This paper introduces new solvers for the computation of low-rank approximate solutions to large-scale linear problems, with a particular focus on the regularization of linear inverse problems. Although Krylov methods incorporating explicit projections onto low-rank subspaces are already used for well-posed systems that arise from discretizing stochastic or time-dependent PDEs, we are mainly concerned with algorithms that solve the so-called nuclear norm regularized problem, where a suitable nuclear norm penalization on the solution is imposed alongside a fit-to-data term expressed in the 2-norm: this has the effect of implicitly enforcing low-rank solutions. By adopting an iteratively reweighted norm approach, the nuclear norm regularized problem is reformulated as a sequence of quadratic problems, which can then be efficiently solved using Krylov methods, giving rise to an inner-outer iteration scheme. Our approach differs from the other solvers available in the literature in that: (a) Kronecker product properties are exploited to define the reweighted 2-norm penalization terms; (b) efficient preconditioned Krylov methods replace gradient (projection) methods; (c) the regularization parameter can be efficiently and adaptively set along the iterations. Furthermore, we reformulate within the framework of flexible Krylov methods both the new inner-outer methods for nuclear norm regularization and some of the existing Krylov methods incorporating low-rank projections. This results in an even more computationally efficient (but heuristic) strategy, that does not rely on an inner-outer iteration scheme. Numerical experiments show that our new solvers are competitive with other state-of-the-art solvers for low-rank problems, and deliver reconstructions of increased quality with respect to other classical Krylov methods.

中文翻译：

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用于低秩正则化的 Krylov 方法

本文介绍了用于计算大规模线性问题的低秩近似解的新求解器，特别关注线性逆问题的正则化。尽管将显式投影结合到低秩子空间的 Krylov 方法已经用于由离散化随机或时间相关 PDE 产生的适定系统，但我们主要关注解决所谓的核范数正则化问题的算法，其中合适的对解决方案的核规范惩罚与 2 范数中表达的数据拟合项一起强加：这具有隐式强制低秩解决方案的效果。通过采用迭代重新加权的范数方法，核范数正则化问题被重新表述为一系列二次问题，然后可以使用 Krylov 方法有效地解决该问题，从而产生内外迭代方案。我们的方法与文献中可用的其他求解器的不同之处在于：（a）利用 Kronecker 乘积属性来定义重新加权的 2 范数惩罚项；(b) 高效的预处理 Krylov 方法取代了梯度（投影）方法；(c) 正则化参数可以沿着迭代有效且自适应地设置。此外，我们在灵活的 Krylov 方法的框架内重新制定了用于核范数正则化的新的内部-外部方法和一些包含低秩投影的现有 Krylov 方法。这导致了计算效率更高（但启发式）的策略，该策略不依赖于内外迭代方案。