Intro to harmonic analysis on groupsRisi Kondor

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The Fourier series (1807)Any (sufficiently smooth) function f on the unit circle (equivalently, any2π–periodic f ) can be decomposed into a sum of sinusoidal waves

f(x) =∞∑

k=−∞cn e

ikx cn =1

2π

∫ 2π

0f(x) e−ikx dx.

• Workhorse of much of applied mathematics.• Exact conditions under which it works get messy.

E.g., for f ∈L2([0, 2π)) almost everywhereconvergence proved only in 1966 (Carleson).

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The Fourier transform

f(x) =

∫f(k) e2πikx dk f(k) =

∫f(x) e−2πikx dx

• Duality between time domain and Fourier domain (wave/particle duality inquantum mechanics)

• Heisenberg uncertainty principle• Easily generalizes to Rp.

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The discrete Fourier transform (DFT)

f(x) =n−1∑k=0

f(k) e2πikx/n f(k) =1

n

n−1∑x=0

f(x) e−2πikx/n

• Unitary transform Cn → Cn (with appropriate normalization).• Can be seen as discretized version of Fourier series, or as the Fourier

transform on 0, 1, 2, . . . , n−1.• Foundation of all of digital signal processing.• Fast Fourier transforms reduce computation time from O(n2) toO(n log n) [Cooley & Tukey, 1965].

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Underlying principles

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1. Analytic

Take a measurable spaceX , a space of functions onX , say L2(X), and aself-adjoint smoothing operator Υ. For example, onX =Rp,Υ may be thetime t diffusion operator

(Υf)(x) =1√4πt

∫f(y) e−∥x−y ∥2/(4t)dy.

Question: How doesΥ filter L2(X) into a nested sequence of spaces

WΩ = f ∈L2(X) | |⟨f,Υf⟩ / ⟨f, f⟩| ≤ Ω ?

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2. Algebraic

Now let a groupG act onX inducing linear operators Tg : L2(X) → L2(X).E.g., on onX =Rp,

(Tgf)(x) = f(x− g) g ∈Rp.

Question: What are the smallest spaces fixed by these operators,

Tg(V ) = V ∀g ∈G ?

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On Rp we are lucky because these two notions match up:

• The diffusion operator is et∇2, where∇2 is the Laplacian

∇2 =∂2

∂x21+

∂2

∂x22+ . . .+

∂2

∂x2p.

• The e2πik·x Fourier basis functions are eigenfunctions of both∆ and Tg : ∇2e2πik·x = −4π2∥k∥2 e2πik·x, Tg e

2πik·x = e2πik·g e2πik·x.

• Therefore WΩ = f | f(k) = 0 if ∥k∥2 ≥Ω (band-limited functions)

Vκ = f | f(k) = 0 if k = κ (isotypics)

Question: Does this correspondence hold more generally?

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Fourier analysis on graphs

On a finite graph G, the analog of∆ is the graph Laplacian

[L]i,j =

1 i ∼ j

− di i= j

0 otherwise.

It does lead to a natural measure of smoothness:

f⊤Lf = −∑i∼j

(fi − fj)2.

Analyzing functions in terms of the eigenfunctions of L is called spectral graphtheory.

However (in general) on graphs there is no analog of translation.

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More properties of the FT on R

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• The Fourier transform is Linear Invertible

∫f(x)g(x)∗dx =

∫f(k) g(k)∗dk (Parseval thm)

Therefore, it is essentially a unitary change of basis.

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• Diagonalizes the derivative operator:

g(x) =d

dxf(x) =⇒ g(k) = 2πik f(k).

• Diagonalizes the Laplacian:

g(x) =d2

dx2f(x) =⇒ g(k) = −4πk2 f(k).

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• Translation theorem:

g(x) = f(x− t) =⇒ g(k) = e−2πikt f(k)

• Scaling theorem:

g(x) = f(λx) =⇒ f ′(k) = |λ|−1 f(k/λ)

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• Convolution theorem:

(f ∗ g)(x) =∫f(x−y)g(y)dy =⇒ f ∗g(k) = f(k) · g(k)

• Cross-correlation theorem:

(f ⋆ g)(x) =

∫f(y)∗g(x+y)dy =⇒ f ⋆ g(k) = f(k)∗ · g(k)

Autocorrelation:

h(x) =

∫f(y)∗f(x+y)dy =⇒ h(k) = ∥f(k)∥2

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Fourier analysis on compact groups

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Fourier tranform on R

f(k) =

∫f(x) e−2πikx dx

Observation: χk(x) = e−2πikx are exactly the characters of R.

