[Paper Review] Complexity of the Fourier transform on the Johnson graph

Rodrigo Iglesias, Mauro Natale|arXiv (Cornell University)|Apr 20, 2017

Tensor decomposition and applications12 references3 citations

TL;DR

This paper presents a fast Fourier transform (FFT)-style algorithm for computing the Fourier transform on the Johnson graph $J(n,k)$, leveraging a factorization of the change-of-basis matrix into $n-1$ sparse orthogonal matrices, each with at most two non-zero entries per column. The method reduces the complexity from $\binom{n}{k}^2$ operations to $O(n\binom{n}{k})$, significantly improving prior bounds when $k$ dominates $\sqrt{n}$, and enables efficient computation of isotypic projections and spectral components.

ABSTRACT

The set $X$ of $k$-subsets of an $n$-set has a natural graph structure where two $k$-subsets are connected if and only if the size of their intersection is $k-1$. This is known as the Johnson graph. The symmetric group $S_n$ acts on the space of complex functions on $X$ and this space has a multiplicity-free decomposition as sum of irreducible representations of $S_n$, so it has a well-defined Gelfand-Tsetlin basis up to scalars. The Fourier transform on the Johnson graph is defined as the change of basis matrix from the delta function basis to the Gelfand-Tsetlin basis. The direct application of this matrix to a generic vector requires $\binom{n}{k}^2$ arithmetic operations. We show that --in analogy with the classical Fast Fourier Transform on the discrete circle-- this matrix can be factorized as a product of $n-1$ orthogonal matrices, each one with at most two nonzero elements in each column. This factorization shows that the number of arithmetic operations required to apply this matrix to a generic vector is bounded above by $2(n-1) \binom{n}{k}$. As a consequence, we show that the problem of computing all the weights of the irreducible components of a given function can be solved in $O(n \binom{n}{k})$ operations, improving the previous bound $O(k^2 \binom{n}{k})$ when $k$ asymptotically dominates $\sqrt{n}$ in a non-uniform model of computation. The same improvement is achieved for the problem of computing the isotypic projection onto a single component. The proof is based on the construction of $n-1$ intermediate bases, each one parametrized by certain pairs composed by a standard Young tableau and a word. The parametrization of each basis is obtained via the Robinson-Schensted insertion algorithm.

Motivation & Objective

• To develop an efficient algorithm for computing the non-abelian Fourier transform on the Johnson graph $J(n,k)$, which arises in spectral analysis of ranked data and coding theory.
• To reduce the algebraic complexity of computing isotypic projections and spectral components of functions on $X = \binom{[n]}{k}$, especially when $k$ grows faster than $\sqrt{n}$.
• To generalize the FFT paradigm—originally for abelian groups like $\mathbb{Z}/2^n\mathbb{Z}$—to non-abelian group actions on combinatorial structures such as the Johnson graph.
• To construct intermediate orthonormal bases parametrized by standard Young tableaux and binary words, enabling a hierarchical, sparse factorization of the Fourier transform matrix.

Proposed method

• The authors define a sequence of $n-1$ intermediate orthonormal bases $B_1, \dots, B_{n-1}$, each parametrized by pairs of a standard Young tableau of height at most two and a word in $\{1,2\}$.
• Each basis transition matrix $[B_{i-1}]_{B_i}$ is constructed to have at most two non-zero entries per column, ensuring low computational cost per step.
• The construction relies on the Robinson-Schensted insertion algorithm, where each horizontal line in the basis labeling corresponds to the sequence of tableaux generated by inserting letters of a binary word.
• The full Fourier transform is realized as a product of $n-1$ such sparse matrices, enabling application to a vector in $O(n\binom{n}{k})$ operations.
• The inverse transform is similarly computed via the same factorization, allowing efficient computation of both forward and inverse transforms.
• The method enables efficient computation of isotypic projections by transforming to the Gelfand-Tsetlin basis, zeroing out components not in the desired isotypic subspace, and transforming back.

Experimental results

Research questions

• RQ1 Can the non-abelian Fourier transform on the Johnson graph be computed in sub-quadratic time relative to the dimension $\binom{n}{k}$, analogous to the classical FFT on cyclic groups?
• RQ2 What structural properties of the symmetric group action on $\binom{[n]}{k}$ allow for a hierarchical, sparse factorization of the change-of-basis matrix to the Gelfand-Tsetlin basis?
• RQ3 How can the Robinson-Schensted insertion algorithm be used to construct intermediate bases that support a fast transform with $O(n\binom{n}{k})$ complexity?
• RQ4 Does the improved complexity bound of $O(n\binom{n}{k})$ for computing isotypic components or their norms outperform existing methods, particularly when $k \gg \sqrt{n}$?
• RQ5 Is there an efficient algorithm to precompute the $n-1$ sparse transition matrices $[B_{i-1}]_{B_i}$ in time polynomial in $n$ and $\binom{n}{k}$?

Key findings

• The Fourier transform on the Johnson graph can be computed in $O(n\binom{n}{k})$ arithmetic operations, a significant improvement over the naive $\binom{n}{k}^2$ bound.
• The transform is factorized as a product of $n-1$ orthogonal matrices, each with at most two non-zero entries per column, enabling efficient application to vectors.
• The isotypic projection onto a single irreducible component can be computed in $O(n\binom{n}{k})$ operations after precomputing the transition matrices.
• The computation of all squared weights $\|f_a\|^2$ of the isotypic components can be performed in at most $(2n-1)\binom{n}{k}$ operations.
• The method improves upon the prior $O(k^2\binom{n}{k})$ bound for computing isotypic components when $k$ asymptotically dominates $\sqrt{n}$ in the non-uniform computation model.
• The authors conjecture that the transition matrices $[B_{i-1}]_{B_i}$ can be precomputed in $O(n^c \binom{n}{k})$ time for some constant $c$, though no such algorithm is provided.