从不完整和不精确测量中稳定恢复信号
Stable Signal Recovery from Incomplete and Inaccurate Measurements
AI总结：
针对不完整和含噪观测，提出通过l1正则化最小化问题实现稳定恢复，证明在均匀不确定性原理和稀疏性条件下，恢复误差受噪声水平控制。
摘要：
假设我们想从不完整且受污染的观测数据 y = A x_0 + e 中恢复一个 n 维实值向量 x_0（例如数字信号或图像）；其中 A 是一个行数远少于列数的 n×m 矩阵（n << m），e 是误差项。是否可能基于数据 y 准确恢复 x_0？为了恢复 x_0，我们考虑 l1 正则化问题 min ||x||_1 满足 ||Ax - y||_2 ≤ ε 的解 x*，其中 ε 是误差项 e 的大小。我们证明，如果 A 满足均匀不确定性原理（列单位范数）且向量 x_0 足够稀疏，那么解在噪声水平内：||x* - x_0||_2 ≤ C ε。第一个例子，假设 A 是高斯随机矩阵，那么只要 x_0 的非零个数与观测数量大致同阶，对于几乎所有这样的 A 都能稳定恢复。第二个例子，假设观测到 x_0 的少量傅里叶样本，那么只要非零个数为 n/[log m]^6 量级，对于几乎任何 p 个系数的集合都能稳定恢复。在误差项为零的情况下，恢复当然是精确的，并且这项工作实际上为早期论文中讨论的精确恢复现象提供了新的见解。该方法也解释了为什么可以非常接近地恢复近似稀疏信号。
英文摘要：
Suppose we wish to recover an n-dimensional real-valued vector x_0 (e.g. a digital signal or image) from incomplete and contaminated observations y = A x_0 + e; A is a n by m matrix with far fewer rows than columns (n << m) and e is an error term. Is it possible to recover x_0 accurately based on the data y? To recover x_0, we consider the solution x* to the l1-regularization problem min \|x\|_1 subject to \|Ax-y\|_2 <= epsilon, where epsilon is the size of the error term e. We show that if A obeys a uniform uncertainty principle (with unit-normed columns) and if the vector x_0 is sufficiently sparse, then the solution is within the noise level \|x* - x_0\|_2 \le C epsilon. As a first example, suppose that A is a Gaussian random matrix, then stable recovery occurs for almost all such A's provided that the number of nonzeros of x_0 is of about the same order as the number of observations. Second, suppose one observes few Fourier samples of x_0, then stable recovery occurs for almost any set of p coefficients provided that the number of nonzeros is of the order of n/[\log m]^6. In the case where the error term vanishes, the recovery is of course exact, and this work actually provides novel insights on the exact recovery phenomenon discussed in earlier papers. The methodology also explains why one can also very nearly recover approximately sparse signals.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
