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.
The other existing dimensionality reduction method which embeds the TT subspace, is the tensor train neighbourhood preserving embedding (TTNPE) @cite_2 . TTNPE solves the exponential explosion on the complexity with the number of modes increasing. However, its robustness to the extreme outliers remains as a concern. Therefore, a dimensionality reduction method for tensors with a large number of modes or dimensions is demanded to propose on the TT subspace and the capability of reducing the sensitivity to the extreme outliers. Our method TTPUDR is thus developed with all the aspects.
@cite_2
2963465654 2962881496 2295543477 16346066
In this paper, we propose a tensor train neighborhood preserving embedding (TTNPE) to embed multidimensional tensor data into low-dimensional tensor subspace. Novel approaches to solve the optimization problem in TTNPE are proposed. For this embedding, we evaluate a novel tradeoff gain among classification, computation, and dimensionality reduction (storage) for supervised learning. It is shown that compared to the state-of-the-arts tensor embedding methods, TTNPE achieves superior tradeoff in classification, computation, and dimensionality reduction in MNIST handwritten digits, Weizmann face datasets, and financial market datasets. Tensor train (TT) decomposition provides a space-efficient representation for higher-order tensors. Despite its advantage, we face two crucial limitations when we apply the TT decomposition to machine learning problems: the lack of statistical theory and of scalable algorithms. In this paper, we address the limitations. First, we introduce a convex relaxation of the TT decomposition problem and derive its error bound for the tensor completion task. Next, we develop a randomized optimization method, in which the time complexity is as efficient as the space complexity is. In experiments, we numerically confirm the derived bounds and empirically demonstrate the performance of our method with a real higher-order tensor. We present TTC, an open-source parallel compiler for multidimensional tensor transpositions. In order to generate high-performance C++ code, TTC explores a number of optimizations, including software prefetching, blocking, loop-reordering, and explicit vectorization. To evaluate the performance of multidimensional transpositions across a range of possible use-cases, we also release a benchmark covering arbitrary transpositions of up to six dimensions. Performance results show that the routines generated by TTC achieve close to peak memory bandwidth on both the Intel Haswell and the AMD Steamroller architectures, and yield significant performance gains over modern compilers. By implementing a set of pruning heuristics, TTC allows users to limit the number of potential solutions; this option is especially useful when dealing with high-dimensional tensors, as the search space might become prohibitively large. Experiments indicate that when only 100 potential solutions are considered, the resulting performance is about 99 of that achieved with exhaustive search. Over the past few years, some embedding methods have been proposed for feature extraction and dimensionality reduction in various machine learning and pattern classification tasks. Among the methods proposed are Neighborhood Preserving Embedding (NPE), Locality Preserving Projection (LPP) and Local Discriminant Embedding (LDE) which have been used in such applications as face recognition and image video retrieval. However, although the data in these applications are more naturally represented as higher-order tensors, the embedding methods can only work with vectorized data representations which may not capture well some useful information in the original data. Moreover, high-dimensional vectorized representations also suffer from the curse of dimensionality and the high computational demand. In this paper, we propose some novel tensor embedding methods which, unlike previous methods, take data directly in the form of tensors of arbitrary order as input. These methods allow the relationships between dimensions of a tensor representation to be efficiently characterized. Moreover, they also allow the intrinsic local geometric and topological properties of the manifold embedded in a tensor space to be naturally estimated. Furthermore, they do not suffer from the curse of dimensionality and the high computational demand. We demonstrate the effectiveness of the proposed tensor embedding methods on a face recognition application and compare them with some previous methods. Extensive experiments show that our methods are not only more effective but also more efficient.