
Oxford / Twitter

Joan Bruna
NYU

Taco Cohen
Qualcomm

Petar Veličković

DeepMind / Cambridge

Geometric Deep Learning

GNNs Beyond Permutation Equivariance
Petar Veličković
Stanford University
CS224W
30 November 2021

1

Learning inhigh dimensions
is hard

The Curse of Dimensionality

Even “simple” target functions (e.g. 1-Lipschitz) require exponential samples in nb. dimensions.

Low-dimensional projections don’t necessarily help!

Shallow MLPs can lose a lot of the ﬁdelity of their inputs.
What can we do?

2Symmetries,
Groups andInvariancesGeometry to the rescue!
●We can inject further assumptions about the geometry of through inductive biases○ Restrict the functions in our hypothesis space to ones that respect the geometry.
○ This can make the high-dimensional problem more tractable!

●

Some popular examples:○ Image data should be processed independently of shifts○ Spherical data should be processed independently of rotations○ Graph data should be processed independently of isomorphism●

We will now attempt to formalise this!

Key elements!
●

We assume data lives on a domain, 𝛀○ e.g. for images, u ∈ 𝛀 are pixels; for graphs they are nodes●

We assume a feature space, C, to be stored in elements of a domain○ For our purposes, C = ℝk●

We can then deﬁne featurised domains using a space of feature functions X(𝛀, C)○ x ∈ X(𝛀, C) is a function s.t. x(u) ∈ C gives features of element u ∈ 𝛀○ For discrete environments we can think of X as a feature matrix (X ∈ ℝ|𝛀| x k)

Principle 1: Symmetry groups●

Symmetry is a transformation that leaves an object invariant (i.e. unchanged)○ Hence they must be composable, invertible, contain identity…

●

In fact, they can be reasoned about using a very elegant mathematical object: the group●

Elements of these groups are domain transformations (e.g. some functions 𝔤 : 𝛀 → 𝛀).

Principle 1: Symmetry groups●

Symmetry is a transformation that leaves an object invariant (i.e. unchanged)○ Hence they must be composable, invertible, contain identity…

●

In fact, they can be reasoned about using a very elegant mathematical object: the group

Principle 1: Symmetry groups●

Symmetry is a transformation that leaves an object invariant (i.e. unchanged)○ Hence they must be composable, invertible, contain identity…

●

In fact, they can be reasoned about using a very elegant mathematical object: the group

∈𝛀
∈𝔊

∈𝛀

Group actions●

We are interested in how these groups affect data
○ Group action
for a group element g, and a domain element u○ E.g. translating or rotating an image, or permuting a set●

We will be interested in linear group actions:●This also allows us to represent group actions using linear algebra;

Invariance and equivariance●

We can largely simplify high-dimensional learning by exploiting the symmetries in 𝔊!

e.g. image classiﬁcation: output class (likely?) won’t depend on image shifts

e.g. image segmentation: segmentation mask must follow any shifts in the input

Principle 2: Scale separation●

Want signal to be stable under slight deformations of thedomain●We derive: highly beneﬁcial to compose local∈𝔊

operations to model larger-scale ones○○

local ops won’t globally propagate errors
e.g. CNNs with 3 x 3 kernels, but very deep●

Accordingly, we would like to support locality in our layers!

●

cf. Fourier Transform vs. Wavelets∉𝔊

3

The Blueprintof Geometric
Deep Learning

The key “building blocks” of Geometric Deep Learning

The key “building blocks” of Geometric Deep Learning
Equivariant local layers

Invariant “tail” (if necessary)

The key “building blocks” of Geometric Deep Learning

Activation function(necessary for deep learning! :) )The key “building blocks” of Geometric Deep Learning

Coarsening layer(Not covered here in detail, butfollows from scale separation!)

All you need to build the architectures that are all you need :)All the fan-favourites are easily derivable...with many (potentially) unexpected and useful extras! :)4

The “5G” ofGeometric
Deep Learning

The “5G” of Geometric Deep Learning
We will use the remainder of this lecture to study a few interesting instances of this blueprint

The “5G” of Geometric Deep Learning
We will use the remainder of this lecture to study a few interesting instances of this blueprint

Throughout CS224W, the focus was primarily on the domain of graphs.

