The higher order singular value decomposition (HOSVD) of tensors is a generalization of matrix SVD. The perturbation analysis of HOSVD under random noise is more delicate than its matrix counterpart. Recent progress has been made in Richard and Montanari [2014], Zhang and Xia [2017] and Liu et al. [2017] demonstrating that minimax optimal singular spaces estimation and low rank tensor recovery ...

Sparsity is a fundamental topic in highdimensional data analysis. Perhaps the most common measures of sparsity are the `norms, for 0 ≤ p < 2. In this paper, we study an alternative measure of sparsity, the truncated `-norm, which is related to other `-norms, but appears to have some unique and useful properties. Focusing on the ndimensional Gaussian location model, we derive exact asymptotic mi...

Hyperspectral images (HSIs) are often corrupted by a mixture of several types of noise during the acquisition process, e.g., Gaussian noise, impulse noise, dead lines, stripes, and many others. Such complex noise could degrade the quality of the acquired HSIs, limiting the precision of the subsequent processing. In this paper, we present a novel tensor-based HSI restoration approach by fully id...

In this project, we propose a novel kernel named Adaptive Data-dependent Matrix Norm Based Gaussian Kernel (ADM-Gaussian kernel) for facial feature extraction. As a popular facial feature extraction method for face recognition, the current kernel method endures some problems. Firstly, the face image must be transformed to the vector, which leads to the large storage requirements and the large c...

Low rank matrix approximations have many applications in different domains. In system theory it has been used in model reduction schemes, in system identification with outputerror models and in static errors-in-variables problems, for instance. The approximations are mostly performed using the singular value decomposition. This is optimal for all unitarily invariant matrix norms, such as the Fr...

Multiplicativity of certain maximal p → q norms of a tensor product of linear maps on matrix algebras is proved in situations in which the condition of complete positivity (CP) is either augmented by, or replaced by, the requirement that the entries of a matrix representative of the map are non-negative (EP). In particular, for integer t, multiplicativity holds for the maximal 2 → 2t norm of a ...

We consider an estimator b βn(t) defined as the element φ ∈ Φ minimizing a contrast process Λn(φ, t) for each t. We give some general results for deriving the weak convergence of √ n(b βn − β) in the space of bounded functions, where, for each t, β(t) is the φ ∈ Φ minimizing the limit of Λn(φ, t) as n → ∞. These results are applied in the context of penalized M-estimation, that is, when Λn(φ, t...

ABSTRACT : The stability operator of a compact oriented minimal hypersurface Mn−1 ⊂ S is given by J = −∆ − ‖A‖ − (n − 1), where ‖A‖ is the norm of the second fundamental form. Let λ1 be the first eigenvalue of J and define β = −λ1 − 2(n − 1). In [S] Simons proved that β ≥ 0 for any non-equatorial minimal hypersurface M ⊂ S. In this paper we will show that β = 0 only for Clifford hypersurfaces. ...

The explicit form of the rate-distortion function has rarely been obtained, except for few cases where the Shannon lower bound coincides with the rate-distortion function for the entire range of the positive rate. From an information geometrical point of view, the evaluation of the rate-distortion function is achieved by a projection to the mixture family defined by the distortion measure. In t...