基于贝叶斯推理的张量向量降维
Vectorial Dimension Reduction for Tensors Based on Bayesian Inference
摘要：
高阶张量的离散性约简是一个具有挑战性的问题。在传统的方法中，高阶张量通过Tucker分解“向量化”以获得低阶张量。这将分别破坏固有的高阶结构或导致不期望的张量。本文介绍了一种张量数据的概率向量降维模型。该模型通过使用同阶基张量的线性组合来表示张量，从而提供了一种将张量直接约简为向量的机制。在该表达式下，模型的投影基基于张量CandeComp/PARAFAC（CP）分解，并且模型中的自由参数的数量仅随着模式的数量而线性增长，而不是指数增长。通过变分EM方法建立了贝叶斯推理。根据经验给出了设置参数（CP分解的因子数和提取的特征的数量）的标准。该模型优于现有的几个PCA为基础的方法和CP分解几个公开可用的数据库中的分类和聚类精度。
英文摘要：
Dimensionality reduction for high-order tensors is a challenging problem. In conventional approaches, higher order tensors are `vectorized` via Tucker decomposition to obtain lower order tensors. This will destroy the inherent high-order structures or resulting in undesired tensors, respectively. This paper introduces a probabilistic vectorial dimensionality reduction model for tensorial data. The model represents a tensor by employing a linear combination of same order basis tensors, thus it offers a mechanism to directly reduce a tensor to a vector. Under this expression, the projection base of the model is based on the tensor CandeComp/PARAFAC (CP) decomposition and the number of free parameters in the model only grows linearly with the number of modes rather than exponentially. A Bayesian inference has been established via the variational EM approach. A criterion to set the parameters (factor number of CP decomposition and the number of extracted features) is empirically given. The model outperforms several existing PCA-based methods and CP decomposition on several publicly available databases in terms of classification and clustering accuracy.
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