Intro to harmonic analysis on groups Risi Kondor

Transcription

1 Risi Kondor

2 Any (sufficiently smooth) function f on the unit circle (equivalently, any 2π periodic f ) can be decomposed into a sum of sinusoidal waves f(x) = k= c n e ikx c n = 1 2π f(x) e ikx dx 2π 0 Workhorse of much of applied mathematics Exact conditions under which it works get messy Eg, for f L 2 ([0, 2π)) almost everywhere convergence proved only in 1966 (Carleson) 2 2/37

3 f(x) = f(k) e 2πikx dk f(k) = f(x) e 2πikx dx Duality between time domain and Fourier domain (wave/particle duality in quantum mechanics) Heisenberg uncertainty principle Easily generalizes to R p 3 3/37

4 f(x) = n 1 k=0 n 1 f(k) e 2πikx/n 1 f(k) = n x=0 f(x) e 2πikx/n Unitary transform C n C n (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(n 2 ) to O(n log n) [Cooley & Tukey, 1965] 4 4/37

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6 Take a measurable space X, a space of functions on X, say L 2 (X), and a self-adjoint smoothing operator Υ For example, on X = R p, Υ may be the time t diffusion operator (Υf)(x) = 1 4πt f(y) e x y 2 /(4t) dy Question: How does Υ filter L 2 (X) into a nested sequence of spaces W Ω = { f L 2 (X) f, Υf / f, f Ω }? 6 6/37

7 Now let a group G act on X inducing linear operators T g : L 2 (X) L 2 (X) Eg, on on X = R p, (T g f)(x) = f(x g) g R p Question: What are the smallest spaces fixed by these operators, T g (V ) = V g G? 7 7/37

8 On R p we are lucky because these two notions match up: The diffusion operator is e t 2, where 2 is the Laplacian 2 = 2 x x x 2 p The e 2πik x Fourier basis functions are eigenfunctions of both and T g : 2 e 2πik x = 4π 2 k 2 e 2πik x, T g e 2πik x = e 2πik g e 2πik x Therefore W Ω = { f f(k) = 0 if k 2 Ω } (band-limited functions) V κ = { f f(k) = 0 if k κ } (isotypics) Question: Does this correspondence hold more generally? 8 8/37

9 On a finite graph G, the analog of is the graph Laplacian 1 i j [L] i,j = d i i = j 0 otherwise It does lead to a natural measure of smoothness: f Lf = i j (f i f j ) 2 Analyzing functions in terms of the eigenfunctions of L is called However (in general) on graphs there is no analog of translation 9 9/37

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11 The Fourier transform is Linear Invertible f(x)g(x) dx = f(k)ĝ(k) dk (Parseval thm) Therefore, it is essentially a unitary change of basis 11 11/37

12 Diagonalizes the derivative operator: g(x) = d f(x) = ĝ(k) = 2πik f(k) dx Diagonalizes the Laplacian: g(x) = d2 dx 2 f(x) = ĝ(k) = 4πk2 f(k) 12 12/37

13 Translation theorem: g(x) = f(x t) = ĝ(k) = e 2πikt f(k) Scaling theorem: g(x) = f(λx) = f (k) = λ 1 f(k/λ) 13 13/37

14 Convolution theorem: (f g)(x) = f(x y)g(y)dy = f g(k) = f(k) ĝ(k) Cross-correlation theorem: (f g)(x) = f(y) g(x + y)dy = f g(k) = f(k) ĝ(k) Autocorrelation: h(x) = f(y) f(x + y)dy = ĥ(k) = f(k) /37

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16 R f(k) = f(x) e 2πikx dx Observation: χ k (x) = e 2πikx are exactly the characters of R 16 16/37

17 The Fourier transform of a function on an LCA group G with Haar measure µ is f(χ) = f(x) χ(x) dµ χ Ĝ G The dual object is itself a group: T Z, R R, and for finite groups Ĝ = G (Pontryagin duality) This covers the Fourier series and the Fourier transform 17 17/37

18 The Fourier transform of a function on a compact group G with Haar measure µ is f(ρ) = f(x) ρ(x) dµ(x) ρ R, G where R 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) If G is finite, R is finite If G 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 Every representation is over C 18 18/37

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20 Forward transform: f(ρ) = f(x) ρ(x) dµ(x) ρ R G Inverse transform: f(x) = 1 [ ] d ρ tr f(ρ) ρ(x 1 ) µ(g) ρ R x G Just as before (with respect to the appropriate scaled matrix norms), this transform is unitary The e ρ i,j (x) = d ρ [ρ(x)] i,j functions form an orthogonormal basis (Peter-Weyl theorem) 20 20/37

21 Given f : G C and t G, define f t (x) = f(t 1 x) Then f t (ρ) = ρ(t) f(ρ) ρ R f t (x) ρ(x) dµ(x) = f(t 1 x) ρ(x) dµ(x) = f(x) ρ(tx) dµ(x) = f(x) ρ(t) ρ(x) dµ(x) = ρ(t) f(ρ) 21 21/37

