4 research outputs found
A Lower Bound for Fourier Transform Computation in a Linear Model Over 2x2 Unitary Gates Using Matrix Entropy
Obtaining a non-trivial (super-linear) lower bound for computation of the
Fourier transform in the linear circuit model has been a long standing open
problem. All lower bounds so far have made strong restrictions on the
computational model. One of the most well known results, by Morgenstern from
1973, provides an lower bound for the \emph{unnormalized}
FFT when the constants used in the computation are bounded. The proof uses a
potential function related to a determinant. The determinant of the
unnormalized Fourier transform is , and thus by showing that it can
grow by at most a constant factor after each step yields the result.
This classic result, however, does not explain why the \emph{normalized}
Fourier transform, which has a unit determinant, should take
steps to compute. In this work we show that in a layered linear circuit model
restricted to unitary gates, one obtains an lower
bound. The well known FFT works in this model. The main argument concluded from
this work is that a potential function that might eventually help proving the
conjectured lower bound for computation of Fourier transform
is not related to matrix determinant, but rather to a notion of matrix entropy
Interesting Open Problem Related to Complexity of Computing the Fourier Transform and Group Theory
The Fourier Transform is one of the most important linear transformations
used in science and engineering. Cooley and Tukey's Fast Fourier Transform
(FFT) from 1964 is a method for computing this transformation in time . From a lower bound perspective, relatively little is known. Ailon shows in
2013 an bound for computing the normalized Fourier Transform
assuming only unitary operations on pairs of coordinates is allowed. The goal
of this document is to describe a natural open problem that arises from this
work, which is related to group theory, and in particular to representation
theory
An n\log n Lower Bound for Fourier Transform Computation in the Well Conditioned Model
Obtaining a non-trivial (super-linear) lower bound for computation of the
Fourier transform in the linear circuit model has been a long standing open
problem for over 40 years.
An early result by Morgenstern from 1973, provides an
lower bound for the unnormalized Fourier transform when the constants used in
the computation are bounded. The proof uses a potential function related to a
determinant. The result does not explain why the normalized Fourier transform
(of unit determinant) should be difficult to compute in the same model. Hence,
the result is not scale insensitive.
More recently, Ailon (2013) showed that if only unitary 2-by-2 gates are
used, and additionally no extra memory is allowed, then the normalized Fourier
transform requires steps. This rather limited result is also
sensitive to scaling, but highlights the complexity inherent in the Fourier
transform arising from introducing entropy, unlike, say, the identity matrix
(which is as complex as the Fourier transform using Morgenstern's arguments,
under proper scaling).
In this work we extend the arguments of Ailon (2013). In the first extension,
which is also the main contribution, we provide a lower bound for computing any
scaling of the Fourier transform. Our restriction is that, the composition of
all gates up to any point must be a well conditioned linear transformation. The
lower bound is , where is the uniform condition
number. Second, we assume extra space is allowed, as long as it contains
information of bounded norm at the end of the computation.
The main technical contribution is an extension of matrix entropy used in
Ailon (2013) for unitary matrices to a potential function computable for any
matrix, using Shannon entropy on "quasi-probabilities"
Tighter Fourier Transform Complexity Tradeoffs
The Fourier Transform is one of the most important linear transformations
used in science and engineering. Cooley and Tukey's Fast Fourier Transform
(FFT) from 1964 is a method for computing this transformation in time . Achieving a matching lower bound in a reasonable computational model is
one of the most important open problems in theoretical computer science.
In 2014, improving on his previous work, Ailon showed that if an algorithm
speeds up the FFT by a factor of , then it must rely on
computing, as an intermediate "bottleneck" step, a linear mapping of the input
with condition number . Our main result shows that a factor
speedup implies existence of not just one but -ill conditioned
bottlenecks occurring at different steps, each causing information
from independent (orthogonal) components of the input to either overflow or
underflow. This provides further evidence that beating FFT is hard. Our result
also gives the first quantitative tradeoff between computation speed and
information loss in Fourier computation on fixed word size architectures. The
main technical result is an entropy analysis of the Fourier transform under
transformations of low trace, which is interesting in its own right