Abstract
Many problems in medical image reconstruction and machine learning can be formulated as nonconvex set theoretic feasibility problems. Among efficient methods that can be put to work in practice, successive projection algorithms have received a lot of attention in the case of convex constraint sets. In the present work, we provide a theoretical study of a general projection method in the case where the constraint sets are nonconvex and satisfy some other structural properties. We apply our algorithm to image recovery in magnetic resonance imaging (MRI) and to a signal denoising in the spirit of Cadzow’s method.
Highlights
Many problems in applied mathematics, engineering, statistics and machine learning can be reduced to finding a point in the intersection of some subsets of a real separable Hilbert space H
Let (Si )i∈ I be a finite family of proximinal subsets [1] of H with a non-empty intersection, we address the problem of finding a point in the intersection of the sets (Si )i∈ I using a successive projection method
Note that one can relax the constraint Card( In ) > 1 in the case where μn is well defined at every iteration, allowing to recover the cyclic projection method
Summary
Many problems in applied mathematics, engineering, statistics and machine learning can be reduced to finding a point in the intersection of some subsets of a real separable Hilbert space H. Important examples are: model order reduction [6], controller design [7], tensor analysis [8], image recovery [9], electrical capacitance tomography [10], MRI [11,12], and stabilisation of quantum systems and application to quantum computation [13,14] Extension of this problem to the nonconvex setting has many applications, related to sparse estimation and low rank constraints, such as in control theory [15], signal denoising [16], phase retrieval [17], structured matrix estimation [18,19] and has great potential impact on the design of scalable algorithm in many machine learning problems such as Deep Neural.
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