Johnson图上傅里叶变换的复杂性
Complexity of the Fourier transform on the Johnson graph
摘要：
$n$ -集合的$k$ -子集的集合$X$具有自然图结构，其中两个$k$ -子集连通当且仅当它们的交集的大小是$k-1$。这被称为约翰逊图。对称群$S_n$作用在$X$上的复变函数空间上，并且该空间具有作为$S_n$的不可约表示之和的无重性分解，因此它具有直到标量的良好定义的Gelfand-Tsetlin基。Johnson图上的傅里叶变换定义为基矩阵从delta函数基到Gelfand-Tsetlin基的变换。将该矩阵直接应用于通用向量需要$\binom{n}{k}^2$算术运算。我们表明，在类比与经典的快速傅里叶变换的离散圆-这个矩阵可以分解为一个产品的$n-1$正交矩阵，每一个最多有两个非零元素在每列。这种因式分解表明，将该矩阵应用于通用向量所需的算术运算的数量上界为$2(n-1) \binom{n}{k}$。因此，我们表明，一个给定的函数的不可约分量的所有权重的计算问题可以解决在$O(n \binom{n}{k})$操作，提高以前的界$O(k^2 \binom{n}{k})$时，$k$渐近主导$\sqrt{n}$在一个非均匀的计算模型。对于计算到单个组件上的同型投影的问题，实现了相同的改进。证明是基于$n-1$中间基地的建设，每一个参数化的某些对组成的一个标准的杨表和一个字。通过Robinson-Schensted插入算法获得每个基的参数化。
英文摘要：
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.
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