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 integrating a 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.

翻译：多通道成像数据是天文学和生物学等科学领域中常见的数据格式。这类三维张量数据所蕴含的结构化信息及其高维特性，使得其分析对统计学家和实践者而言既具有吸引力又充满挑战。特别地，低秩标量-张量回归模型已获得广泛关注，并在Yu等人（2018）的工作中被重新表述为具有多线性核的张量高斯过程模型。本文通过将一种名为张量收缩的降维技术与张量高斯过程相结合，拓展了该模型在多通道成像数据标量-张量回归任务中的应用。此项研究源于高维多通道成像数据下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低维张量，随后对该潜在张量数据应用多线性张量高斯过程进行预测。在进行张量收缩时，我们引入各向异性全变分正则化以获得稀疏且平滑的潜在张量。接着，我们提出一种交替近端梯度下降算法用于参数估计。通过广泛的模拟研究以及将其应用于太阳耀斑预报问题，我们验证了该方法的有效性。