Functional Bayesian Tucker Decomposition for Continuous-indexed Tensor Data Analysis

Kernekoncepter

Generalizing Tucker decomposition to continuous-indexed tensor data using Functional Bayesian Tucker Decomposition.

Resumé

The content introduces Functional Bayesian Tucker Decomposition (FunBaT) to handle continuous-indexed tensor data. It addresses the limitations of standard tensor models in dealing with real-world continuous data. The method utilizes Gaussian processes as functional priors and state-space models to reduce computational costs. An efficient inference algorithm is developed for scalable posterior approximation. Results show superior performance in both synthetic and real-world applications.
Segments:

Introduction to Tensor Decomposition

Explanation of tensor decomposition models like CP, TT, and Tucker.

Limitations of Standard Tensor Models

Challenges posed by structured grid data and discrete indexes.

Proposal of FunBaT Methodology

Generalizing Tucker decomposition to continuous-indexed data using GPs.

Algorithm Overview

Description of the inference algorithm based on CEP.

Data Extraction Techniques

Utilization of Matérn kernels for GP priors and state-space modeling.

Experimental Results

Evaluation on synthetic and real-world datasets, showcasing superior performance compared to baselines.

Statistik

"We use Gaussian processes (GP) as functional priors to model the latent functions."
"An efficient inference algorithm is developed for scalable posterior approximation based on advanced message-passing techniques."

Citater

"We propose FunBaT: Functional Bayesian Tucker decomposition, which generalizes the standard Tucker decomposition to the functional field under the probabilistic framework."
"Our method outperforms state-of-the-art methods by large margins in terms of reconstruction error."

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Functional Bayesian Tucker Decomposition for Continuous-indexed Tensor Data

by Shikai Fang,... kl. arxiv.org 03-20-2024

https://arxiv.org/pdf/2311.04829.pdf

Dybere Forespørgsler

How does FunBaT compare to other tensor decomposition methods in terms of scalability

FunBaT demonstrates superior scalability compared to other tensor decomposition methods due to its efficient inference algorithm based on advanced message-passing techniques. The use of Conditional Expectation Propagation (CEP) allows for the decoupling of the likelihood term, enabling parallel computation and reducing computational complexity. This approach results in a linear time complexity of O(NKR), where N is the number of observations, K is the number of modes, and R is the pre-set mode rank. Additionally, FunBaT's space complexity is also linear with respect to both data size and tensor mode, further enhancing its scalability.

What are the implications of using continuous indexes over discretized indexes in tensor decomposition

Using continuous indexes over discretized indexes in tensor decomposition has significant implications for capturing fine-grained information without losing detail. By representing real-world data as continuous coordinates rather than discrete bins or ranges, FunBaT can model complex relationships more accurately and effectively. Continuous indexing preserves the inherent structure and patterns present in the data while allowing for probabilistic interpolation at arbitrary indexes. This approach enables more precise modeling of phenomena represented by continuous variables such as geographic coordinates or temporal sequences.

How can FunBaT be applied to other domains beyond climate modeling

FunBaT's flexibility and adaptability make it applicable to various domains beyond climate modeling where multi-aspect data analysis is required. For example:

Healthcare: FunBaT could be used to analyze patient records structured as multi-mode tensors (patients, doctors, treatments) to identify patterns in healthcare outcomes.
Finance: In financial analytics, FunBaT could handle multi-dimensional datasets related to market trends (stocks, commodities) using continuous indices like time series or asset classes.
Image Processing: Applying FunBaT to image tensors with spatial dimensions could help extract latent features from images based on pixel values across different modes.
Recommendation Systems: Utilizing FunBaT for collaborative filtering tasks involving user-item interactions represented as tensors can enhance recommendation accuracy by capturing nuanced preferences through continuous indexing.
By leveraging its functional Bayesian framework and scalable inference algorithm, FunBaT offers a versatile solution for analyzing diverse types of multi-aspect data across various domains beyond climate modeling.