Abstract
In this paper, we establish necessary and sufficient conditions for the existence of line segments (or flats) in the sphere of the nuclear norm via the notion of simultaneous polarization and a refined expression for the subdifferential of the nuclear norm. This is then leveraged to provide (point-based) necessary and sufficient conditions for uniqueness of solutions for minimizing the nuclear norm over an affine subspace. We further establish an alternative set of sufficient conditions for uniqueness, based on the interplay of the subdifferential of the nuclear norm and the range of the problem-defining linear operator. Finally, we show how to transfer the uniqueness results for the original problem to a whole class of nuclear norm-regularized minimization problems with a strictly convex fidelity term.
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Notes
Of course, we assume throughout that this problem is feasible.
Nuclear norm minimization contains \(\ell _1\)-minimization as a special case since \(x\in {\mathbb {R}}^n\) can be identified with a diagonal matrix \(\textrm{diag}(x)\) for which \(\Vert \textrm{diag}(x)\Vert _*=\Vert x\Vert _1\).
The nuclear norms of X and \(X^T\) are equal; the linear equation \({\mathcal {A}}(X)=b\) can always be rewritten in terms of \(X^T\).
Flats, in the context of Riemannian geometry, are (uncurved) Euclidean submanifolds.
Sometimes this is also called the ‘angular’ part of the polar decomposition.
See Sect. 3.1 for a discussion of \(W(\bar{X})\).
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The authors would like to thank two anonymous referees for their extremely valuable comments which substantially improved the presentation of the material.
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Communicated by Nicolas Hadjisavvas.
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Hoheisel, T., Paquette, E. Uniqueness in Nuclear Norm Minimization: Flatness of the Nuclear Norm Sphere and Simultaneous Polarization. J Optim Theory Appl 197, 252–276 (2023). https://doi.org/10.1007/s10957-023-02167-7
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DOI: https://doi.org/10.1007/s10957-023-02167-7
Keywords
- Nuclear norm
- Singular value decomposition
- Polar decomposition
- Convex analysis
- Convex subdifferential
- Fenchel conjugate
- Low-rank minimization