a generalized fft for clifford algebras, 2003.leopardi/gfft-2003-07-07.pdf · gft for clifford...

A generalized FFT for Clifford algebrasPaul Leopardi

[email protected]

School of Mathematics, University of New South Wales.

For presentation at ICIAM, Sydney, 2003-07-07.

8 16

8

16

Real representation matrix R2

http://glucat.sf.net – p.1/27

GFT for finite groups

(Beth 1987, Diaconis and Rockmore 1990; Clausen and Baum 1993)

The generalized Fourier transform (GFT) for a finite group G is therepresentation map D from the group algebra CG to a faithfulcomplex matrix representation of G, such that D is the direct sum ofa complete set of irreducible representations.

D : CG → C(M), D =⊕n

k=1Dk,

where Dk : CG → C(mk), andn

∑

k=1

mk = M.

The generalized FFT is any fast algorithm for the GFT.


GFT for supersolvable groups

(Baum 1991; Clausen and Baum 1993)

Definition 1.

The 2-linear complexity L2(X), of a linear operator X counts

non-zero additions A(X), and non-zero multiplications, except

multiplications by 1 or −1.

The GFT D, for supersolvable groups has

L2(D) = O(|G|log2|G|).


GFT for Clifford algebras

(Hestenes and Sobczyck 1984; Wene 1992; Felsberg et al. 2001)

The GFT for a real universal Clifford algebra, Rp,q, is therepresentation map P p,q from the real framed representation to thereal matrix representation of Rp,q.

P p,q : RPς(−q,p) → R(2N(p,q)), where

• ς(a, b) := {a, a + 1, . . . , b} \ {0}.

• Pς(−q, p) is the power set of ς(−q, p) , a set of index setswith cardinality 2p+q.

• The real framed representation RPς(−q,p) is the set of maps fromPς(−q, p) to R, isomorphic as a real vector space to the set of2p+q tuples of real numbers indexed by subsets of ς(−q, p).

This is not the “discrete Clifford Fourier transform” of Felsberg, et al.


Clifford algebras and supersolvable groups

(Braden 1985; Lam and Smith 1989)

The real universal Clifford algebra Rp,q, is a quotient of the groupalgebra RGp,q, by an ideal, where Gp,q is a 2-group, here called theframe group. Gp,q is supersolvable.The GFT for Clifford algebras is related to that for supersolvablegroups:

CGp,qD

−−−→ D (CGp,q) ⊆ C(M)

project

y

y

project

RGp,qD

−−−→ D (RGp,q) ⊆ C(M)

quotient

y

y

quotient

Rp,q −−−→P p,q

P p,q (Rp,q) ⊆ R(2N(p,q))


GFT for the neutral Clifford algebra Rn,n

(Braden 1985; Lam and Smith 1989)

The GFT for the neutral Clifford algebra Rn,n is a map:

P n : RPς(−n,n) → R(2n). |Pς(−n, n)| = 4n.

The frame group Gn,n is an extraspecial 2-group. |Gn,n| = 22n+1.

For Rn,n we might expect L2(P n) = O(n4n) this way:

Rn,nP n,n

−−−→ R(2n)

y

x

CGn,n −−−→D

D (CGn,n)

but there are explicit algorithms for both forward and inverse GFT.


Kronecker product

Definition 2.If A ∈ R(r) and B ∈ R(s) , then

(A ⊗ B)j,k = Aj,kB

if A ⊗ B is treated as an r × r block matrix with s × s blocks.

A well known property of the Kronecker product is:

Lemma 3.If A, C ∈ R(r) and B, D ∈ R(s) , then

(A ⊗ B)(C ⊗ D) = AC ⊗ BD


Generating set for Rn,n

(Porteous 1969)

Definition 4. Here and in what follows, define:

In := unit matrix of dimension 2n, I := I1

J :=

[

0 −1

1 0

]

, K :=

[

0 1

1 0

]

.

Lemma 5.If S is an orthonormal generating set for R(2n−1), then

{−JK ⊗ A | A ∈ S} ∪ {J ⊗ In−1, K ⊗ In−1}

is an orthonormal generating set for R(2n).


