Geometric Deep Learning HT26, Mathematical background
Flashcards
Group representations
Suppose we have a group $G$ acting on a set $\Omega$, and that $\Omega$ comes equipped with a space of signals $\mathcal X(\Omega)$. @Describe how this yields an action of $G$ on the space $\mathcal X(\Omega)$.
(this is required so that the property $(g \cdot (h \cdot x))(u) = ((gh) \cdot x) (u)$ holds).
@Define a representation of a group $G$ over a field $k$.
A homomorphism
\[\rho : G \to \text{GL}(V)\]where $V$ is a $k$-vector space and $\text{GL}(V)$ is the group of invertible linear maps $V \to V$.
A representation of a group $G$ over a field $k$ is a homomorphism
\[\rho : G \to \text{GL}(V)\]
where $V$ is a $k$-vector space and $\text{GL}(V)$ is the group of invertible linear maps $V \to V$.
@Define what is meant by the degree of the representation.
A representation of a group $G$ over a field $k$ is a homomorphism
\[\rho : G \to \text{GL}(V)\]
where $V$ is a $k$-vector space and $\text{GL}(V)$ is the group of invertible linear maps $V \to V$.
Suppose further that $V$ is finite dimensional and we choose a basis for $V$. How can we then interpret this representation?
It turns group elements into a matrix multiplication, and multiplying matrices corresponds to multiplying group elements.
@Define an $n$-dimensional real representation of a group $G$. What does it mean for such a representation to be unitary or orthogonal?
A homomorphism
\[\rho : G \to \mathbb R^{n \times n}\]assigning to each $g \in G$ an invertible matrix $\rho(g)$ (and hence satisfying the condition that $\rho(gh) = \rho(g) \rho(h)$ for all $g, h \in G$. It is unitary or orthogonal if the matrix $\rho(g)$ is unitary or orthogonal for all $g \in G$.