Concepts

What’s vector space?

A vector space on a field \(F\) is a set \(V\) with two binary operators that satisfy:

  • For vector addition:
    • Associativity: \(\forall u,v,w\in V, (u+v)+w=u+(v+w)\)
    • Commutavility: \(\forall u,v\in V, u+v=v+u\)
    • Identity: \(\exists 0\in V\) so that \(\forall u\in V, u+0=u\)
    • Inverse: \(\forall u\in V, \exists -v \in V\) so that \(v + (-v) = 0\)
  • For scalar multiplication:
    • Compatibility between scalar multiplication and field multiplication: a(bv)=(ab)v
    • Identity: \(1v=v\)
    • Distributivity (w.r.t. vector addition): \(a(u+v)=au+av\)
    • Distributivity (w.r.t. field addition): \((a+b)v=av+bv\)

What’s group?

\((G, \cdot)\), where \(G\) is a set, and \(\cdot\) is the group product (i.e., a binary operator), so that

  • Closure: \(\forall g,h\in G, g\cdot h \in G\)
  • Identity: \(\exists e\in G\), s.t., \(e\cdot g=g,\forall g\in G\)
  • Inverse: \(\forall g\in G,\exists g^{-1}\), s.t., \(g^{-1}\cdot g = g\cdot g^{-1} = e\)
  • Associativity: \(\forall g,h,i\in G, (g\cdot h)\cdot i = g\cdot(h\cdot i)\)

What’s Lie group? (not that formal)

It is continuous group that is also a differentiable manifold, where continuous group means \(G\) is infinite, and its group operator is continuous.

What’s subgroup?

When we say \((H,\cdot)\) is a subgroup of \((G,\cdot)\), \(H\subset G\) should preserve the closure property of \(\cdot\).

What’s group homomorphism?

Given two gruops \((G,\ast)\) and \((H,\cdot)\), a group homomorphism from the former to the latter is a function \(f\) that satisfies:

\[\forall g,\tilde{g}\in G,(h=f(g)\text{ and }\tilde{h}=f(\tilde{g}))\rightarrow h\cdot\tilde{h} = f(g\ast\tilde{g})\]

What’s group action?

Given a group \((G,\cdot)\), a group action on a space \(X\) is a binary operator \(\odot\) from \(G\times X\) to \(X\), so that:

\[\forall g,\tilde{g}\in G,x\in X, g\odot(\tilde{g}\odot x) = (g\cdot\tilde{g})\odot x\]

What’s representation?

Given a group \((G,\cdot)\), a representation parameterized by \(g\in G\) is a linear and invertible function \(\rho(g)\) mapping from a vector space \(V\) to itself, so that:

\[\forall g,\tilde{g}\in G,v\in V, \rho(g)\rho(\tilde{g})v = \rho(g\cdot\tilde{g})v\]

What’s matrix representation?

When the dimension of \(V\), denoted by \(d\) is finite, it is equivalent to consider \(\Re^{d}\). Then any linear transformation can be expressed by \(d\times d\) matrix. So a matrix representation \(D(g)\) is a \(d\times d\) matrix that respects the properties a representation should have and acts on \(v\in \Re^{d}\) by matrix-vector multiplication. It is often said \(D(g)\in\text{GL}(d,\Re)\), where “GL” is short for general linear group.

What’s left-regular representation?

Suppose the vector space \(V\) consists of functions in \(\mathbb{L}_2(X)\), and a group \(G\) has group action on \(X\) denoted by \(\odot\). Then a left-regular representation \(\mathcal{L}_{g}\) of \(G\) acting on \(\mathbb{L}_2(X)\) is representation that satisfies:

\[\forall g\in G,\forall f\in V, \forall x\in X,[\mathcal{L}_{g}f](x)=f(g^{-1}\odot x)\]

Examples and Interpretations

Why do we need group product?

Recall that, in chap1, we interpret cross-correlation as the inner product between the kernel (transformed by a group) and the signal. Taking input at different positions by the cross-correlation means transforming the kernel by different group members. Thus, group product can be interpreted as the composition of two transformations, and the inverse of \(g\) means the transformation cancelling out \(g\)’s effect.

Examples include translation group \(G=(\Re^d,+)\) and rotation group \(\text{SO}(2)=([0,2\pi),+_{\text{mod }2\pi})\), which is often parameterized as:

\(R_{\theta}=\left[\begin{array}{cc} cos\theta & -sin\theta \\ sin\theta & cos\theta \\ \end{array}\right],\) and group product corresponds to matrix multiplication.

So why do we need group structure?

Obviously, the translation group with its group product as “+” and vanilla scalar-vector multiplication forms a vector space. As for SO(2), with its group product as “+” and scalar multiplication (and then mod \(2\pi\)), it also forms a vector space. At least, their group product is our very familiar arithmatic operations (matrix multiplication between two rotation matrices is commutative). However, for SE(\(d\)), its group product is:

\[(x,\theta) \cdot (\tilde{x},\tilde{\theta})=(x+R_{\theta}\tilde{x},R_{\theta}R_{\tilde{\theta}}),\]

which is not commutative, and thus SE(\(d\)) cannot be a vector space by regarding group product as vector addition and supplementing the definition of scalar-vector multiplication.

How to interpret “group action is a group homomorphism”?

It is a mapping from \(G\) to \(\{f_g \| g\in G\}\), and the group product of \(\{f_g \| g\in G\}\) is function composition, namely, \(\forall f_g, f_h, f_g \cdot f_h = f_g(f_{h}())\). Then we need to prove \(f_{g\cdot h}=f_g \cdot f_h\) (note that \(\cdot\) at the LHS and RHS are that of respective group), which is correct by the definition of group action.

What’s the essential property that makes a group representation left-regular?