Interesting Open Problem Related to Complexity of Computing the Fourier Transform and Group Theory
aa r X i v : . [ c s . CC ] J u l Interesting Open Problem Related to Complexity of Computing theFourier Transform and Group Theory
Nir AilonCS, Technion [email protected] ∗ July 18, 2019
Abstract
The Fourier Transform is one of the most important linear transformations used in scienceand engineering. Cooley and Tukey’s Fast Fourier Transform (FFT) from 1964 is a method forcomputing this transformation in time O ( n log n ). From a lower bound perspective, relativelylittle is known. Ailon shows in 2013 an Ω( n log n ) bound for computing the normalized FourierTransform assuming only unitary operations on pairs of coordinates is allowed. The goal of thisdocument is to describe a natural open problem that arises from this work, which is related togroup theory, and in particular to representation theory. The (discrete) normalized Fourier transform is a complex linear mapping sending an input x ∈ C n to y = F x ∈ C n , where F is an n × n unitary matrix defined by F ( k, ℓ ) = n − / e − i πkℓ/n . (1)If n is a power of 2, then Walsh-Hadamard transform is a real, orthogonal mapping H , withthe element in position ( k, ℓ ) given by: H ( k, ℓ ) = n − / ( − h k,ℓ i , (2)where h k, ℓ i is the dot-product modulo 2 of the binary representations of the integers k − ℓ − F and H (resp.) are defined by the characters of underlying abeliangroups Z /n Z (integers modulo n , under addition) and the log n dimensional binary cube ( Z / Z ) log n (log n dimensional vector space over bits). Given an input vector x ∈ C n , it is possible to compute F x and Hx (resp.) in time O ( n log n ) using the Fast Fourier-Transform [Cooley and Tukey(1964)], or the Walsh-Hadamard transform (resp.).As for computational lower bounds, it is trivial that computing both F x and Hx requires alinear number of steps, because each coordinate of the output depends on all the input coordinates.There has not been much prior work on better bounds. We refer the reader to [Ailon(2013)] fora brief history of this line of work, and concentrate on a recent lower bound. ∗ This Ongoing Work is Funded by ERC Grant SpeedInfTradeoff n log n ) operations for computing F x (or Hx ) given x , assuming that at each step the computer can perform a unitary operation affectingat most 2 rows. In other words, the algorithm, running in m steps, is viewed as a product R m · R m − . . . R · R of matrices R t , each R t a block-diagonal matrix with n − × A t : R t = i t j t i t A t (1 , A t (1 , j t A t (2 , A t (2 ,
2) 1 . . . 1 . (3)The justification for this model of computation is threefold: • Similarly to matrices of the form (3), any basic operation of a modern computer (e.g., additionof numbers) acts on only a fixed number of inputs. • The Fast Fourier-Transform, as well as the Walsh-Hadamard transform, operate in this model,and • The set of matrices of the form (3) generate the group of unitary matrices.Thus the question of computational complexity of of the Fourier transform becomes that of com-puting distances between elements of a group, namely the unitary group, with respect to a set ofgenerators that is computationally simple.Obtaining the lower bound of Ω( n log n ) in [Ailon(2013)] is done by defining a potential functionΦ for unitary matrices, as follows:Φ( U ) = − X i,j | U ( i, j ) | log | U ( i, j ) | . With this potential function, one shows that:(a) Φ(Id) = 0(b) Φ( F ) = Φ( H ) = Ω( n log n )(c) | Φ( M t ) − Φ( M t − ) | ≤
2, where M t = R t · R t − . . . R is the state of the algorithm after t steps.Indeed, if the potentail Φ grows from 0 to Ω( n log n ) changing (in absolute value) by no morethan 2 at each step, then the number of steps must be Ω( n log n ). Showing ( c ) is done using two1bservations. The first is that M t defers from M t − in at most 2 rows i t and j t , and that for eachcolumn k , due to unitarity of A t , | M t ( i t , k ) | + | M t ( j t , k ) | = | M t − ( i t , k ) | + | M t − ( j t , k ) | . The next observation is that any 4 numbers x, y, z, w satisfying x + y = z + w =: r , also satisfy | ( x log x + y log y ) − ( z log z + w log w ) | ≤ r . Combining the observations, we conclude that the total change in the potential function can be atmost X k ( | M t ( i t , k ) | + | M t ( j t , k ) | ) = 2 . The advantage of the method just described is that it reduces a computational problem to that ofcomputing distance between two elements of a group, with respect to a chosen set of generators ofthe group. We now define a more general problem within the same group theoretical setting.Consider the 2 n × n matrix G defined as G = (cid:18) F − F ∗ (cid:19) . (One may replace F with H , but we work with F henceforth.) The matrix G is skew-Hermitian.Let Id denote the 2 n × n identity matrix, and finally define for a real angle α the following matrix: X α = (cos α ) Id +(sin α ) G .
