We study a general class of nonlinear iterative algorithms which includes power iteration, belief propagation and approximate message passing, and many forms of gradient descent. When the input is a random matrix with i.i.d. entries, we use Boolean Fourier analysis to analyze these algorithms as low-degree polynomials in the entries of the input matrix. Each symmetrized Fourier character represents all monomials with a certain shape as specified by a small graph, which we call a Fourier diagram. We prove fundamental asymptotic properties of the Fourier diagrams: over the randomness of the input, all diagrams with cycles are negligible; the tree-shaped diagrams form a basis of asymptotically independent Gaussian vectors; and, when restricted to the trees, iterative algorithms exactly follow an idealized Gaussian dynamic. We use this to prove a state evolution formula, giving a "complete" asymptotic description of the algorithm's trajectory. The restriction to tree-shaped monomials mirrors the assumption of the cavity method, a 40-year-old non-rigorous technique in statistical physics which has served as one of the most important techniques in the field. We demonstrate how to implement cavity method derivations by 1) restricting the iteration to its tree approximation, and 2) observing that heuristic cavity method-type arguments hold rigorously on the simplified iteration. Our proofs use combinatorial arguments similar to the trace method from random matrix theory. Finally, we push the diagram analysis to a number of iterations that scales with the dimension $n$ of the input matrix, proving that the tree approximation still holds for a simple variant of power iteration all the way up to $n^{\Omega(1)}$ iterations.

翻译：我们研究了一类广泛的非线性迭代算法，其涵盖了幂迭代、置信传播、近似消息传递以及多种形式的梯度下降。当输入是一个具有独立同分布元素的随机矩阵时，我们使用布尔傅里叶分析将这些算法视为输入矩阵元素中的低次多项式。每个对称化的傅里叶特征表示所有具有特定形状（由一个小图指定，我们称之为傅里叶图）的单项式。我们证明了傅里叶图的基本渐近性质：在输入的随机性下，所有包含环的图都是可忽略的；树形图构成了一组渐近独立的高斯向量基；并且，当限制在树形结构上时，迭代算法精确地遵循理想化的高斯动态。我们利用这一点证明了一个状态演化公式，从而对算法的轨迹给出了一个“完整”的渐近描述。对树形单项式的限制反映了空腔方法的假设，这是统计物理学中已有40年历史的非严格技术，也是该领域最重要的技术之一。我们展示了如何通过以下步骤实现空腔方法的推导：1）将迭代限制在其树近似上，以及 2）观察到启发式的空腔方法类论证在简化后的迭代上严格成立。我们的证明使用了类似于随机矩阵理论中迹方法的组合论证。最后，我们将图分析推至与输入矩阵维度 $n$ 成比例的迭代次数，证明对于幂迭代的一个简单变体，即使迭代次数高达 $n^{\Omega(1)}$，树近似仍然成立。