[Paper Review] Generating Random Vectors in (Z/pZ)^d Via an Affine Random Process

Martin Hildebrand, Joseph McCollum|ArXiv.org|Jan 20, 2007

Random Matrices and Applications7 references4 citations

TL;DR

This paper studies affine random processes on the finite group $(\mathbb{Z}/p\mathbb{Z})^d$, defined by $\mathbf{X}_{n+1} = T\mathbf{X}_n + \mathbf{B}_n \pmod{p}$, where $\mathbf{B}_n$ is uniformly distributed over the standard basis vectors and zero. It proves that if $T$ has no complex eigenvalues of magnitude 1, then $\mathbf{X}_n$ becomes uniformly distributed in $O((\log p)^2)$ steps; if $T$ has an eigenvalue that is a root of unity, convergence requires $\Omega(p^b)$ steps for some $b \leq 2$. The results are established using Fourier analysis on finite groups and the Upper Bound Lemma.

ABSTRACT

This paper considers some random processes of the form X_{n+1}=TX_n+B_n (mod p) where B_n and X_n are random variables over (Z/pZ)^d and T is a fixed d x d integer matrix which is invertible over the complex numbers. For a particular distribution for B_n, this paper improves results of Asci to show that if T has no complex eigenvalues of length 1, then for integers p relatively prime to det(T), order (log p)^2 steps suffice to make X_n close to uniformly distributed where X_0 is the zero vector. This paper also shows that if T has a complex eigenvalue which is a root of unity, then order p^b steps are needed for X_n to get close to uniform where b is a value which may depend on T and X_0 is the zero vector.

Motivation & Objective

• To analyze the mixing time of affine random processes on $({\mathbb{Z}}/p{\mathbb{Z}})^d$ driven by a fixed matrix $T$ and i.i.d. noise $\mathbf{B}_n$.
• To determine conditions under which the process $\mathbf{X}_n$ converges to uniform distribution on $({\mathbb{Z}}/p{\mathbb{Z}})^d$.
• To improve upon prior bounds—especially those of Asci—by leveraging spectral properties of $T$, particularly the magnitude of its eigenvalues.
• To establish sharp thresholds for convergence speed based on whether $T$'s eigenvalues lie on the unit circle in $\mathbb{C}$.

Proposed method

• The process is modeled as $\mathbf{X}_{n+1} = T\mathbf{X}_n + \mathbf{B}_n \pmod{p}$ with $\mathbf{X}_0 = \mathbf{0}$ and $\mathbf{B}_n$ taking values in $\{\mathbf{0}, \mathbf{e}_1, \dots, \mathbf{e}_d\}$ with equal probability $1/(d+1)$.
• Fourier analysis on the finite abelian group $({\mathbb{Z}}/p{\mathbb{Z}})^d$ is used, with characters $\rho_{\mathbf{c}}(\mathbf{b}) = e^{2\pi i \sum b_i c_i / p}$.
• The Fourier transform of the distribution $P_n$ is analyzed via $\hat{P}_n(\mathbf{c}) = \sum_{\mathbf{s}} P_n(\mathbf{s}) \rho_{\mathbf{c}}(\mathbf{s})$, and the behavior of $|\hat{P}_n(\mathbf{c})|$ for $\mathbf{c} \neq \mathbf{0}$ is tracked over time.
• The Upper Bound Lemma is applied: $\|P_n - U\|^2 \leq \frac{1}{4} \sum_{\rho^*} d_\rho \operatorname{Tr}(\hat{P}_n(\rho) \hat{P}_n(\rho)^*)$, where $\rho^*$ runs over non-trivial irreducible representations.
• For eigenvalues of $T$ not on the unit circle, the decay of $|\hat{P}_n(\mathbf{c})|$ is shown to be geometric, leading to $O((\log p)^2)$ convergence.
• For eigenvalues that are roots of unity, the process exhibits periodic structure; the analysis shows that the projection $p(\mathbf{X}_n)$ on a suitable linear functional performs a random walk with bounded support, leading to slow mixing.

Experimental results

Research questions

• RQ1 What is the mixing time of the affine process $\mathbf{X}_{n+1} = T\mathbf{X}_n + \mathbf{B}_n \pmod{p}$ on $({\mathbb{Z}}/p{\mathbb{Z}})^d$ when $T$ has no eigenvalues of magnitude 1 in $\mathbb{C}$?
• RQ2 How does the presence of eigenvalues that are roots of unity affect the convergence rate to uniformity?
• RQ3 Can the $O((\log p)^2)$ mixing time bound be improved to $O((\log p)\log\log p)$ for general $T$?
• RQ4 What is the maximal possible value of $b$ in the $\Omega(p^b)$ lower bound when $T$ has a root-of-unity eigenvalue?
• RQ5 To what extent do the results extend to random processes on finite fields of size $p^d$?

Key findings

• If $T$ has no complex eigenvalues of magnitude 1, then $\|P_n - U\| \to 0$ as $p \to \infty$ when $n \geq C(\log p)^2$ for some constant $C > 0$ independent of $p$, provided $p$ is coprime to $\det(T)$.
• When $T$ has an eigenvalue that is a root of unity, $\|P_n - U\| \to 1$ as $p \to \infty$ if $n \leq p^b$ for some $b \leq 2$, indicating slow mixing.
• The convergence rate is bounded below by $\Omega(p^b)$ for some $b > 0$ when $T$ has a root-of-unity eigenvalue, and this $b$ depends on the structure of $T$ and the initial state.
• For $T$ with no eigenvalues of magnitude 1, the variation distance $\|P_n - U\|$ decays to zero faster than $1/p$ when $n = O((\log p)^2)$, implying rapid mixing.
• The proof relies on bounding the Fourier transform of the distribution and using the Upper Bound Lemma to relate spectral decay to total variation distance.
• The results are sharp in the sense that $b \leq 2$ in the lower bound, and this bound is tight in the sense that $b$ can be arbitrarily close to 2 depending on $T$.