An optimization-based framework for deconvolution: theoretical guarantees and practical algorithms

反卷积是一个反问题，它包括梳理数据中不同信号源的贡献。虽然这些问题在应用科学领域很常见，但本项目将重点关注三个例子。在神经科学中，用细胞外电极记录神经元活动，测量相邻细胞的动作电位或峰电位。尖峰排序，或多波形反卷积，是识别每个细胞对应的信号，并对它们进行反卷积以揭示单独的尖峰模式的问题。超分辨率荧光显微镜允许人们从低分辨率数据中获得复杂细胞结构的高分辨率图像或视频。在计算机视觉中，模糊的图像，比如从手机中拍摄的图像，可以通过清晰图像与运动模糊内核的卷积得到很好的近似。该项目的目的是开发和分析算法来解决这些问题，特别强调调整这些方法，以便它们可以有效地应用于大量数据。最近关于稀疏性约束下的欠定线性逆问题的文献大多集中在随机感知方案上，这些方案不允许对卷积问题进行建模，例如超分辨率、神经科学中的尖峰排序或计算机视觉中的盲反卷积。该项目的主要目标是为确定性反卷积问题开发基于优化的方法，并推导出其性能和对噪声的鲁棒性的理论保证。这将需要开发新的证明技术，它不依赖于概率工具，以表征卷积算子的条件和l1范数最小化问题的最优性条件。在这些基于优化的方法的基础上，实用的算法将被设计为在大数据体系中运行，在大数据体系中，在计算上难以对数据进行统一的复杂处理。

Deconvolution is an inverse problem that consists of teasing apart the contributions of different signal sources in data. While these problems are common across the applied sciences, this project will focus on three examples. In neuroscience, recordings of neuron activity using extracellular electrodes measure the action potentials or spikes of adjacent cells. Spike sorting, or equivalently multi-waveform deconvolution, is the problem of identifying the signals corresponding to each cell and deconvolving them to reveal the separate spiking patterns. Super-resolution fluorescence microscopy allows one to obtain images or videos of complex cell structures at high resolution from low-resolution data. In computer vision, blurred pictures, such as ones taken from a cellphone, are well approximated by the convolution of a sharp image with a motion-blur kernel. The aim of this project is to develop and analyze algorithms to tackle these problems, with special emphasis on adapting these methods so that they can be applied efficiently to large amounts of data. Most recent literature on underdetermined linear inverse problems under sparsity constraints focuses on randomized sensing schemes, which do not allow to model convolution problems such as super-resolution, spike sorting in neuroscience, or blind deconvolution in computer vision. The main goal of this project is to develop optimization-based methods for deterministic deconvolution problems, as well as to derive theoretical guarantees on their performance and their robustness to noise. This will require developing novel proof techniques, which do not rely on probabilistic tools, to characterize the conditioning of convolution operators and the optimality conditions of L1-norm minimization problems. Building upon these optimization-based methods, practical algorithms will be designed to operate in big-data regimes where it is not computationally tractable to apply sophisticated processing uniformly across the data.