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7/14 Introduction to Geometric Deep Learning

Introduction to Group Actions

To discuss Geometric Deep Learning, we first need to lay down the basic notions of group actions.

Groups

Definition 1. A group is a pair \((G, \cdot)\) of

  1. a set \(G\), and
  2. a binary operation \(G\times G\xrightarrow{(a, b)\mapsto ab}G\),

such that the following conditions are satisfied:

  1. Associativity: For all \(a,b,c\in G\), we have \((ab)c=a(bc)\).
  2. Identity element: There exists \(1\in G\) such that for all \(a\in G\), we have \(1a=a1=a\). Note that this in particular implies that the identity element is unique.
  3. Inverses: For all \(a\in G\), there exists \(a^{-1}\in G\) such that \(aa^{-1}=a^{-1}a=1\). Note that this in particular implies that the inverse of \(a\in G\) is unique.

Example 2. Let \(n\) be a positive integer. Then the permutation group \(S_n\) is the group of bijections \([n]\xrightarrow\pi[n]\) of an \(n\)-element set \([n]=\{0,\dotsc,n-1\}\). The group structure is given by composition.

Example 3. Addition of integers equips \(\mathbf Z\) with a group structure. We denote this by \((\mathbf Z, +)\).

Group Actions

Definition 4. Let \(G\) be a group and let \(X\) be a set. Then a right action of \(G\) on \(X\) is a map \(X\times G\xrightarrow{(x, a)\mapsto xa}X\) such that the following condition is satisfied:

For \(a,b\in G\) and \(x\in X\), we have \((xa)b=x(ab)\).

In this case, we also call \(X\) a right \(G\)-set.

Example 5. For any group \(G\) and any set \(X\), the trivial action of \(G\) on \(X\) is the action \(X\times G\xrightarrow{(x, a)\mapsto x}X\).

Example 6. The right translation action of the group \(G\) on itself as a set is the action \(G\times G\to G\) given by multiplication.

Example 7. Let \(m,n\) be positive integers. Then the permutation group \(S_m\) acts on the set \(\mathbf R^{m\times n}\) of \(m\times n\) matrices on the right by permuting the rows.

Example 8. Let \(X\) denote the set

$$ (\mathbf R^n)^\mathbf Z={\mathbf x_i\in\mathbf R^n:i\in\mathbf Z} $$ of infinite sequences of \(n\)-dimensional vectors. Then the additive group \(G:=(\mathbf Z, +)\) of integers acts on \(X\) from the right via translation:

\[ (\mathbf x_i:i\in\mathbf Z).k := (\mathbf x_{i+k}:i\in\mathbf Z) \text{ for }(\mathbf x_i\in\mathbf Z)\text{ and }k\in\mathbf Z. \]

Equivariant and Invariant Maps, Quotients

Definition 9. Let \(G\) be a group, and let \(X\xrightarrow fY\) be a map between right \(G\)-sets. Then we say that \(f\) is \(G\)-equivariant, if for each \((x,a)\in X\times G\), we have \(f(xa)=f(x)a\). If \(Y\) is equipped with the trivial \(G\)-action, then we say that \(f\) is \(G\)-invariant.

Theorem 10. Let \(G\) be a group and \(X\) a right \(G\)-set. Then there exists a set \(X/G\) and an invariant map \(X\xrightarrow qX/G\), such that for all sets \(Y\) and invariant maps \(X\xrightarrow{f}Y\), there exists a unique map of sets \(X/G\xrightarrow gY\) such that we have \(f=g\circ q\).

Definition 11. We call the set \(X/G\) in Theorem 9 the quotient and \(q\) the quotient map or universal invariant map.

Remark 12. The property stated in Definition 10 is a so-called universal property. It implies uniqueness of quotients up to unique isomorphism: if \(X\xrightarrow hZ\) is another quotient, then there exists a unique isomorphism \(X/G\xrightarrow\phi Z\) such that \(h=\phi\circ q\).

Example 13. Let \(\mathbf R^{m\times n}\xrightarrow f\mathbf R^n\) be the map that takes the mean or the maximum of each column. Then it is invariant with respect to the action of \(S_m\) on \(\mathbf R^{m\times n}\) by permuting the rows. One can construct the quotient space \(\mathbf R^{m\times n}/S_m\) as the set of \(m\)-element sets of \(n\)-dimensional vectors.

Example 14. Let \(f\) denote the map that sends a sequence \((\mathbf x_i:i\in\mathbf Z)\) to the directed labeled graph \((V, E, W_V)\) where:

  1. The vertices are the integers: \(V=\mathbf Z\).
  2. The edges point 1 step to the right: \(E=\{(i, i+1):i\in\mathbf Z\}\).
  3. The node with index \(i\in\mathbf Z\) has the label \(\mathbf x_i\).

Then the map \(f\) is the quotient map with respect to the action given in Example 8.

The Objective of Geometric Deep Learning

Recall that we build models

\[ \mathscr X\xrightarrow f\mathscr Y, \]

where \(\mathscr X\) is the feature space, and \(\mathscr Y\) is the target space or the space of distributions of targets. What Geometric Deep Learning is concerned with is how can we make the model respect symmetries. That is, we suppose that \(\mathscr X\) has an action by a group \(G\), and we want \(f\) to be invariant / equivariant (if \(G\) acts on \(\mathscr Y\) also). For example:

  1. We can view text as a sequence of tokens, parts viewed as atomic, let them be characters or words (more on this later). For a first step, we disregard token order. Therefore, the feature space has a permutation action.

  2. When we're classifying images, then translating an image usually does not change its class. Therefore, the features have an action by translation. Shortly, we'll introduce Convolutional Neural Networks: models that respect translation.

Respecting symmetries brings huge benefits:

  1. Parameter sharing: If we can systematically exploit that multiple inputs can be viewed as being the same, then we need much less parameters in our models.
  2. Pseudo-augmented training: If the model is designed in a way that respects symmetries, then processing a training sample amounts to processing many equivalent samples with respect to symmetry.

A work-in-progress book and additional resources on Geometric Deep Learning are available at https://geometricdeeplearning.com/