Our strategy for the rest of the lecture●

We start by seeing how GNNs ﬁt in this paradigm○ This will involve re-deriving / re-introducing some concepts you’ve seen in the course●

Then, we will see how we can use the blueprint to expand GNNs into other domains○ Also, it will give us an insight into “the world beyond”○ GNNs beyond permutation equivariance!

●

Our discussion will span many architectures you (may) know of :)○ Deep Sets
○ Transformers○ CNNs○ Spherical CNNs
○ Mesh CNNs

Architectures of interest

5

Geometric DL
Perspective on
Graph Neural
Networks

Learning on sets: Setup●

For now, assume the graph has no edges (e.g. our domain is just the set of nodes, V).

●

Let xi ∈ ℝk be the features of node i.

●

We can stack them into a node feature matrix of shape n x k:●That is, the ith row of X corresponds to xi●

Note that, by doing so, we have speciﬁed a node ordering!
○ We would like the result of any neural networks to not depend on this.

What do we want?

What do we want?

What do we want?

Permutations and permutation matrices●

It will be useful to think about the operations that change the node order○ Such operations are known as permutations (there are n! of them)○ e.g. a permutation (2, 4, 1, 3) means y1 ← x2, y2 ← x4, y3 ← x1, y4 ← x3.

●

To stay within linear algebra, each permutation deﬁnes an n x n matrix (group action!)○ Such matrices are called permutation matrices○ They have exactly one 1 in every row and column, and zeros everywhere else
○ Their effect when left-multiplied is to permute the rows of X, like so:Permutation invariance●

We want to design functions f(X) over sets that will not depend on the order●

Equivalently, applying a permutation matrix shouldn’t modify the result!

●

We arrive at a useful notion of permutation invariance. We say that f(X) is permutation
invariant if, for all permutation matrices P:●

One very generic form is the Deep Sets model (Zaheer et al., NeurIPS’17):
where 𝜓 and 𝜙 are (learnable) functions, e.g. MLPs.
○

The sum aggregation is critical! (other choices possible, e.g. max or avg)

Permutation equivariance●

Permutation-invariant models are a good way to obtain set-level outputs●

What if we would like answers at the node level?
○ We want to still be able to identify node outputs, which a permutation-invariant
aggregator would destroy!

●

We may instead seek functions that don’t change the node order○ i.e. if we permute the nodes, it doesn’t matter if we do it before or after the function!

●

Accordingly, we say that f(X) is permutation equivariant if, for all permutation matrices P:

Learning on graphs●

Now we augment the set of nodes with edges between them.
○ That is, we consider general E ⊆ V x V.

●

We can represent these edges with an adjacency matrix, A, such that:●

Further additions (e.g. edge features) are possible but ignored for simplicity.

●

Our main desiderata (permutation {in,equi}variance) still hold!

What’s changed?

What’s changed?

Permutation invariance and equivariance on graphs●

The main difference: node permutations now also accordingly act on the edges●

We need to appropriately permute both rows and columns of A
○ When applying a permutation matrix P, this amounts to PAPT●

We arrive at updated deﬁnitions of suitable functions f(X, A) over graphs:

Invariance:
Equivariance:Locality on graphs: neighbourhoods●

Recall: it is also highly beneﬁcial to design geometrically stable (local) equivariant layers●

Graphs give us a context for locality: a node’s neighbourhood○ For a node i, its (1-hop) neighbourhood is commonly deﬁned as follows:

N.B. we do not explicitly consider directed edges, and often we assume i ∈ Ni●

Accordingly, we can extract the multiset of features in the neighbourhoodand deﬁne a local function, g, as operating over this multiset: g(xi, XNi).

A recipe for graph neural networks●

Now we can construct permutation equivariant functions, f(X, A), by appropriately applying
the local function, g, over all neighbourhoods:●

To ensure equivariance, we need g to not depend on the order of the vertices in XNi○ Hence, g should be permutation invariant!

A recipe for graph neural networks, visualised

How to use GNNs?

How to use GNNs?

How to use GNNs?

How to use GNNs?

How to use GNNs?

