Inheritance Properties and Sum-of-Squares Decomposition of Hankel Tensors: Theory and Algorithms

In this paper, we show that if a lower-order Hankel tensor is positive semi-definite (or positive definite, or negative semi-definite, or negative definite, or SOS), then its associated higher-order Hankel tensor with the same generating vector, where the higher order is a multiple of the lower order, is also positive semi-definite (or positive definite, or negative semi-definite, or negative definite, or SOS, respectively). Furthermore, in this case, the extremal H-eigenvalues of the higher order tensor are bounded by the extremal H-eigenvalues of the lower order tensor, multiplied with some constants. Based on this inheritance property, we give a concrete sum-of-squares decomposition for each strong Hankel tensor. Then we prove the second inheritance property of Hankel tensors, i.e., a Hankel tensor has no negative (or non-positive, or positive, or nonnegative) H-eigenvalues if the associated Hankel matrix of that Hankel tensor has no negative (or non-positive, or positive, or nonnegative, respectively) eigenvalues. In this case, the extremal H-eigenvalues of the Hankel tensor are also bounded by the extremal eigenvalues of the associated Hankel matrix, multiplied with some constants. The third inheritance property of Hankel tensors is raised as a conjecture

Fast Hankel Tensor-Vector Products and Application to Exponential Data Fitting

This paper is contributed to a fast algorithm for Hankel tensor-vector products. For this purpose, we first discuss a special class of Hankel tensors that can be diagonalized by the Fourier matrix, which is called \emph{anti-circulant} tensors. Then we obtain a fast algorithm for Hankel tensor-vector products by embedding a Hankel tensor into a larger anti-circulant tensor. The computational complexity is about $\mathcal{O}(m^2 n \log mn)$ for a square Hankel tensor of order $m$ and dimension $n$, and the numerical examples also show the efficiency of this scheme. Moreover, the block version for multi-level block Hankel tensors is discussed as well. Finally, we apply the fast algorithm to exponential data fitting and the block version to 2D exponential data fitting for higher performance.Comment: 20 pages, 4 figure

On Some Sufficient Conditions for Strong Ellipticity

We establish several sufficient conditions for the strong ellipticity of any fourth-order elasticity tensor in this paper. The first presented sufficient condition is an extension of positive definite matrices, which states that the strong ellipticity holds if the unfolding matrix of this fourth-order elasticity tensor can be modified into a positive definite one by preserving the summations of some corresponding entries. An alternating projection algorithm is proposed to verify whether an elasticity tensor satisfies the first condition or not. Conditions for some special cases beyond the first sufficient condition are further investigated, which includes some important cases for the isotropic and some particular anisotropic linearly elastic materials

P-Tensors, P0_0-Tensors, and Tensor Complementarity Problem

The concepts of P- and P$_0$-matrices are generalized to P- and P$_0$-tensors of even and odd orders via homogeneous formulae. Analog to the matrix case, our P-tensor definition encompasses many important classes of tensors such as the positive definite tensors, the nonsingular M-tensors, the nonsingular H-tensors with positive diagonal entries, the strictly diagonally dominant tensors with positive diagonal entries, etc. As even-order symmetric PSD tensors are exactly even-order symmetric P$_0$-tensors, our definition of P$_0$-tensors, to some extent, can be regarded as an extension of PSD tensors for the odd-order case. Along with the basic properties of P- and P$_0$-tensors, the relationship among P$_0$-tensors and other extensions of PSD tensors are then discussed for comparison. Many structured tensors are also shown to be P- and P$_0$-tensors. As a theoretical application, the P-tensor complementarity problem is discussed and shown to possess a nonempty and compact solution set

Robust Elastic Net Regression

We propose a robust elastic net (REN) model for high-dimensional sparse regression and give its performance guarantees (both the statistical error bound and the optimization bound). A simple idea of trimming the inner product is applied to the elastic net model. Specifically, we robustify the covariance matrix by trimming the inner product based on the intuition that the trimmed inner product can not be significant affected by a bounded number of arbitrarily corrupted points (outliers). The REN model can also derive two interesting special cases: robust Lasso and robust soft thresholding. Comprehensive experimental results show that the robustness of the proposed model consistently outperforms the original elastic net and matches the performance guarantees nicely

