解决观测数据未对齐时的张量分解问题，适用于多种类型数据。
Tensor Decomposition with Unaligned Observations
- 用再生核希尔伯特空间表示未对齐的模式，建模灵活。
- 设计通用损失函数，支持二值、整数和正数数据类型。
- 提出高效优化算法，适合大规模或高维数据场景。
本文提出一种处理未对齐观测的典型多线性（CP）张量分解方法。将存在未对齐观测的模式通过再生核希尔伯特空间（RKHS）中的函数进行建模，引入一种通用损失函数，能有效处理二值、整数值及正数值等不同类型的数据。同时，提出了用于计算该张量分解的优化算法，并采用随机梯度法提升计算效率；针对ℓ₂损失函数，还设计了采样算法以进一步加速。通过合成数据与早期儿童微生物组数据集的实例验证了所提方法的有效性。
原文摘要 · Abstract (English)
This paper presents a canonical polyadic (CP) tensor decomposition that addresses unaligned observations. The mode with unaligned observations is represented using functions in a reproducing kernel Hilbert space (RKHS). We introduce a versatile loss function that effectively accounts for various types of data, including binary, integer-valued, and positive-valued types. Additionally, we propose an optimization algorithm for computing tensor decompositions with unaligned observations, along with a stochastic gradient method to enhance computational efficiency. A sketching algorithm is also introduced to further improve efficiency when using the $\ell_2$ loss function. To demonstrate the efficacy of our methods, we provide illustrative examples using both synthetic data and an early childhood human microbiome dataset.
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