xvi Notation

Notation

U, u Domain, point on domain

x, x(u) ∈ X (U, C) Signal on the domain of the form x : U → C

f (x) ∈ F(X ) Functions on signals on the domain of the form f :

X (U, C) → Y (when domain U and target C are clear from

the context)

G, g Group, element of the group

g.u, ρ(g) Group action, group representation

X ∈ C

|U |×s

Matrix representing a signal on a discrete domain

x

u

∈ C

s

Vector representing a discrete domain signal X on element

u ∈ U

x

uj

∈ C Scalar representing the jth component of a discrete domain

signal X on element u ∈ U

F(X) Function on discrete domain signals that returns another

discrete domain signal, as a matrix

τ : U → U Automorphism of the domain

η : U → U

′

Isomorphism between two different domains

σ : C → C

′

Activation function (point-wise non-linearity)

G = (V, E) Graph with nodes V and edges E

M = (V, E, F) Mesh with nodes V, edges E, and faces F

x ⋆ θ Convolution with ﬁlter θ

S

v

Shift operator

φ

i

Basis function

T

u

U, TU Tangent space at u, tangent bundle

X ∈ T

u

U Tangent vector

g

u

(X, Y) = ⟨X, Y⟩

u

Riemannian metric

ℓ(γ), ℓ

uv

Length of a curve γ, discrete metric on edge (u, v)