Hankel Tensor Decompositions and Ranks

Nie, Jiawang; Ye, Ke

doi:10.1137/18m1168285

Cited by 15 publications

(8 citation statements)

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“…Moment and localizing matrices are important tools for solving polynomial optimization [12,16,25,43]. They are also useful in tensor decompositions [44,47]. We refer to [28,29,31,32] for the books and surveys about polynomial and moment optimization.…”

Section: If the Cardinality |Suppmentioning

confidence: 99%

The Multi-Objective Polynomial Optimization

Nie,

Yang

2021

Preprint

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The multi-objective optimization is to optimize several objective functions over a common feasible set. Since the objectives usually do not share a common optimizer, people often consider (weakly) Pareto points. This paper studies multi-objective optimization problems that are given by polynomial functions. First, we study the convex geometry for (weakly) Pareto values and give a convex representation for them. Linear scalarization problems (LSPs) and Chebyshev scalarization problems (CSPs) are typical approaches for getting (weakly) Pareto points. For LSPs, we show how to use tight relaxations to solve them, how to detect existence or nonexistence of proper weights. For CSPs, we show how to solve them by moment relaxations. Moreover, we show how to check if a given point is a (weakly) Pareto point or not and how to detect existence or nonexistence of (weakly) Pareto points. We also study how to detect unboundedness of polynomial optimization, which is used to detect nonexistence of proper weights or (weakly) Pareto points.

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Section: If the Cardinality |Suppmentioning

confidence: 99%

The Multi-Objective Polynomial Optimization

Nie,

Yang

2021

Preprint

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“…They can be used to compute symmetric tensor decompositions and low rank approximations [40,42], which are closely related to truncated moment problems and polynomial optimization [20,[37][38][39]43]. There are special versions of symmetric tensors and their decompositions [19,44,45].…”

Section: Preliminarymentioning

confidence: 99%

Learning Diagonal Gaussian Mixture Models and Incomplete Tensor Decompositions

2021

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This paper studies how to learn parameters in diagonal Gaussian mixture models. The problem can be formulated as computing incomplete symmetric tensor decompositions. We use generating polynomials to compute incomplete symmetric tensor decompositions and approximations. Then the tensor approximation method is used to learn diagonal Gaussian mixture models. We also do the stability analysis. When the first and third order moments are sufficiently accurate, we show that the obtained parameters for the Gaussian mixture models are also highly accurate. Numerical experiments are also provided.

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“…They can be used to compute symmetric tensor decompositions and low rank approximations [36,38], which are closely related to truncated moment problems and polynomial optimization [17,33,34,35,39]. There are special versions of symmetric tensors and their decompositions [16,40,41].…”

Section: Preliminarymentioning

confidence: 99%