The blind deconvolution problem seeks to recover a pair of vectors from a set of rank one bilinear measurements. We consider a natural nonsmooth formulation of the problem and show that under standard statistical assumptions, its moduli of weak convexity, sharpness, and Lipschitz continuity are all dimension independent. This phenomenon persists even when up to half of the measurements are corrupted by noise. Consequently, standard algorithms, such as the subgradient and prox-linear methods, converge at a rapid dimension-independent rate when initialized within constant relative error of the solution. We then complete the paper with a new initialization strategy, complementing the local search algorithms. The initialization procedure is both provably efficient and robust to outlying measurements. Numerical experiments, on both simulated and real data, illustrate the developed theory and methods.
The task of recovering a low-rank matrix from its noisy linear measurements plays a central role in computational science. Smooth formulations of the problem often exhibit an undesirable phenomenon: the condition number, classically defined, scales poorly with the dimension of the ambient space. In contrast, we here show that in a variety of concrete circumstances, nonsmooth penalty formulations do not suffer from the same type of ill-conditioning. Consequently, standard algorithms for nonsmooth optimization, such as subgradient and prox-linear methods, converge at a rapid dimension-independent rate when initialized within constant relative error of the solution. Moreover, nonsmooth formulations are naturally robust against outliers. Our framework subsumes such important computational tasks as phase retrieval, blind deconvolution, quadratic sensing, matrix completion, and robust PCA. Numerical experiments on these problems illustrate the benefits of the proposed approach.
The aim of this paper is to frame the stakeholder-driven system mapping approach in the context of climate change, building on stakeholder knowledge of system boundaries, key elements and interactions within a system and to introduce a decision support tool for managing and visualising this knowledge into insightful system maps with policy implications. Design/methodology/approach-This methodological framework is based on the concepts of market maps. The process of eliciting and visualising expert knowledge is facilitated by means of a reference implementation in MATLAB, which allows for designing technological innovation systems models in either a structured or a visual format. Findings-System mapping can contribute to evaluating systems for climate change by capturing knowledge of expert groups with regard to the dynamic interrelations between climate policy strategies and other system components, which may promote or hinder the desired transition to low carbon societies. Research limitations/implications-This study explores how system mapping addresses gaps in analytical tools and complements the systems of innovation framework. Knowledge elicitation, however, must be facilitated and build upon a structured framework such as technological innovation systems. Practical implications-This approach can provide policymakers with significant insight into the strengths and weaknesses of current policy frameworks based on tacit knowledge embedded in stakeholders. Social implications-The developed methodological framework aims to include societal groups in the climate policy-making process by acknowledging stakeholders' role in developing transition pathways. The system map codifies stakeholder input in a structured and transparent manner. Originality/value-This is the first study that clearly defines the system mapping approach in the frame of climate policy and introduces the first dedicated software option for researchers and decision makers to use for implementing this methodology.
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