The three “flavours” of GNN layers

e.g. GraphSAGE, GCN, SGC

e.g. MoNet, GAT, GATv2

e.g. IN, MPNN, GraphNet

Architectures of interest

6Permutationequivariant
NNs are GNNs

What to do when there is no graph?
●

Before we go beyond permutation equivariance, a brief note on how we can use the GDL
blueprint to immediately relate permutation equivariant NNs as special cases of GNNs●

So far, we’ve assumed something (seemingly) very innocent: that the graph is given to us!

What to do when there is no graph?
●

Before we go beyond permutation equivariance, a brief note on how we can use the GDL
blueprint to immediately relate permutation equivariant NNs as special cases of GNNs●

So far, we’ve assumed something (seemingly) very innocent: that the graph is given to us!

●

In practice, the given graph may often be suboptimal for the task we’re solving○ For various connectivity querying on graphs, maintaining the right set of edges canmake a difference between linear-time and (amortised) constant-time complexity!

●

Taken to the extreme: what to do when there is no graph?
○ Assume we’re given a node feature matrix, but no adjacency●

Let’s brieﬂy cover two “special case” solutions…

Option 1: Assume no edges
Deep Sets (Zaheer et al., NeurIPS’17)

No edges (set)Option 2: Assume all edges
Interaction Nets (Battaglia et al., NeurIPS’16)
Relational Nets (Santoro et al., NeurIPS’17)
Let the GNN decide which edges matter!
Using conv-GNNs no longer makes sense.
If we use attentional GNNs we recover:

Does this look familiar?
All edges (fully-connected graph)A note on Transformers
Transformers are Graph Neural Networks!
●●

Fully-connected graphAttentional ﬂavour

The sequential structural information is injected throughthe positional embeddings. Dropping them yields afully-connected GAT model.
Attention can be seen as inferring soft adjacency.
See Joshi (The Gradient; 2020).

Remark: the “truth” likely lies in between●

Empty graph ignores a potential wealth of information●

Full graph can be hard to scale (quadratic complexity, large neighbourhoods)●

Ideally, we want to infer the adjacency matrix A, then use it as edges for a GNN!
○ Commonly termed “latent graph inference”.
○ Choosing edges is a discrete decision -- inherently hard to backpropagate!

●

Out of scope for this lecture. Some interesting pointers include:○ Neural Relational Inference (Kipf, Fetaya et al., ICML’18)○ Dynamic Graph CNN (Wang et al., ACM TOG’18)○ Differentiable Graph Module (Kazi et al., MICCAI’20)○ Pointer Graph Networks (Veličković et al., NeurIPS’20)

Architectures of interest

7

The Fourier
connection:
CNNs andSpectral GNNs

Look to the Fourier transform●

The convolution theorem deﬁnes a very attractive identity:“convolution in the time domain is multiplication in the frequency domain”●

This could give us a ‘detour’ to deﬁning convolutions on graphs○ Pointwise multiplication is easy!
○ But what are the ‘domains’ in this case?

●We will ﬁrst see how graphs arise in discrete sequences (grids).

What’s changed?
●

Grids (e.g. images, text, speech) can still be seen as a graph○ Pixels connected to immediate neighbours in the grid●

What would a graph neural network look like in this case?

●

Simplifying assumption: cyclical grids●

Now every node has identical structure and degree○ Allows us to more directly deﬁne a convolution over a neighbourhood●

Also, coordinates matter: we get a much stronger translation equivariance requirement!

Rethinking the convolution on sequences*for easier handling of boundary conditions●

We can imagine a sequence as a cyclical grid graph, and a convolution over it:●

NB this deﬁnes a circulant matrix C([b, c, 0, 0, …, 0, a]) s.t. H = f(X) = CXProperties of circulants, and their eigenvectors●

Circulant matrices commute!
○ That is, C(v)C(w) = C(w)C(v), for any parameter vectors v, w.

●

Matrices that commute are jointly diagonalisable.
○ That is, the eigenvectors of one are eigenvectors of all of them!

●

Conveniently, the eigenvectors of circulants are the discrete Fourier basis●

This can be easily computed by studying C([0, 1, 0, 0, 0, …]), which is the shift matrix.