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Locally Compact Abelian groups

The Fourier transform of a function on an LCA groupG with Haar measure µ is

f(χ) =

∫Gf(x)χ(x) dµ χ∈ G.

• The dual object is itself a group: T ↔ Z, R ↔ R, and for finite groupsG ∼= G (Pontryagin duality).

• This covers the Fourier series and the Fourier transform.

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Compact non-Abelian groups

The Fourier transform of a function on a compact groupG with Haar measure µis

f(ρ) =

∫Gf(x) ρ(x) dµ(x) ρ ∈ R,

whereR is a complete set of inequivalent irreducible representations (irreps).

• Now the dual object is no longer a group, but a set of representations(Tannaka–Krein duality).

• IfG is finite,R is finite. IfG is compact,R is countable.• Each Fourier component f(ρ) is a matrix.

In the following, we will always assume that each ρ is unitary. Everyrepresentation is over C.

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Properties

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InvertibilityForward transform:

f(ρ) =

∫Gf(x) ρ(x) dµ(x) ρ ∈ R.

Inverse transform:

f(x) =1

µ(G)

∑ρ∈R

dρ tr[f(ρ) ρ(x−1)

]x∈G.

• Just as before (with respect to the appropriate scaled matrix norms), thistransform is unitary.

• The eρi,j(x) =√dρ [ρ(x)]i,j functions form an orthogonormal basis

(Peter-Weyl theorem).

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Left-translation

Theorem. Given f : G→ C and t∈G, define f t(x) = f(t−1x). Then

f t(ρ) = ρ(t) · f(ρ) ρ∈R.

Proof.∫f t(x) ρ(x) dµ(x) =

∫f(t−1x) ρ(x) dµ(x) =

∫f(x) ρ(tx) dµ(x) =

∫f(x) ρ(t) ρ(x) dµ(x) = ρ(t) f(ρ)

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Left-translation

• Convolution theorem:

(f ∗ g)(x) =∫f(xy−1)g(y)dµ(y) =⇒ f ∗g(ρ) = f(ρ) · g(ρ)

• Cross-correlation theorem:

(f ⋆ g)(x) =

∫f(xy)g(y)∗dµ(y) =⇒ f ⋆ g(ρ) = f(ρ) · g(ρ)†

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Right-translation

Theorem. Given f : G→ C and t∈G, define f (t)(x) = f(xt−1). Then

f (t)(ρ) = f(ρ) · ρ(t) ρ∈R.

Proof.∫f (t)(x) ρ(x) dµ(x) =

∫f(xt−1) ρ(x) dµ(x) =

∫f(x) ρ(xt) dµ(x) =

∫f(x) ρ(x) ρ(t) dµ(x) = f(ρ) ρ(t)

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Invariant subspaces

• To left-translation:

Wρ,j = span eρi,j | j = 1, . . . , dρ ρ∈R j = 1, . . . , dρ.

• To right-translation:

Wρ,i = span eρi,j | i = 1, . . . , dρ ρ∈R i = 1, . . . , dρ.

• To left- and right-translation:

Vρ = span eρi,j | i, j = 1, . . . , dρ ρ∈R.

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The group algebra

The group algebra C[G] is a space with orthonormal basis ex | x∈G anda notion of multipilication defined by

exey = exy ∀x, y ∈G.

Letting ⟨f, ex⟩ = f(x) and extending to the rest of C[G] by linearity, for anyf, g ∈C[G],

(fg)(x) =

∫f(xy−1)g(y)dµ(y) = (f ∗ g)(x).

The group algebra of any compact group is semi-simple, i.e., it decomposesinto a direct sum of simple algebras.

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The group algebra

• The group algebra decomposes into a sum of simple algebras:

C[G] =⊕ρ

Vρ. (1)

Each Vρ is called an isotypic, and corresponds to a single Fourier matrixf(ρ). This decomposition is unique.

• Each Vρ further decomposes into a sum of dρ leftG–modules

Vρ =W ρ1 ⊕W ρ

2 ⊕ . . .⊕W ρdρ

(2)

corresponding to each column of f(ρ). This decomposition is not unique,i.e., it depends on the choice ofR.

