用于多通道成像分析的带收缩的张量高斯过程
Tensor Gaussian Process with Contraction for Multi-Channel Imaging Analysis
AI总结：
本文针对多通道成像数据的高维张量回归问题，提出将张量收缩降维与张量高斯过程相结合的建模方法，并通过引入各向异性全变差正则化和交替近端梯度下降算法，在模拟和太阳耀斑预报中验证了其有效性。
摘要：
多通道成像数据是天文学和生物学等科学领域中一种普遍的数据格式。这些三维张量数据的结构化信息和高维特性，使得其分析对统计学家和从业者而言既引人入胜又充满挑战。特别是低秩标量对张量回归模型受到了广泛关注，并在Yu等人（2018）中被重新表述为具有多线性核的张量高斯过程（Tensor-GP）模型。在本文中，我们通过引入一种称为张量收缩的综合性降维技术，扩展了Tensor-GP模型，用于多通道成像数据的标量对张量回归任务。这一工作的动机来源于具有高维多通道成像数据的太阳耀斑预报问题。我们首先为每个数据张量估计一个潜在的、降维后的张量，然后在潜在张量数据上应用多线性Tensor-GP进行预测。在进行张量收缩时，我们引入各向异性全变差正则化，以获得稀疏且平滑的潜在张量。随后，我们提出一种交替近端梯度下降算法用于估计。我们通过广泛的模拟研究并将该方法应用于太阳耀斑预报问题，验证了所提方法的有效性。
英文摘要：
Multi-channel imaging data is a prevalent data format in scientific fields such as astronomy and biology. The structured information and the high dimensionality of these 3-D tensor data makes the analysis an intriguing but challenging topic for statisticians and practitioners. The low-rank scalar-on-tensor regression model, in particular, has received widespread attention and has been re-formulated as a tensor Gaussian Process (Tensor-GP) model with multi-linear kernel in Yu et al.(2018). In this paper, we extend the Tensor-GP model by introducing an integrative dimensionality reduction technique, called tensor contraction, with a Tensor-GP for a scalar-on-tensor regression task with multi-channel imaging data. This is motivated by the solar flare forecasting problem with high dimensional multi-channel imaging data. We first estimate a latent, reduced-size tensor for each data tensor and then apply a multi-linear Tensor-GP on the latent tensor data for prediction. We introduce an anisotropic total-variation regularization when conducting the tensor contraction to obtain a sparse and smooth latent tensor. We then propose an alternating proximal gradient descent algorithm for estimation. We validate our approach via extensive simulation studies and applying it to the solar flare forecasting problem.
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