The DFT and the convolution theorem●

If we stack these Fourier basis vectors into a matrix:○ We recover the discrete Fourier transform (DFT), as multiplication by 𝚽*. (conjugate transpose)●

We can now eigendecompose any circulant as C(𝜃) = 𝚽𝚲𝚽*○ Where 𝚲 is a diagonal matrix of its eigenvalues,●

The convolution theorem naturally follows:●

Now, as long as we know 𝚽, we can express our convolution usingrather thanWhat we have covered so far

Key idea: we don’t need to know the circulant if we know its eigenvalues!

Credits to Michael BronsteinWhat about graphs?
●

On graphs, convolutions of interest need to be more generic than circulants.
○ But we can still use the concept of joint eigenbases!
○ If we know a “graph Fourier basis”, 𝚽, we can only focus on learning the eigenvalues.

●

For grids, we wanted our convolutions to commute with shifts.
○ We can think of the shift matrix as an adjacency matrix of the grid
○ This generalises to non-grids!
○ For the grid convolution on n nodes, 𝚽 was always the same (n-way DFT).
○ Now every graph will have its own 𝚽!

●

Want our convolution to commute with A, but we cannot always eigendecompose A!

●

Instead, use the graph Laplacian matrix, L = D - A, where D is the degree matrix.
○ Captures all adjacency properties in mathematically convenient way!

Example LaplacianGraph Fourier Transform●

Assuming undirected graphs, L is:○ Symmetric (LT = L)
○ Positive semi-deﬁnite (xTLx ≥ 0 for all x ∈ ℝ|V|)○ This means we will be able to eigendecompose it!

●

This allows us to re-express L = 𝚽𝚲𝚽*, as before.
○ Changing the eigenvalues in 𝚲 expresses any operation that commutes with L.
○ Commonly referred to as the graph Fourier transform (Bruna et al., ICLR’14)●

Now, to convolve with some feature matrix X, do as follows (the diagonal can be learnable):Spectral GNNs in practice●

However, directly learning the eigenvalues is typically inappropriate:○ Not localised, doesn’t generalise to other graphs, computationally expensive, etc.

●

Instead, a common solution is to make the eigenvalues related to 𝚲, the eigenvalues of L
○ Commonly by a degree-k polynomial function, pk○ Yielding○ Popular choices include:■ Cubic splines (Bruna et al., ICLR’14)■ Chebyshev polynomials (Defferrard et al., NeurIPS’16)■ Cayley polynomials (Levie et al., Trans. Sig. Proc.’18)●

NB by using a polynomial in L, we have deﬁned a conv-GNN!
○ With coefﬁcients deﬁned by cij = (pk(L))ij○ Most efﬁcient spectral approaches “spatialise” themselves in similar ways
○ The “spatial-spectral” divide is often not really a divide!

The Transformer positional encodings and beyond●

Lastly, another look at Transformers.

●

Transformers signal that the input is a sequence of words by using positional embeddings○ Sines/cosines sampled depending on position●

Very similar to the DFT eigenvectors!

●

Positional embeddings could hence be interpreted as eigenvectors of the grid graph○ Which is the only assumed ‘underlying’ connectivity between the words●

We can use this idea to run Transformers over general graph structures!
○ Just feed some eigenvectors of the graph Laplacian (columns of 𝚽)○ See the Graph Transformer from Dwivedi & Bresson (2021)○ Rapidly emerging area of research!

Architectures of interest

8

The Group
connection:
Spherical CNNs

Convolutions in the Euclidean domain*NB all our ﬁndings will be applicable to discretisations●

Now consider the continuous Euclidean domain 𝛀 = ℝ, where convolution takes the form:●

The output of the grid convolution is another function on the grid
○ But this is not the case for every domain!

●

Now we can see how to generalise the convolution to more general 𝔊

Group convolutions●

We can deﬁne inner products on general domains 𝛀:

GridsGroups●

NB: the group convolution is, generally, a function over elements of 𝔊!!!
○ It was also a real-input function under the translation group...
○ ...because the group itself is also 𝔊 = ℝ (scalar shifts!!!)Example: Spherical convolution●

Consider signal deﬁned on a sphere, 𝛀 = S2
○ Very relevant, e.g. for earth maps, astrophysics, …

●

We want to be rotation equivariant○ This means using the rotation group 𝔊 = SO(3)○ Image CNNs cannot support this!

