Tensor decomposition is a fundamental framework to analyze data that can be represented by multi-dimensional arrays. In practice, tensor data is often accompanied by temporal information, namely the time points when the entry values were generated. This information implies abundant, complex temporal variation patterns. However, current methods always assume the factor representations of the entities in each tensor mode are static, and never consider their temporal evolution. To fill this gap, we propose NONparametric FActor Trajectory learning for dynamic tensor decomposition (NONFAT). We place Gaussian process (GP) priors in the frequency domain and conduct inverse Fourier transform via Gauss-Laguerre quadrature to sample the trajectory functions. In this way, we can overcome data sparsity and obtain robust trajectory estimates across long time horizons. Given the trajectory values at specific time points, we use a second-level GP to sample the entry values and to capture the temporal relationship between the entities. For efficient and scalable inference, we leverage the matrix Gaussian structure in the model, introduce a matrix Gaussian posterior, and develop a nested sparse variational learning algorithm. We have shown the advantage of our method in several real-world applications.

翻译：Tensor 分解是分析数据的基本框架, 可以用多维阵列来代表数据。 在实践中, 粒子数据往往伴有时间信息, 即输入值生成的时间点。 此信息意味着大量复杂的时间变化模式。 然而, 目前的方法总是假定每个发子模式中实体的系数表示是静态的, 并且从不考虑它们的时间演变。 为了填补这一空白, 我们建议使用非对称的进阶轨迹学习, 用于动态的进阶分解( NONFAT ) 。 我们将高山进程( GP) 的前列放在频率域中, 并且通过高斯- Laguerre 二次曲线来反向 Fourier 变换, 以抽样轨迹函数。 这样, 我们就能克服数据的广度, 并在很长的时间跨度范围内获得稳健的轨迹估计。 鉴于特定时间点的轨迹值, 我们使用二级的GP来抽样输入值, 并捕捉各实体之间的时间关系 。 为了高效和可缩的推论, 我们利用模型中的矩阵结构结构, 引入高斯结构结构结构, 来测试我们所显示的数个星系的模型应用 。