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A generalized FFT for Clifford algebrasPaul Leopardi
School of Mathematics, University of New South Wales.
For presentation at ICIAM, Sydney, 2003-07-07.
8 16
8
16
Real representation matrix R2
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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.
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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|).
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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.
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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))
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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.
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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
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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).
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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 .
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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. �
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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−).
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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.
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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. �
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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 .
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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.
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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. �
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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).
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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 .
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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. �
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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. �
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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.
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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].
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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].
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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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