Constrained Low Rank Approximation of the Hermitian Nonnegative-Definite Matrix

In this paper, we consider a constrained low rank approximation problem: , where E is a given complex matrix, p is a positive integer, and is the set of the Hermitian nonnegative-definite least squares solution to the matrix equation . We discuss the range of p and derive the corresponding explicit solution expression of the constrained low rank approximation problem by matrix decompositions. And an algorithm for the problem is proposed and the numerical example is given to show its feasibility.

Preservation of Linear Constraints in Approximation of Tensors

For an arbitrary tensor(multi-index array) with linear constraints at each direction,it is proved that the factors of any minimal canonical tensor approximation to this tensor satisfy the same linear constraints for the corresponding directions.

Best rank one approximation of real symmetric tensors can be chosen symmetric

We show that a best rank one approximation to a real symmetric tensor, which in principle can be nonsymmetric, can be chosen symmetric. Furthermore, a symmetric best rank one approximation to a symmetric tensor is unique if the tensor does not lie on a certain real algebraic variety.

Degrees of freedom in low rank matrix estimation

The objective of this paper is to quantify the complexity of rank and nuclear norm constrained methods for low rank matrix estimation problems. Specifically, we derive analytic forms of the degrees of freedom for these types of estimators in several common settings. These results provide efficient ways of comparing different estimators and eliciting tuning parameters. Moreover, our analyses reveal new insights on the behavior of these low rank matrix estimators. These observations are of great theoretical and practical importance. In particular, they suggest that, contrary to conventional wisdom, for rank constrained estimators the total number of free parameters underestimates the degrees of freedom, whereas for nuclear norm penalization, it overestimates the degrees of freedom. In addition, when using most model selection criteria to choose the tuning parameter for nuclear norm penalization, it oftentimes suffices to entertain a finite number of candidates as opposed to a continuum of choices. Numerical examples are also presented to illustrate the practical implications of our results.

Improved polyreference time domain method for modal identification using local or global noise removal techniques

Modal identification involves estimating the modal parameters, such as modal frequencies, damping ratios, and mode shapes, of a structural system from measured data. Under the condition that noisy impulse response signals associated with multiple input and output locations have been measured, the primary objective of this study is to apply the local or global noise removal technique for improving the modal identification based on the polyreference time domain (PTD) method. While the traditional PTD method improves modal parameter estimation by over-specifying the computational model order to absorb noise, this paper proposes an approach using the actual system order as the computational model order and rejecting much noise prior to performing modal parameter estimation algorithms. Two noise removal approaches are investigated: a "local" approach which removes noise from one signal at a time, and a "global" approach which removes the noise of multiple measured signals simultaneously. The numerical investigation in this article is based on experimental measurements from two test setups: a cantilever beam with 3 inputs and 10 outputs, and a hanged plate with 4 inputs and 32 outputs. This paper demonstrates that the proposed noise-rejection method outperforms the traditional noise-absorption PTD method in several crucial aspects.

Stationary flow fields prediction of variable physical domain based on proper orthogonal decomposition and kriging surrogate model

In this paper a new flow field prediction method which is independent of the governing equations, is developed to predict stationary flow fields of variable physical domain. Predicted flow fields come from linear superposition of selected basis modes generated by proper orthogonal decomposition（POD）. Instead of traditional projection methods, kriging surrogate model is used to calculate the superposition coefficients through building approximate function relationships between profile geometry parameters of physical domain and these coefficients. In this context,the problem which troubles the traditional POD-projection method due to viscosity and compressibility has been avoided in the whole process. Moreover, there are no constraints for the inner product form, so two forms of simple ones are applied to improving computational efficiency and cope with variable physical domain problem. An iterative algorithm is developed to determine how many basis modes ranking front should be used in the prediction. Testing results prove the feasibility of this new method for subsonic flow field, but also prove that it is not proper for transonic flow field because of the poor predicted shock waves.