22 Convolution theorem: (f g)(x) = f(xy 1 )g(y)dµ(y) = f g(ρ) = f(ρ) ĝ(ρ) Cross-correlation theorem: (f g)(x) = f(xy)g(y) dµ(y) = f g(ρ) = f(ρ) ĝ(ρ) 22 22/37

23 Given f : G C and t G, define f (t) (x) = f(xt 1 ) Then f (t) (ρ) = f(ρ) ρ(t) ρ R 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) 23 23/37

24 To left-translation: W ρ,j = span{ e ρ i,j j = 1,, d ρ } ρ R j = 1,, d ρ To right-translation: W ρ,i = span{ e ρ i,j i = 1,, d ρ } ρ R i = 1,, d ρ To left- and right-translation: V ρ = span{ e ρ i,j i, j = 1,, d ρ } ρ R 24 24/37

25 The group algebra C[G] is a space with orthonormal basis { e x x G } and a notion of multipilication defined by e x e y = e xy x, y G Letting f, e x = f(x) and extending to the rest of C[G] by linearity, for any f, 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, ie, it decomposes into a direct sum of simple algebras 25 25/37

26 The group algebra decomposes into a sum of simple algebras: C[G] = ρ V ρ (1) Each V ρ is called an, and corresponds to a single Fourier matrix f(ρ) This decomposition is unique Each V ρ further decomposes into a sum of d ρ left G modules V ρ = W ρ 1 W ρ 2 W ρ d ρ (2) corresponding to each column of f(ρ) This decomposition is not unique, ie, it depends on the choice of R The Fourier transform is a projection of f onto a basis adapted to (1) and (2) 26 26/37

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28 So far we have considered: f is a function on a compact group G G acts on G by t: x tx inducing f f t, where f t (x) = f(t 1 x) (similarly for the right-action and right-translation) In practice it is often more common that: f is a function on a set X G acts on X transitively by t: x tx, inducing f f t, where f t (x) = f(t 1 x) Example: The rotation group SO(3) and the sphere S 2 The symmetric group acting on a matrix by permuting rows/columns 28 28/37

29 Assume that G acts on X transitively by x gx Pick some x 0 X The subset of G fixing x 0 is a subgroup H of G Each set gh = { gh h H } is called a left H coset The set of left H cosets we denote G/H { gh gh G/H } forms a partition of G yx 0 = y x 0 if and only if y, y belong to the same coset Therefore, we have a bijection X G/H X is called a of G Example: S 2 SO(3)/SO(2) 29 29/37

30 Now L(X) is only a G-module, not a C[G] algebra However, we can still ask, how it decomposes into a sum of invariant modules The Fourier transform of f : X C is the FT of the induced function f G (g) = f(gx 0 ), ie, f(ρ) = f(gx 0 ) ρ(g) dµ(g) ρ R G 30 30/37

31 We say that the repesentation ρ of G is adapted to H G, if ρ H is of the block diagonal form ρ H (h) = ρ (h) h H (3) ρ R ρ for some multiset R ρ of irreps of H We use m tr (ρ) to denote the multiplicity of the trivial representation in the decomposition (3) 31 31/37

32 If f : X C and f is expressed in a basis adapted to H G, then f(ρ) has at most m tr (ρ) non-zero columns f(ρ) = g G/H [ h H g G/H f(gx 0 ) ρ(gh) dµ(g)dµ(h) = ][ ] f(gx 0 ) ρ(g) dµ(g) ρ(h)dµ(h) h H }{{} 32 32/37

33 h H ρ(h)dµ(h) = ρ R h H ρ (h)dµ(h) If ρ is the trivial irrep of H, then h H ρ (h)dµ(h) = µ(h) However, by orthogonality of the Fourier basis functions, for any other irrep, h H ρ (h)dµ(h) = /37

34 The rotation group SO(3) is parametrized by the Euler angles (θ, ϕ, ψ), and the irreps are given by the D (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 m l (θ, ϕ) = 2l + 1 (l m)! 4π (l + m)! P m l (cos θ) e imϕ, are the spherical harmonics Clearly, the Wigner matrices are adapted to the subgroup SO(2) that rotates ψ In particular, l D (l) SO(2) (ψ) = χ m (ψ) χ m (ψ) = e im ψ, m = l so the multiplicity of the trivial irrep χ 0 is /37

35 The Fourier transform of f : S 2 C is f(l) = f(rx 0 ) D (l) (R) dµ(r) l = 0, 1, 2,, SO(3) but by our theorem only the middle column of each of these matrices is non-zero, which yields exactly the celebrated spherical harmonic expansion m f(θ, ϕ) = l=0 m= l f m l Yl m (θ, ϕ) 35 35/37

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37 Invariants to group actions: Computer vision Graph invariants Band limited approximations on S n : Multi-object tracking Wavelets on S n Learning on S n : Ranking problems Optimization on S n : Fast QAP solvers 37 37/37