Understanding V2V Driving Scenarios through Traffic Primitives

Semantically understanding complex drivers' encountering behavior, wherein two or multiple vehicles are spatially close to each other, does potentially benefit autonomous car's decision-making design. This paper presents a framework of analyzing various encountering behaviors through decomposing driving encounter data into small building blocks, called driving primitives, using nonparametric Bayesian learning (NPBL) approaches, which offers a flexible way to gain an insight into the complex driving encounters without any prerequisite knowledge. The effectiveness of our proposed primitive-based framework is validated based on 976 naturalistic driving encounters, from which more than 4000 driving primitives are learned using NPBL - a sticky HDP-HMM, combined a hidden Markov model (HMM) with a hierarchical Dirichlet process (HDP). After that, a dynamic time warping method integrated with k-means clustering is then developed to cluster all these extracted driving primitives into groups. Experimental results find that there exist 20 kinds of driving primitives capable of representing the basic components of driving encounters in our database. This primitive-based analysis methodology potentially reveals underlying information of vehicle-vehicle encounters for self-driving applications

Elasticity M\mathscr{M}-tensors and the Strong Ellipticity Condition

In this paper, we establish two sufficient conditions for the strong ellipticity of any fourth-order elasticity tensor and investigate a class of tensors satisfying the strong ellipticity condition, the elasticity $\mathscr{M}$-tensor. The first sufficient condition is that the strong ellipticity holds if the unfolding matrix of this fourth-order elasticity tensor can be modified into a positive definite one by preserving the summations of some corresponding entries. Second, an alternating projection algorithm is proposed to verify whether an elasticity tensor satisfies the first condition or not. Besides, the elasticity $\mathscr{M}$-tensor is defined with respect to the M-eigenvalues of elasticity tensors. We prove that any nonsingular elasticity $\mathscr{M}$-tensor satisfies the strong ellipticity condition by employing a Perron-Frobenius-type theorem for M-spectral radii of nonnegative elasticity tensors. Other equivalent definitions of nonsingular elasticity $\mathscr{M}$-tensors are also established.Comment: arXiv admin note: text overlap with arXiv:1705.0508

Efficient Face Alignment via Locality-constrained Representation for Robust Recognition

Practical face recognition has been studied in the past decades, but still remains an open challenge. Current prevailing approaches have already achieved substantial breakthroughs in recognition accuracy. However, their performance usually drops dramatically if face samples are severely misaligned. To address this problem, we propose a highly efficient misalignment-robust locality-constrained representation (MRLR) algorithm for practical real-time face recognition. Specifically, the locality constraint that activates the most correlated atoms and suppresses the uncorrelated ones, is applied to construct the dictionary for face alignment. Then we simultaneously align the warped face and update the locality-constrained dictionary, eventually obtaining the final alignment. Moreover, we make use of the block structure to accelerate the derived analytical solution. Experimental results on public data sets show that MRLR significantly outperforms several state-of-the-art approaches in terms of efficiency and scalability with even better performance

Computing the pp-Spectral Radii of Uniform Hypergraphs with Applications

The $p$-spectral radius of a uniform hypergraph covers many important concepts, such as Lagrangian and spectral radius of the hypergraph, and is crucial for solving spectral extremal problems of hypergraphs. In this paper, we establish a spherically constrained maximization model and propose a first-order conjugate gradient algorithm to compute the $p$-spectral radius of a uniform hypergraph (CSRH). By the semialgebraic nature of the adjacency tensor of a uniform hypergraph, CSRH is globally convergent and obtains the global maximizer with a high probability. When computing the spectral radius of the adjacency tensor of a uniform hypergraph, CSRH stands out among existing approaches. Furthermore, CSRH is competent to calculate the $p$-spectral radius of a hypergraph with millions of vertices and to approximate the Lagrangian of a hypergraph. Finally, we show that the CSRH method is capable of ranking real-world data set based on solutions generated by the $p$-spectral radius model

Isotropic Polynomial Invariants of the Hall Tensor

The Hall tensor emerges from the study of the Hall effect, an important magnetic effect observed in electric conductors and semiconductors. The Hall tensor is third order and three dimensional, whose first two indices are skew-symmetric. In this paper, we investigate the isotropic polynomial invariants of the Hall tensor by connecting it with a second order tensor via the third order Levi-Civita tensor. We propose a minimal isotropic integrity basis with 10 invariants for the Hall tensor. Furthermore, we prove that this minimal integrity basis is also an irreducible isotropic function basis of the Hall tensor