Example: Spherical convolution●

Consider signal deﬁned on a sphere, 𝛀 = S2
○ Very relevant, e.g. for earth maps, astrophysics, …

●

We want to be rotation equivariant○ This means using the rotation group 𝔊 = SO(3)●We can represent the points on a sphere with 3-dim unit vectors, u○ The group action (rotation) transforms it with a 3 x 3 orthogonal matrix, R●

We recover our spherical convolution:

Example: SO(3) convolution●

The output of a spherical convolution is a function over all 3D rotations in SO(3)○ So, to “stack more layers”, we need to deﬁne a convolution over 𝛀 = 𝔊 = SO(3)●Luckily, our blueprint still works: we can deﬁne a 𝔊-action over elements of 𝔊 by function
composition:●

The corresponding group action can be deﬁned as follows:●

Consequently, we can build 𝔊-equivariant layers over 𝔊!

●

Start with spherical convolution at the input layer, then stack SO(3)-convolutions!

Example: SO(3) convolution, cont’d●SO(3) group actions deﬁned by 3x3 rotation matrices R●

Hence, group action○ for some two rotation matrices R and Q.

●

Recall the expression for the group convolution:●

Putting it all together, we obtain the following two-layer convolution over spheres:●This forms the essence of Spherical CNNs (Cohen et al., ICLR’18)

is expressible as x(R-1Q)

Relationship to GNNs●

The spherical CNN can still be related to GNNs
○ In practice, the sphere is discretised and the ﬁlter 𝜓 is local○ The non-zero elements of 𝜓 are the neighbourhood of point u○ The integration is akin to message passing! (as many points will have zero product)●

Further, the 𝔊-convolution described here works over any global symmetry group 𝔊

Caveat 1: Tractability●

The 𝔊-convolution described here works over any symmetry group 𝔊●

But it is only tractable for very small groups○ e.g. SO(3) was describable with a 3x3 orthogonal matrices●

It may be tempting to apply this idea to graphs○ But the permutation group Σn has n! entries to maintain●

This hints at existence of “graph convolutions” not captured by our ‘ﬂavours’○ (though it likely captures the most tractable ones :) )●Some of these convolutions may tradeoff expressivity for stability / complexity○ Suggests a “way out” of the Weisfeiler-Lehman expressivity limit…

Caveat 2: Homogeneity●

Both the spherical and the circular grid domains are homogeneous●

For any two points u, v ∈ 𝛀, there exists some 𝔤 ∈ 𝔊 s.t. 𝔤.u = v●

In a sense, “all points look the same”○ This does not hold for graphs!

●

This allowed ‘sliding’ ﬁlters across 𝛀 to build convolutions●

What can we do in the more general case?

Architectures of interest

9

Geometric
Graphs,Geodesics andGaugesGeometric graphs●

Thus far, we have assumed our graphs to be a discrete, unordered, set of nodes and edges●

In many cases, this is not the entire story!
○ The graph, in fact, may often be endowed with some spatial geometry○ It will be useful for us to exploit this geometry!

●

Molecules are a classical case (with their three-dimensional conformers)

Learning over geometric graphs●

Simpliﬁed setup: nodes endowed with features, fu and coordinates xu ∈ ℝ3●

An equivariant message passing layer transforms them separately○ Yielding updated features f’u and coordinates x’u●

We can now express a group of symmetries 𝔊 we would like to be resistant to○ In the case of molecules, a standard group is the Euclidean group, E(3)○ Roto-translations and reﬂections●

For any 3D orthogonal matrix R and translation vector b, we deﬁne a group action 𝔤 ∈ E(3):and applying them to coordinates typically should not affect how features are processed!

E(n)-equivariant GNNs●

As for permutation equivariance, there exist many GNNs that obey E(n)-equivariance●

One elegant solution was exposed by Satorras et al. (ICML’21):●

The actions of E(n) do not change distance between nodes (they are isometric)○ Hence if xu ← Rxu + b…
○ f’u does not change, and x’u ← Rx’u + b, as expected!