Generalized Fourier transform

Definition 6. Use Lemma 5 to define the GFT of eachgenerator of Rn,n

P ne{−n} := J ⊗ In−1, P ne{n} := K ⊗ In−1,

for 1 − n 6 k 6 n − 1, P ne{k} := −JK ⊗ P n−1e{k}.

We can now compute P neT as: P neT =∏

k∈T

P ne{k}

Each basis matrix is monomial and each non-zero is −1 or 1.

We can now define P n : RPς(−n,n) → R(2n), by:

P na =∑

T ⊆ς(−n,n)

aT P neT , for a =∑

T ⊆ς(−n,n)

aT eT .


Bound for linear complexity of GFT

Theorem 7.L2(P n) is bounded by d3/2,

where d is the dimension of R(2n) ∼= Rn,n.

ProofSince P neT is of size 2n × 2n and is monomial, it has 2n

non-zeros.

R(2n) has 4n basis elements.

A(P n) is therefore bounded by

4n × 2n = (4n)3/2 = d3/2,

where d is the dimension of R(2n) ∼= Rn,n.

There are no non-trivial multiplications. �


Z2 grading and ⊗ are keys to GFT

(Lam 1973)The algebras Rp,q are Z2 -graded. Each a ∈ Rp,q can be split intoodd and even parts, a = a+ + a−, with odd × odd = even, etc.Scalars are even and the generators are odd.

We can express P n in terms of its actions on the even and odd parts ofa multivector: P na = P na+ + P na−, a = a+ + a− ∈ Rn,n.

Lemma 8. For all b ∈ Rn−1,n−1, we have

P nb+ = I ⊗ P n−1b+, P nb− = −JK ⊗ P n−1b−,

so that

(P na−)(P nb−) = (−JK ⊗ P n−1a−)(−JK ⊗ P n−1b−)

= I ⊗ (P n−1a−)(P n−1b−)

= I ⊗(

P n−1(a−b−)

)

= P n(a−b−).


Recursive expression for P n

Theorem 9. For n > 0, for the GFT P n as per Definition 6,

for a ∈ Rn,n, a = a+ + a−, with

a+ = a+∅ + e{−n}a

+−n + a+

n e{n} + e{−n}a+−n,ne{n},

a− = a−∅ + e{−n}a

−−n + a−

n e{n} + e{−n}a−−n,ne{n},

we have P na = P na+ + P na−,

P na+ = I ⊗ P n−1a+∅ + K ⊗ P n−1a

+−n+

− J ⊗ P n−1a+n + JK ⊗ P n−1a

+−n,n, and

P na− = −JK ⊗ P n−1a−∅ + J ⊗ P n−1a

−−n+

K ⊗ P n−1a−n + I ⊗ P n−1a

−−n,n.


Base cases and linear complexity

Theorem 10.

P 0a+ = [a+], P 0a− = 0.

Proof If a ∈ R0,0 then a is even, so a− = 0. �

Theorem 11. For n ≥ 0,

L2(P n) 6 n4n =1

2d log2 d,

where d = 4n is the dimension of Rn,n.

Proof (Sketch)Count non-zero additions at each level of recursion.

You will obtain at most 4n additions at each of n levels. �


Real framed inner product

Recall that if a ∈ Rn,n, then a can be expressed as

a =∑

T ⊆ς(−n,n)

aT eT

The basis {eT | T ⊆ ς(−n, n)} is orthornormal with respect to thereal framed inner product

a • b :=∑

T ⊆ς(−n,n)

aT bT .

We have

eS • eT = δS,T and aT = a • eT .


Normalized Frobenius inner product

Since the GFT P n is an isomorphism, it preserves this inner product.That is, there is an inner product • : R(2n) × R(2n) → R, suchthat, for a, b ∈ Rn,n,

P na • P nb = a • b,

so P na • P neT = a • eT = aT .

