Fast Discrete Polynomial Transforms with Applications to Data Analysis for Distance Transitive Graphs

• Authors: J. R. Driscoll;D. M. Healy, Jr.;D. N. Rockmore
• Venue: SIAM Journal on Computing
• Year: 1997

Let $\poly = \{P_0,\dots,P_{n-1}\}$ denote a set of polynomials with complex coefficients. Let $\pts = \{z_0,\dots,z_{n-1}\}\subset \cplx$ denote any set of {\it sample points}. For any $f = (f_0,\dots,f_{n-1}) \in \cplx^n$, the {\it discrete polynomial transform} of $f$ (with respect to $\poly$ and $\pts$) is defined as the collection of sums, $\{\fhat(P_0),\dots,\fhat(P_{n-1})\}$, where $\fhat(P_j) = \langle f,P_j \rangle = \sum_{i=0}^{n-1} f_iP_j(z_i)w(i)$ for some associated weight function $w$. These sorts of transforms find important applications in areas such as medical imaging and signal processing.In this paper, we present fast algorithms for computing discrete orthogonal polynomial transforms. For a system of $N$ orthogonal polynomials of degree at most $N-1$, we give an $O(N\log^2 N)$ algorithm for computing a discrete polynomial transform at an arbitrary set of points instead of the $N^2$ operations required by direct evaluation. Our algorithm depends only on the fact that orthogonal polynomial sets satisfy a three-term recurrence and thus it may be applied to any such set of discretely sampled functions. In particular, sampled orthogonal polynomials generate the vector space of functions on a distance transitive graph. As a direct application of our work, we are able to give a fast algorithm for computing subspace decompositions of this vector space which respect the action of the symmetry group of such a graph. This has direct applications to treating computational bottlenecks in the spectral analysis of data on distance transitive graphs, and we discuss this in some detail.