快速傅里叶优化：稀疏性至关重要
Fast Fourier Optimization: Sparsity Matters
AI总结：
本文通过将快速傅里叶变换（FFT）的递归结构转化为优化问题中的稀疏约束，提出了一种高效的快速傅里叶优化方法，并在高对比度成像等实际问题中展示了其显著优势。
摘要：
许多有趣且实际重要的优化问题，涉及光学、信号处理、雷达和声学等领域，都包含对函数傅里叶变换的约束。众所周知，快速傅里叶变换（FFT）是一种递归算法，可以显著提高离散傅里叶变换的计算效率。然而，由于其递归性，它难以嵌入到线性优化问题中。在本文中，我们解释了快速傅里叶变换背后的主要思想，并展示了如何以使其能够编码为优化问题中的约束的方式进行调整。我们展示了一个来自高对比度成像领域的实际问题。在这个问题上，显著的改进转化为能够解决具有更精细离散点网格的问题的能力。正如我们将要展示的，一般来说，优化约束的“快速傅里叶”版本会产生一个更大但更稀疏的约束矩阵，因此可以将快速傅里叶变换视为一种使优化问题中的约束稀疏化的方法，这通常是一件好事。
英文摘要：
Many interesting and fundamentally practical optimization problems, ranging from optics, to signal processing, to radar and acoustics, involve constraints on the Fourier transform of a function. It is well-known that the {\em fast Fourier transform} (fft) is a recursive algorithm that can dramatically improve the efficiency for computing the discrete Fourier transform. However, because it is recursive, it is difficult to embed into a linear optimization problem. In this paper, we explain the main idea behind the fast Fourier transform and show how to adapt it in such a manner as to make it encodable as constraints in an optimization problem. We demonstrate a real-world problem from the field of high-contrast imaging. On this problem, dramatic improvements are translated to an ability to solve problems with a much finer grid of discretized points. As we shall show, in general, the "fast Fourier" version of the optimization constraints produces a larger but sparser constraint matrix and therefore one can think of the fast Fourier transform as a method of sparsifying the constraints in an optimization problem, which is usually a good thing.
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