This is the normalized Frobenius inner product.Lemma 12.For A, B ∈ R(2n), the normalized Frobenius inner product:

A • B := 2−n tr AT B = 2−n2n

∑

j,k=1

Aj,kBj,k,

satisfies P na • P nb = a • b, for a, b ∈ Rn,n.


Inverse GFT

Since P na • P neT = aT , we can define Qn := P −1n by

Definition 13.

Qn : R(2n) → RPς(−n,n)

For A ∈ R(2n), T ⊆ ς(−n, n),

(QnA)T := A • P neT .

Naive algorithm for Qn evaluates A • P neT for eachT ⊆ ς(−n, n).

Theorem 14. L2(Qn) 6 d3/2 + d log d, where d = 4n.

Proof (Sketch)

A(Qn) 6 2n × 4n, and the naive algorithm also needs at most 4n

divisions by 2n. �


Left Kronecker quotient

The left Kronecker quotient is a binary operation which is an inverseoperation to the Kronecker matrix product.

Definition 15.

; : R(r) × R(rs) → R(s),

for A ∈ R(r), nnz(A) 6= 0, C ∈ R(rs),

(A ; C)j,k :=1

nnz(A)

∑

Aj,k 6=0

Cj,k

Aj,k,

where C is treated as an r × r block matrix with s × s blocks,

ie. as if C ∈ R(s)(r).


Left Kronecker quotient

Theorem 16.The left Kronecker quotient is an inverse operation to theKronecker matrix product, when applied from the left, as follows:

For A ∈ R(r), nnz(A) 6= 0, B ∈ R(s),we have A ; (A ⊗ B) = B.

Proof

A ; (A ⊗ B) =1

nnz(A)

∑

Aj,k 6=0

Aj,kB

Aj,k

=1

nnz(A)

∑

Aj,k 6=0

B

= B

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Left Kronecker quotient and orthogonality

Lemma 17. For A ∈ R(2n), B ∈ R(2n), C ∈ R(2ns),if nnz(A) = 2n then A ; (B ⊗ C) = (A′ • B)C, where

A′j,k =

1

Aj,k, if Aj,k 6= 0, 0 otherwise.

Proof

A ; (B ⊗ C) =1

nnz(A)

∑

Aj,k 6=0

Bj,kC

Aj,k

=1

2n

2n

∑

j,k=1

A′j,kBj,kC

= (A′ • B)C

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Left Kronecker quotient and orthogonality

Lemma 18. If n > 0 and

A ∈ R(2n+m) =∑

T ⊆ς(−n,n)

(P neT ) ⊗ AT , where

AT ∈ R(2m), then (P neT ) ; A = AT , for T ⊆ ς(−n, n).

Corollary 19.

If n > 0, AI , AJ , AK , AJK ∈ R(2n−1),

and A = I ⊗ AI + J ⊗ AJ + K ⊗ AK + JK ⊗ AJK ,

then I ; A = AI , J ; A = AJ ,

K ; A = AK , JK ; A = AJK .


Recursive expression for Qn

Theorem 20.For n > 0, A ∈ R(2n), Qn as per Definition 13,

Qn(A) = Qn−1(I ; A)+ − Qn−1(JK ; A)−

+ e{−n}

(

Qn−1(JK ; A)+ + Qn−1(I ; A)−)

e{n}

+ e{−n}

(

Qn−1(K ; A)− + Qn−1(J ; A)+)

+(

−Qn−1(J ; A)− + Qn−1(K ; A)+)

e{n}.

For n = 0, we have Q0[a] = a.

Proof (Sketch)

Start with Theorems 9 and 10 and apply Corollary 19. �


Linear complexity of inverse GFT

Theorem 21.L2(Qn) 6 2n4n = d log2 d,

where d = 4n is the dimension of Rn,n.

ProofQn uses ; four times. Each time needs at most 4n−1 additions.

Qn also uses Qn−1 four times. So,

A(Qn) 6 4n + 4A(Qn−1) 6 n4n =1

2d log2 d.

For Qn, each of the four uses of ; needs 4n−1 divisions by 2.