It is easy to verify that X α is unitary for all α . It is also easy to verify that X α ′ X α = X α + α ′ . (4)Also, using the potential function Φ defined above, we see thatΦ( X α ) = Θ( α n log n ) . Hence, using the argument as above, the number of steps required to compute X α must be at leastΩ( α n log n ). However, it is unreasonable that it should be possible to compute X α faster than thetime it takes to compute F , by a factor of 1 /α . Indeed, given an input x ∈ C n , we could simplyembed it as ˜ x ∈ C n by padding with n y = X α ˜ x and then retrieve y = F x from ˜ y and ˜ x by a simple arithmetic manipulation. Hence, we conjecture that the number of stepsrequired to compute X α should be not much smaller than Ω( n log n ). Ω( αn log n ) . It is possible to get a better bound than Ω( α n log n ), as follows. Instead of starting the computationat state Id and finishing at X α , we can opportunistically choose a starting point M (and finish at X α M ). The author conjectures Ω(( n log n ) / log(1 /α )) to be the correct bound.
2f we choose the state M = X π/ − α/ then it is trivial to verify that the computation ends atstate X α M , which equals X π/ α/ by (4). We then observe thatΦ( X π/ − α/ ) = Θ (cid:0) sin ( π/ − α/ · n log n (cid:1) Φ( X π/ α/ ) = Θ (cid:0) sin ( π/ α/ · n log n (cid:1) , and hence, (cid:12)(cid:12) Φ( X π/ α/ ) − Φ( X π/ − α/ ) (cid:12)(cid:12) = Ω (cid:0) (sin ( π/ α/ − sin ( π/ − α/ n log n (cid:1) = Ω ( αn log n ) . Is it possible to get a stronger lower bound than αn log n ? One approach for solving this problemmight be using group representation theory. If Ψ : U ( n ) U ( n ′ ) is any unitary representation of U ( n ), then we could define a new potential function Φ ◦ Ψ on U ( n ), and use it to obtain possiblybetter lower bounds.An interesting representation is related to determinants. We let the order k determinant repre-sentation of a unitary matrix U be the matrix Ψ kdet ( U ) of shape (cid:0) nk (cid:1) × (cid:0) nk (cid:1) , defined by(Ψ kdet ( U )) I,J = det U I,J , where I, J are subsets of size exactly k of [ n ], U I,J is the k -by- k submatrix defined by row set I and column set J . The fact that Ψ kdet ( U ) is a unitary matrix coming from a group representationis non-trivial, and we refer the reader to resources on representation theory for more details.So far I have not been able to make progress on the problem using this (quite natural) repre-sentation, but I am not convinced that this direction is futile either. α = π/ is Interesting Note that although the main problem proposed here is to understand the asymptotic behviour ofthe complexity of X α , as α tends to 0, even the case of finding a lower bound for computation of X π/ is not trivial, in the sense that it is not clear how (and whether it is at all possible) to get abound better than √ n log n , which is the best possible using the “vanilla” entropy function Φ. References [Ailon(2013)] Nir Ailon. A lower bound for Fourier transform computation in a linear model over2x2 unitary gates using matrix entropy.
Chicago J. of Theo. Comp. Sci. , 2013.[Cooley and Tukey(1964)] J. W Cooley and J. W Tukey. An algorithm for the machine computationof complex Fourier series.