The Fourier transform is a projection of f onto a basis adapted to (1) and (2).

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Fourier analysis on homogeneous spaces

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Group actions

• So far we have considered: f is a function on a compact groupG. G acts onG by t : x 7→ tx inducing f 7→ f t, where f t(x) = f(t−1x)

(similarly for the right-action and right-translation).

• In practice it is often more common that: f is a function on a setX . G acts onX transitively by t : x 7→ tx, inducing f 7→ f t, wheref t(x) = f(t−1x).

Example: The rotation group SO(3) and the sphere S2. The symmetric groupacting on a matrix by permuting rows/columns.

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Homogeneous spacesAssume thatG acts onX transitively by x 7→ gx.

• Pick some x0 ∈X .• The subset ofG fixing x0 is a subgroupH ofG.• Each set gH = gh | h∈H is called a leftH–coset.• The set of leftH–cosets we denoteG/H .• gH | gH ∈G/H forms a partition ofG.• yx0 = y′x0 if and only if y, y′ belong to the same coset.• Therefore, we have a bijection

X ↔ G/H.

X is called a homogeneous space ofG.

Example: S2 ∼ SO(3)/SO(2).

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FT on homogeneous spaces

Now L(X) is only aG-module, not a C[G] algebra. However, we can still ask,how it decomposes into a sum of invariant modules.

Definition. The Fourier transform of f : X→C is the FT of the inducedfunction f ↑G(g) = f(gx0), i.e.,

f(ρ) =

∫Gf(gx0) ρ(g) dµ(g) ρ ∈ R

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Adapted bases

We say that the repesentation ρ ofG is adapted toH ≤ G, if ρ↓H is of theblock diagonal form

ρ↓H(h) =⊕ρ′∈Rρ

ρ′(h) h∈H (3)

for some multiset Rρ of irreps ofH .

We usemtr(ρ) to denote the multiplicity of the trivial representation in thedecomposition (3).

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FT on homogeneous spaces

Theorem. If f : X → C and f is expressed in a basis adapted toH ≤ G,then f(ρ) has at mostmtr(ρ) non-zero columns.

Proof.

f(ρ) =

∫g∈G/H

∫h∈H

f(gx0) ρ(gh) dµ(g)dµ(h) =

[∫g∈G/H

f(gx0) ρ(g) dµ(g)

][∫h∈H

ρ(h)dµ(h)︸ ︷︷ ︸]

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Proof (continued)∫h∈H

ρ(h)dµ(h) =⊕ρ′∈R

∫h∈H

ρ′(h)dµ(h)

If ρ′ is the trivial irrep ofH , then∫h∈H ρ

′(h)dµ(h) = µ(H). However, byorthogonality of the Fourier basis functions, for any other irrep,∫h∈H ρ

′(h)dµ(h) = 0.

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ExampleThe rotation group SO(3) is parametrized by the Euler angles (θ, ϕ, ψ), andthe irreps are given by theD(0), D(1), D(2), . . .Wigner matrices, where

[D(ℓ)]m,m′ = e−im′ψ Y m

ℓ (θ, ϕ), m,m′ = −ℓ, . . . , ℓ

where

Y mℓ (θ, ϕ) =

√2ℓ+ 1

4π

(ℓ−m)!

(ℓ+m)!Pml (cos θ) eimϕ,

are the spherical harmonics. Clearly, the Wigner matrices are adapted to thesubgroup SO(2) that rotates ψ. In particular,

D(ℓ) ↓SO(2) (ψ) =

ℓ⊕m′=−ℓ

χm′(ψ) χm′(ψ) = e−im′ψ,

so the multiplicity of the trivial irrep χ0 is 1.

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Example

The Fourier transform of f : S2 → C is

f(ℓ) =

∫SO(3)

f(Rx0)D(ℓ)(R) dµ(R) ℓ = 0, 1, 2, . . . ,

but by our theorem only the middle column of each of these matrices isnon-zero, which yields exactly the celebrated spherical harmonic expansion

f(θ, ϕ) =∞∑ℓ=0

m∑m=−ℓ

fmℓ Y mℓ (θ, ϕ).

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Example

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APPLICATIONS

• Invariants to group actions: Computer vision Graph invariants

• Band–limited approximations on Sn: Multi-object tracking

• Wavelets on Sn• Learning on Sn:

Ranking problems

• Optimization on Sn: Fast QAP solvers.

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