Tensor Decomposition Meets RKHS: Efficient Algorithms for Smooth and Misaligned Data

TL;DR

This paper introduces CP-HiFi, a tensor decomposition method that combines finite and infinite-dimensional modes using RKHS, enabling smoothness enforcement and handling of misaligned data without grid constraints.

Contribution

It proposes a novel hybrid tensor decomposition approach that integrates RKHS modes, allowing for smoothness and flexibility with irregular and misaligned data.

Findings

Enables tensor decomposition with continuous functions in RKHS.

Handles misaligned data without requiring regular grids.

Demonstrates effectiveness on synthetic examples.

Abstract

The canonical polyadic (CP) tensor decomposition decomposes a multidimensional data array into a sum of outer products of finite-dimensional vectors. Instead, we can replace some or all of the vectors with continuous functions (infinite-dimensional vectors) from a reproducing kernel Hilbert space (RKHS). We refer to tensors with some infinite-dimensional modes as quasitensors, and the approach of decomposing a tensor with some continuous RKHS modes is referred to as CP-HiFi (hybrid infinite and finite dimensional) tensor decomposition. An advantage of CP-HiFi is that it can enforce smoothness in the infinite dimensional modes. Further, CP-HiFi does not require the observed data to lie on a regular and finite rectangular grid and naturally incorporates misaligned data. We detail the methodology and illustrate it on a synthetic example.