Beyond Moore-Penrose Part I: Generalized Inverses that Minimize Matrix Norms

Beyond Moore-Penrose Part I: Generalized Inverses that Minimize Matrix Norms
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超越摩尔-彭罗斯第一部分：最小化矩阵范数的广义逆

DOI：
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发表时间：
2017
期刊：
arXiv.org
影响因子：
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通讯作者：
R. Gribonval
R. Gribonval
中科院分区：
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文献类型：
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作者：
Ivan Dokmanić;R. Gribonval

文献摘要

被引文献

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这是第一篇论文的两个长系列中，我们研究线性广义逆，最小化矩阵范数。这种广义逆是著名的摩尔-彭罗斯伪逆（MPP），它恰好最小化Frobenius范数。释放与Frobenius最优性相关的自由度使我们能够推广其他有趣的性质。在第一部分中，我们研究范数最小化广义逆的基本性质，特别是唯一性和与MPP的关系。我们首先表明，MPP最大限度地减少了许多规范以外的酉不变，从而进一步支持其作为一个强大的选择在许多情况下的作用。然后，我们集中在一些规范，一般不最小化的MPP，但其最小化是相关的线性逆问题和稀疏表示。特别地，我们研究混合范数和诱导的$ell^p g $ norms.一个有趣的代表是稀疏伪逆，我们在第二部分中更详细地研究。接下来，我们将注意力从范数转移到具有有趣行为的矩阵。我们展示了一类广义逆总是MPP-即使是正常的结果在不同的逆规范和一类，其中许多广义逆相吻合，但不与MPP。最后，我们讨论了范数最小化广义逆的有效计算。
This is the first paper of a two-long series in which we study linear generalized inverses that minimize matrix norms. Such generalized inverses are famously represented by the Moore-Penrose pseudoinverse (MPP) which happens to minimize the Frobenius norm. Freeing up the degrees of freedom associated with Frobenius optimality enables us to promote other interesting properties. In this Part I, we look at the basic properties of norm-minimizing generalized inverses, especially in terms of uniqueness and relation to the MPP. We first show that the MPP minimizes many norms beyond those unitarily invariant, thus further bolstering its role as a robust choice in many situations. We then concentrate on some norms which are generally not minimized by the MPP, but whose minimization is relevant for linear inverse problems and sparse representations. In particular, we look at mixed norms and the induced $ell^p ightarrow ell^q$ norms. An interesting representative is the sparse pseudoinverse which we study in much more detail in Part II. Next, we shift attention from norms to matrices with interesting behaviors. We exhibit a class whose generalized inverse is always the MPP—even for norms that normally result in different inverses—and a class for which many generalized inverses coincide, but not with the MPP. Finally, we discuss efficient computation of norm-minimizing generalized inverses.