So L2(Qn) 6 2n4n = d log2 d. �


Benchmark for GluCat implementation

(Lounesto et al. 1987; Lounesto 1992; Raja 1996)• Generic library of universal Clifford algebra templates• For details, see http://glucat.sf.net

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References 1

References

[Ablamowicz 1996] R. Ablamowicz, P. Lounesto, J. M. Parra (eds), Clifford

algebras with numeric and symbolic computations, Birkhäuser,

1996.

[Baum 1991] Ulrich Baum, “Existence and efficient construction of fast

Fourier transforms on supersolvable groups”, Computational

Complexity 1 (1991), no. 3, 235–256

[Bayro Corrochano & Sobczyk] E. Bayro Corrochano, G. Sobczyk, Geometric algebra with

applications in science and engineering. Birkhäuser, 2001.

[Beth 1987] Thomas Beth, “On the computational complexity of the general

discrete Fourier transform”, Theoretical Computer Science, 51

(1987), no. 3, 331–339.

[Braden 1985] H. W. Braden, “N-dimensional spinors: Their properties in

terms of finite groups”, J. Math. Phys. 26 (4), April 1985.

American Institute of Physics.


References 2

References

[Bulow et al. 2001] Thomas Bülow, Michael Felsberg, Gerald Sommer,

“Non-commutative hypercomplex Fourier transforms of

multidimensional signals”, pp187–207 of [Sommer 2001].

[Chernov 2001] Vladimir M. Chernov, “Clifford algebras as projections of group

algebras”, pp461–476 of [Bayro Corrochano & Sobczyk].

[Clausen & Baum 1993] Michael Clausen, Ulrich Baum, Fast Fourier transforms.

Bibliographisches Institut, Mannheim, 1993.

[Diaconis & Rockmore 1990] Persi Diaconis, Daniel Rockmore, “Efficient computation of the

Fourier transform on finite groups”, J. Amer. Math. Soc. 3 (1990),

no. 2, 297–332.

[Felsberg et al. 2001] Michael Felsberg, Thomas Bülow, ; Gerald Sommer,

Vladimir M. Chernov, “Fast algorithms of hypercomplex Fourier

transforms”, pp231–254 of [Sommer 2001].


References 3

References

[Hestenes & Sobczyk 1984] David Hestenes, Garret Sobczyk, Clifford algebra to geometric

calculus : a unified language for mathematics and physics, D.

Reidel, 1984.

[Lam 1973] T. Y. Lam, The algebraic theory of quadratic forms, W. A.

Benjamin, Inc., 1973.

[Lam & Smith 1989] T. Y. Lam, Tara L. Smith, “On the Clifford-Littlewood-Eckmann

groups: a new look at periodicity mod 8”, Rocky Mountains

Journal of Mathematics, vol 19, no 3, Summer 1989.

[Lounesto 1987] P. Lounesto, R. Mikkola, V. Vierros, CLICAL User Manual:

Complex Number, Vector Space and Clifford Algebra Calculator

for MS-DOS Personal Computers, Institute of Mathematics,

Helsinki University of Technology, 1987.

[Lounesto 1992] P. Lounesto, “Clifford algebra calculations with a microcomputer”,

pp39–55 of [Micali et al. 1989].


References 4

References

[Micali et al. 1989] A. Micali, R. Boudet, J. Helmstetter, (eds), Clifford algebras and their

applications in mathematical physics : proceedings of second workshop

held at Montpellier, France, 1989, Kluwer Academic Publishers, 1992.

[Porteous 1969] I. Porteous, Topological geometry, Van Nostrand Reinhold, 1969.

[Raja 1996] A. Raja, “Object-oriented implementations of Clifford algebras in C++: a

prototype”, in [Ablamowicz 1996].

[Sommer 2001] G. Sommer (ed.), Geometric Computing with Clifford Algebras, Springer,

2001.

[Wene 1995] G. P. Wene, “The Idempotent stucture of an infinite dimensional Clifford

algebra”, pp161–164 of [Micali et al. 1989].


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