We consider estimating a piecewise‐constant image, or a gradient‐sparse signal on a general graph, from noisy linear measurements. We propose and study an iterative algorithm to minimize a penalized least‐squares objective, with a penalty given by the “ ${\ell }_0$‐norm” of the signal’s discrete graph gradient. The method uses a non‐convex variant of proximal gradient descent, applying the alpha‐expansion procedure to approximate the proximal mapping in each iteration, and using a geometric decay of the penalty parameter across iterations to ensure convergence. Under a cut‐restricted isometry property for the measurement design, we prove global recovery guarantees for the estimated signal. For standard Gaussian designs, the required number of measurements is independent of the graph structure, and improves upon worst‐case guarantees for total‐variation (TV) compressed sensing on the 1‐D line and 2‐D lattice graphs by polynomial and logarithmic factors respectively. The method empirically yields lower mean‐squared recovery error compared with TV regularization in regimes of moderate undersampling and moderate to high signal‐to‐noise, for several examples of changepoint signals and gradient‐sparse phantom images.

中文翻译：

---

迭代Alpha展开，用于根据线性测量估算梯度稀疏信号

我们考虑从噪声线性测量中估计分段恒定图像或一般图形上的梯度稀疏信号。我们提出并研究一种迭代算法，以最小化受罚最小二乘目标，并给出“${\ell }_0$信号的离散图梯度的“范数”。该方法使用近端梯度下降的非凸变体，在每次迭代中应用alpha扩展过程来近似于近端映射，并在迭代过程中使用惩罚参数的几何衰减以确保收敛。在用于测量设计的限制切割等距特性下，我们证明了估计信号的整体恢复保证。对于标准的高斯设计，所需的测量次数与图形结构无关，并且通过多项式和对数因子改进了对一维线图和二维晶格图上的总变化（TV）压缩感测的最坏情况保证分别。