The structure of completely positive matrices according to their CP-rank and CP-plus-rank

Bomze, Immanuel M.; Dickinson, Peter J. C.; Still, Georg J.

doi:10.1016/j.laa.2015.05.021

Cited by 16 publications

(14 citation statements)

References 15 publications

(19 reference statements)

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“…In particular, the rate of success for ModMAP and SpFeasDC decreases to zero when

ω

to 1, that is, when

A_{ω}

becomes nearly identical to A . For this experiment, we set, according to the lower bounds for the cp‐rank derived in Reference 5, r : = 11 for

ω : = 1

and r : = 12 otherwise. We present in Tables 7 and 8 the numerical performances of the algorithms applied to the nonnegative factorization of the matrices A 0.99 and A 1.00 = A , respectively.…”

Section: Particular Instances and Numerical Experimentsmentioning

confidence: 99%

“…The problem of computing the cp‐rank of a matrix is in general open (see 4 ). However, upper bounds for the cp‐rank have been derived by Bomze, Dickinson and Still in Reference 5, Theorem 4.1, namely, for

A \in {𝒞 𝒫}_{n}

, it holds

cpr (A) \leq {cp}_{n} : = \{\begin{cases} n & for n \in ({}, 2, 3, 4), \\ \frac{1}{2} n ((), n + 1) prefix- 4 & for n \geq 5 . \end{cases}

Moreover, if

A \in int ({𝒞 𝒫}_{n})

, then

{cpr}^{+} (A) \leq {cp}_{n}^{+} : = \{\begin{cases} n + 1 & for n \in ({}, 2, 3, 4), \\ \frac{1}{2} n ((), n + 1) prefix- 3 & for n \geq 5 . \end{cases}

Notice that there exists matrices

A \in int ({𝒞 𝒫}_{n})

such that

cpr (A) \neq {cpr}^{+} (A)

.…”

Section: Introductionmentioning

confidence: 99%

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Factorization of completely positive matrices using iterative projected gradient steps

Boţ

Nguyen

2021

Numerical Linear Algebra App

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We aim to factorize a completely positive matrix by using an optimization approach which consists in the minimization of a nonconvex smooth function over a convex and compact set. To solve this problem we propose a projected gradient algorithm with parameters that take into account the effects of relaxation and inertia. Both projection and gradient steps are simple in the sense that they have explicit formulas and do not require inner loops. Furthermore, no expensive procedure to find an appropriate starting point is needed. The convergence analysis shows that the whole sequence of generated iterates converges to a critical point of the objective function and it makes use of the Łojasiewicz inequality. Its rate of convergence expressed in terms of the Łojasiewicz exponent of a regularization of the objective function is also provided. Numerical experiments demonstrate the efficiency of the proposed method, in particular in comparison to other factorization algorithms, and emphasize the role of the relaxation and inertial parameters.

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“…In particular, the rate of success for ModMAP and SpFeasDC decreases to zero when

ω

to 1, that is, when

A_{ω}

becomes nearly identical to A . For this experiment, we set, according to the lower bounds for the cp‐rank derived in Reference 5, r : = 11 for

ω : = 1

and r : = 12 otherwise. We present in Tables 7 and 8 the numerical performances of the algorithms applied to the nonnegative factorization of the matrices A 0.99 and A 1.00 = A , respectively.…”

Section: Particular Instances and Numerical Experimentsmentioning

confidence: 99%

A \in {𝒞 𝒫}_{n}

, it holds

cpr (A) \leq {cp}_{n} : = \{\begin{cases} n & for n \in ({}, 2, 3, 4), \\ \frac{1}{2} n ((), n + 1) prefix- 4 & for n \geq 5 . \end{cases}

Moreover, if

A \in int ({𝒞 𝒫}_{n})

, then

{cpr}^{+} (A) \leq {cp}_{n}^{+} : = \{\begin{cases} n + 1 & for n \in ({}, 2, 3, 4), \\ \frac{1}{2} n ((), n + 1) prefix- 3 & for n \geq 5 . \end{cases}

Notice that there exists matrices

A \in int ({𝒞 𝒫}_{n})

such that

cpr (A) \neq {cpr}^{+} (A)

.…”

Section: Introductionmentioning

confidence: 99%

Factorization of completely positive matrices using iterative projected gradient steps

Boţ

Nguyen

2021

Numerical Linear Algebra App

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“…We would like to understand the cp-ranks and cp `-ranks in the interior of CP 5 . It is known [8,Theorem 5.1] that they agree generically on an open subset of the interior of CP 5 . For that, we consider a few extra subsets of CP n .…”

Section: 7mentioning

confidence: 99%

The cone of $5\times 5$ completely positive matrices

Pfeffer¹,

Samper²

2021

Preprint

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We study the cone of completely positive (cp) matrices for the first interesting case n " 5. This is a semialgebraic set for which the polynomial equalities and inequlities that define its boundary can be derived. We characterize the different loci of this boundary and we examine the two open sets with cp-rank 5 or 6. A numerical algorithm is presented that is fast and able to compute the cp-factorization even for matrices in the boundary. With our results, many new example cases can be produced and several insightful numerical experiments are performed that illustrate the difficulty of the cp-factorization problem.

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“…Steps towards clarifying if it is tight for n = 6 have been taken in [18,28]. The second lower bound appears in [7], where it is also observed that it is superior to the first one for n ≥ 15, and follows from a bound given in [10,Theorem 2.2], where also some possible improvements are indicated, which however only affect the constant 3 2 . In this article we will show…”

mentioning

confidence: 99%

Lower bounds for maximal cp-ranks of completely positive matrices and tensors

Schachinger

2020

ELA

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Let $p_n$ denote the maximal cp-rank attained by completely positive $n\times n$ matrices. Only lower and upper bounds for $p_n$ are known, when $n\ge6$, but it is known that $p_n=\frac{n^2}2\big(1+o(1)\big)$, and the difference of the current best upper and lower bounds for $p_n$ is of order $\mathcal{O}\big(n^{3/2}\big)$. In this paper, that gap is reduced to $\mathcal{O}\big(n\log\log n\big)$. To achieve this result, a sequence of generalized ranks of a given matrix A has to be introduced. Properties of that sequence and its generating function are investigated. For suitable A, the $d$th term of that sequence is the cp-rank of some completely positive tensor of order $d$. This allows the derivation of asymptotically matching lower and upper bounds for the maximal cp-rank of completely positive tensors of order $d>2$ as well.

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The structure of completely positive matrices according to their CP-rank and CP-plus-rank

Cited by 16 publications

References 15 publications

Factorization of completely positive matrices using iterative projected gradient steps

Factorization of completely positive matrices using iterative projected gradient steps

The cone of $5\times 5$ completely positive matrices

Lower bounds for maximal cp-ranks of completely positive matrices and tensors

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