Locality preserving projections (LPP) are a classical dimensionality reduction method based on data graph information. However, LPP is still responsive to extreme outliers. LPP aiming for vectorial data may undermine data structural information when it is applied to multidimensional data. Besides, it assumes the dimension of data to be smaller than the number of instances, which is not suitable for high-dimensional data. For high-dimensional data analysis, the tensor-train decomposition is proved to be able to efficiently and effectively capture the spatial relations. Thus, we propose a tensor-train parameterization for ultra dimensionality reduction (TTPUDR) in which the traditional LPP mapping is tensorized in terms of tensor-trains and the LPP objective is replaced with the Frobenius norm to increase the robustness of the model. The manifold optimization technique is utilized to solve the new model. The performance of TTPUDR is assessed on classification problems and TTPUDR significantly outperforms the past methods and the several state-of-the-art methods.
For most of the applications, in order to achieve the computational efficiency and be less information redundant, the researchers often restrict the tensor ranks to be smaller than the size of their corresponding tensor mode, i.e., @math for @math @cite_2 .
@cite_2
2132267493 2030927653 2953204310 2951021721
There has been continued interest in seeking a theorem describing optimal low-rank approximations to tensors of order 3 or higher that parallels the Eckart-Young theorem for matrices. In this paper, we argue that the naive approach to this problem is doomed to failure because, unlike matrices, tensors of order 3 or higher can fail to have best rank- @math approximations. The phenomenon is much more widespread than one might suspect: examples of this failure can be constructed over a wide range of dimensions, orders, and ranks, regardless of the choice of norm (or even Bregman divergence). Moreover, we show that in many instances these counterexamples have positive volume: they cannot be regarded as isolated phenomena. In one extreme case, we exhibit a tensor space in which no rank-3 tensor has an optimal rank-2 approximation. The notable exceptions to this misbehavior are rank-1 tensors and order-2 tensors (i.e., matrices). In a more positive spirit, we propose a natural way of overcoming the ill-posedness of the low-rank approximation problem, by using weak solutions when true solutions do not exist. For this to work, it is necessary to characterize the set of weak solutions, and we do this in the case of rank 2, order 3 (in arbitrary dimensions). In our work we emphasize the importance of closely studying concrete low-dimensional examples as a first step toward more general results. To this end, we present a detailed analysis of equivalence classes of @math tensors, and we develop methods for extending results upward to higher orders and dimensions. Finally, we link our work to existing studies of tensors from an algebraic geometric point of view. The rank of a tensor can in theory be given a semialgebraic description; in other words, it can be determined by a system of polynomial inequalities. We study some of these polynomials in cases of interest to us; in particular, we make extensive use of the hyperdeterminant @math on @math . In this paper we propose novel methods for completion (from limited samples) and de-noising of multilinear (tensor) data and as an application consider 3-D and 4- D (color) video data completion and de-noising. We exploit the recently proposed tensor-Singular Value Decomposition (t-SVD)[11]. Based on t-SVD, the notion of multilinear rank and a related tensor nuclear norm was proposed in [11] to characterize informational and structural complexity of multilinear data. We first show that videos with linear camera motion can be represented more efficiently using t-SVD compared to the approaches based on vectorizing or flattening of the tensors. Since efficiency in representation implies efficiency in recovery, we outline a tensor nuclear norm penalized algorithm for video completion from missing entries. Application of the proposed algorithm for video recovery from missing entries is shown to yield a superior performance over existing methods. We also consider the problem of tensor robust Principal Component Analysis (PCA) for de-noising 3-D video data from sparse random corruptions. We show superior performance of our method compared to the matrix robust PCA adapted to this setting as proposed in [4]. In this paper we propose novel methods for completion (from limited samples) and de-noising of multilinear (tensor) data and as an application consider 3-D and 4- D (color) video data completion and de-noising. We exploit the recently proposed tensor-Singular Value Decomposition (t-SVD)[11]. Based on t-SVD, the notion of multilinear rank and a related tensor nuclear norm was proposed in [11] to characterize informational and structural complexity of multilinear data. We first show that videos with linear camera motion can be represented more efficiently using t-SVD compared to the approaches based on vectorizing or flattening of the tensors. Since efficiency in representation implies efficiency in recovery, we outline a tensor nuclear norm penalized algorithm for video completion from missing entries. Application of the proposed algorithm for video recovery from missing entries is shown to yield a superior performance over existing methods. We also consider the problem of tensor robust Principal Component Analysis (PCA) for de-noising 3-D video data from sparse random corruptions. We show superior performance of our method compared to the matrix robust PCA adapted to this setting as proposed in [4]. Recovering a low-rank tensor from incomplete information is a recurring problem in signal processing and machine learning. The most popular convex relaxation of this problem minimizes the sum of the nuclear norms of the unfoldings of the tensor. We show that this approach can be substantially suboptimal: reliably recovering a @math -way tensor of length @math and Tucker rank @math from Gaussian measurements requires @math observations. In contrast, a certain (intractable) nonconvex formulation needs only @math observations. We introduce a very simple, new convex relaxation, which partially bridges this gap. Our new formulation succeeds with @math observations. While these results pertain to Gaussian measurements, simulations strongly suggest that the new norm also outperforms the sum of nuclear norms for tensor completion from a random subset of entries. Our lower bound for the sum-of-nuclear-norms model follows from a new result on recovering signals with multiple sparse structures (e.g. sparse, low rank), which perhaps surprisingly demonstrates the significant suboptimality of the commonly used recovery approach via minimizing the sum of individual sparsity inducing norms (e.g. @math , nuclear norm). Our new formulation for low-rank tensor recovery however opens the possibility in reducing the sample complexity by exploiting several structures jointly.