Tensor Decomposition with Unaligned Observations

TL;DR

This paper introduces a novel tensor decomposition method that handles unaligned observations by using RKHS functions, versatile loss functions, and efficient optimization algorithms, demonstrated on synthetic and microbiome data.

Contribution

It proposes a new CP tensor decomposition approach for unaligned data, incorporating RKHS representations, versatile loss functions, and scalable optimization techniques.

Findings

Effective handling of unaligned observations in tensor data.

Versatile loss functions for different data types.

Improved computational efficiency through stochastic gradient and sketching algorithms.

Abstract

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.