Finding the Largest Low-Rank Clusters With Ky Fan $2$-$k$-Norm and $\ell_1$-Norm

Doan, Xuan Vinh; Vavasis, Stephen A.

doi:10.1137/140962097

Cited by 18 publications

(20 citation statements)

References 14 publications

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“…The following proposition on the subgradiential of · r has been shown in [44] for r ∈ N. It is straightforward to extend it to the real-valued case. It is readily seen that Proposition 4 is equivalent to ∂ A r * = 1 A r conv(svd r (A)).…”

Section: A Subdifferentialsmentioning

confidence: 89%

Low-Rank Optimization With Convex Constraints

Grussler

Rantzer

Giselsson

2018

IEEE Trans. Automat. Contr.

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The problem of low-rank approximation with convex constraints, which appears in data analysis, system identification, model order reduction, low-order controller design and low-complexity modelling is considered. Given a matrix, the objective is to find a low-rank approximation that meets rank and convex constraints, while minimizing the distance to the matrix in the squared Frobenius norm. In many situations, this non-convex problem is convexified by nuclear norm regularization. However, we will see that the approximations obtained by this method may be far from optimal. In this paper, we propose an alternative convex relaxation that uses the convex envelope of the squared Frobenius norm and the rank constraint. With this approach, easily verifiable conditions are obtained under which the solutions to the convex relaxation and the original non-convex problem coincide. An SDP representation of the convex envelope is derived, which allows us to apply this approach to several known problems. Our example on optimal low-rank Hankel approximation/model reduction illustrates that the proposed convex relaxation performs consistently better than nuclear norm regularization and may outperform balanced truncation.

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Section: A Subdifferentialsmentioning

confidence: 89%

Low-Rank Optimization With Convex Constraints

Grussler

Rantzer

Giselsson

2018

IEEE Trans. Automat. Contr.

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“…The convergence results show that the DCA algorithm improves the objective of the ratio minimization problem (5). The DCA algorithm can stop if…”

Section: Difference Of Convex Algorithmsmentioning

confidence: 96%

“…Given the result in Theorem 1, the exact recovery of a low-rank matrix using (5) or (6) relies on the uniqueness of the low-rank solution of A(X) = b. Recht et al [11] generalized the restricted isometry property of vectors introduced by Candès and Tao [4] to matrices and use it to provide sufficient conditions on the uniqueness of these solutions.…”

Section: Ky Fan 2-k-norm-based Modelsmentioning

confidence: 99%

“…When k = 1, the Ky Fan 2-k-norm is the spectral norm, A = σ 1 (A), the largest singular value of A, whose dual norm is the nuclear norm. Similar to the nuclear norm, the dual Ky Fan 2-knorm with k > 1 can be used to compute the k-approximation of a matrix A (Proposition 2.9, [5]), which demonstrates its low-rank property. Motivated by this low-rank property of the (dual) Ky Fan 2-k-norm, which is more general than that of the nuclear norm, and its usage in other applications, in this paper, we propose a Ky Fan 2-k-norm-based non-convex approach for the matrix recovery problem which aims to recover matrices which are not recoverable by the convex relaxation formulation (1).…”

Section: Introductionmentioning

confidence: 99%

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Low-Rank Matrix Recovery with Ky Fan 2-k-Norm

Doan

Vavasis

2019

Advances in Intelligent Systems and Computing

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“…For other values of k, the representation of B • s,k can be significantly more complicated, e.g. see [8]. 6.…”

Section: Variational Gram Functionsmentioning

confidence: 99%

Variational Properties of Matrix Functions via the Generalized Matrix-Fractional Function

Burke¹,

Gao²,

Hoheisel³

2019

SIAM J. Optim.

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We show that many important convex matrix functions can be represented as the partial infimal projection of the generalized matrix fractional (GMF) and a relatively simple convex function. This representation provides conditions under which such functions are closed and proper as well as formulas for the ready computation of both their conjugates and subdifferentials. Particular instances yield all weighted Ky Fan norms and squared gauges on R n×m , and as an example we show that all variational Gram functions are representable as squares of gauges. Other instances yield weighted sums of the Frobenius and nuclear norms. The scope of applications is large and the range of variational properties and insight is fascinating and fundamental. An important byproduct of these representations is that they lay the foundation for a smoothing approach to many matrix functions on the interior of the domain of the GMF function.

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Finding the Largest Low-Rank Clusters With Ky Fan $2$-$k$-Norm and $\ell_1$-Norm

Cited by 18 publications

References 14 publications

Low-Rank Optimization With Convex Constraints

Low-Rank Optimization With Convex Constraints

Low-Rank Matrix Recovery with Ky Fan 2-k-Norm

Variational Properties of Matrix Functions via the Generalized Matrix-Fractional Function

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