ADMiRA: 最小秩近似的原子分解
ADMiRA: Atomic Decomposition for Minimum Rank Approximation
AI总结：
针对低秩矩阵压缩感知的逆问题，提出基于原子分解的贪婪算法ADMiRA，扩展了向量情况下的CoSaMP算法，在近似低秩和含噪测量下给出性能保证，并在矩阵补全中表现出竞争力。
摘要：
我们处理压缩感知中低秩矩阵的逆问题。我们的方法是将逆问题表述为具有指定目标秩的近似问题。通过对目标秩进行简单搜索，即可得到满足给定数据近似界限的最小秩解。我们提出了一种原子分解，在稀疏向量和低秩矩阵的简约表示之间建立了类比。通过这种原子分解，将向量情况下解决逆问题的高效贪婪算法扩展到矩阵情况。特别地，我们提出了一种高效且有保证的算法，名为ADMiRA，它扩展了向量情况下的对应算法CoSaMP。性能保证以秩受限等距性质给出，并限制了迭代次数和近似解的误差，适用于解近似低秩且测量含噪的一般情况。对于稀疏测量算子（如矩阵补全问题中出现的算子），ADMiRA的计算量与测量数成线性关系。矩阵补全问题的数值实验表明，尽管该情况下的测量算子不满足秩受限等距性质，ADMiRA仍是一种有竞争力的矩阵补全算法。
英文摘要：
We address the inverse problem that arises in compressed sensing of a low-rank matrix. Our approach is to pose the inverse problem as an approximation problem with a specified target rank of the solution. A simple search over the target rank then provides the minimum rank solution satisfying a prescribed data approximation bound. We propose an atomic decomposition that provides an analogy between parsimonious representations of a sparse vector and a low-rank matrix. Efficient greedy algorithms to solve the inverse problem for the vector case are extended to the matrix case through this atomic decomposition. In particular, we propose an efficient and guaranteed algorithm named ADMiRA that extends CoSaMP, its analogue for the vector case. The performance guarantee is given in terms of the rank-restricted isometry property and bounds both the number of iterations and the error in the approximate solution for the general case where the solution is approximately low-rank and the measurements are noisy. With a sparse measurement operator such as the one arising in the matrix completion problem, the computation in ADMiRA is linear in the number of measurements. The numerical experiments for the matrix completion problem show that, although the measurement operator in this case does not satisfy the rank-restricted isometry property, ADMiRA is a competitive algorithm for matrix completion.
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