●

Hence, this layer is E(n)-equivariant over scalar features fu

Vector-structured features●

However, what if some of the node features (fu) depend on the geometry?
○ e.g. they could be vector forces○ Rotating the molecule should rotate these vectors too!

●

The model on the previous slide does not take this into account!

Vector-structured features●

However, what if some of the node features (fu) depend on the geometry?
○ e.g. they could be vector forces○ Rotating the molecule should rotate these vectors too!

●

The model on the previous slide does not take this into account!

●

Satorras et al. do propose a variant of their model that works with vectors:●

But the issue will keep re-appearing as we “tensor up” our features!
○ Is there a way to talk about the general “set of solutions” for E(n)-equivariance?

Towards a general solution: Tensor Field Networks●

We actually can factorise the group action of SE(3) on inputs of any tensor order l!

where Dl are the Wigner D-matrices, and Q is a 𝔤-speciﬁc change-of-basis matrix.
●

This can be exploited to write a generic SE(3)-equivariant model (the TFN)

where the matrices WJ are precomputed based on the above parametrisation

SE(3)-Transformers (Fuchs, Worrall, et al., NeurIPS’20)
Restrict TFNs to only act over neighbourhood Ni. Also adds Transformer attention.

Gives all possible equivariant (attentional) GNNs, but we still need to pre-compute WJ

SE(3)-Transformers (Fuchs, Worrall, et al., NeurIPS’20)

Manifolds●

More generally, we can deﬁne manifold domains○ Highly relevant e.g. for computer graphics,
protein design, fMRI processing●

In fact, manifold-oriented models end up not at all
dissimilar to equivariant message passing!

●

Detailed exposition and extended theory aredeferred to the draft bookManifolds●

More generally, we can deﬁne manifold domains○ Highly relevant e.g. for computer graphics,
protein design, fMRI processing●

In fact, manifold-oriented models end up not at all
dissimilar to equivariant message passing!

●

Key concept: parallel transport, which allows us
to transport tangent vectors along a curve topreserve direction○ NB. It is path-dependent (e.g. BC ≠ ABC)

Manifolds●

More generally, we can deﬁne manifold domains○ Highly relevant e.g. for computer graphics,
protein design, fMRI processing●

In fact, manifold-oriented models end up not at all
dissimilar to equivariant message passing!

●

Key concept: parallel transport, which allows us
to transport tangent vectors along a curve topreserve direction○ NB. It is path-dependent●

Now we can deﬁne (with some caveats) “sliding” aﬁlter along more general surfaces

Manifolds●

More generally, we can deﬁne manifold domains○ Highly relevant e.g. for computer graphics,
protein design, fMRI processing●

In fact, manifold-oriented models end up not at all
dissimilar to equivariant message passing!

●

Key concept: parallel transport, which allows us
to transport tangent vectors along a curve topreserve direction○ NB. It is path-dependent●

Now we can deﬁne (with some caveats) “sliding” aﬁlter along more general surfaces

Manifolds●

More generally, we can deﬁne manifold domains○ Highly relevant e.g. for computer graphics,
protein design, fMRI processing●

In fact, manifold-oriented models end up not at all
dissimilar to equivariant message passing!

●

Key concept: parallel transport, which allows us
to transport tangent vectors along a curve topreserve direction○ NB. It is path-dependent●

Now we can deﬁne (with some caveats) “sliding” aﬁlter along more general surfaces

Some architectures recovered from blueprint●

Geodesic CNN (Masci et al., CVPR’15)●

Gauge-equivariant Mesh CNN (de Haan et al., ICLR’21)

Some architectures recovered from blueprint●

Geodesic CNN (Masci et al., CVPR’15)●

Gauge-equivariant Mesh CNN (de Haan et al., ICLR’21)○ GEM-CNNs feel a lot like message passing! (with precomputed transport matrices)Architectures of interest

A

Furtherresourcesgeometricdeeplearning.com

156-page proto book on arXiv

African Master’s in Machine Intelligence course (12h)Thank you!
Questions / Feedback?
petarv@deepmind.com | https://petar-v.com

With many thanks to Joan Bruna, Michael Bronstein and Taco Cohen