Introduction to Tensor Decompositions and their Applications in Machine Learning (1711.10781v1)

Abstract: Tensors are multidimensional arrays of numerical values and therefore generalize matrices to multiple dimensions. While tensors first emerged in the psychometrics community in the $20^{\text{th}}$ century, they have since then spread to numerous other disciplines, including machine learning. Tensors and their decompositions are especially beneficial in unsupervised learning settings, but are gaining popularity in other sub-disciplines like temporal and multi-relational data analysis, too. The scope of this paper is to give a broad overview of tensors, their decompositions, and how they are used in machine learning. As part of this, we are going to introduce basic tensor concepts, discuss why tensors can be considered more rigid than matrices with respect to the uniqueness of their decomposition, explain the most important factorization algorithms and their properties, provide concrete examples of tensor decomposition applications in machine learning, conduct a case study on tensor-based estimation of mixture models, talk about the current state of research, and provide references to available software libraries.

• The paper introduces tensor decomposition techniques, notably CPD and Tucker, that enable unique extraction of latent structures in high-dimensional data.
• It details efficient algorithms, including ALS and the tensor power method, to address computational challenges in machine learning.
• The study connects theoretical insights with practical tools by reviewing software libraries and case studies for effective mixture model estimation.

Analyzing Tensor Decompositions in Machine Learning

The paper "Introduction to Tensor Decompositions and their Applications in Machine Learning" offers an in-depth exploration of tensor decompositions, presenting their theoretical underpinnings, algorithmic strategies, and practical applications within machine learning. The authors demonstrate considerable expertise in elucidating a complex topic, rendering it accessible yet rigorous for fellow researchers in the domain.

Tensors, as multidimensional generalizations of matrices, inherently possess a structure that makes their decompositions particularly advantageous for specific machine learning tasks, especially those involved with high-dimensional data. The paper commences by establishing foundational tensor concepts, elucidating the key differences from matrices, particularly in terms of decomposition rigidity. A core assertion is that tensors, unlike matrices, can provide unique decompositions absent of stringent conditions due to their dimensionality properties.

The text explores two primary tensor decomposition techniques: the Canonical Polyadic Decomposition (CPD) and the Tucker Decomposition. These methodologies generalize concepts akin to the matrix Singular Value Decomposition (SVD), albeit catering to higher-order data structures. CPD serves prominently in latent variable modeling due to its ability to approximate data efficiently as a sum of rank-one tensors, highlighted by its formulation for unique solutions under mild assumptions. Conversely, the Tucker Decomposition excels in scenarios necessitating data compression or dimensionality reduction, aligning with principal component analysis strategies in high-dimensional contexts.

A key contribution of the paper is the exploration of decomposition algorithms, with the Alternating Least Squares (ALS) algorithm discussed extensively as a primary means of achieving CPD. The tensor power method is particularly underscored in the context of latent structure extraction from high-order moments, which is crucial for probabilistic models in unsupervised learning paradigms.

The paper presents a compelling case paper on the estimation of mixture models, showcasing an application of tensor methods in parameter estimation for Gaussian Mixture Models (GMMs) and topic models. The authors leverage the method of moments, alongside tensor power transformations, to attain a tractable solution for extracting latent model parameters. This is particularly significant for scenarios where traditional Maximum Likelihood Estimation (MLE) proves computationally infeasible due to NP-hard complexities.

The authors provide a practical guide through discussions on available software libraries, such as TensorLy and the N-way Toolbox, which facilitate the application of these theoretical insights within real-world data problems. This is complemented by a vision of future research avenues, urging exploration of tensor formulations for broader machine learning applications and advocating for decompositions under increasingly relaxed assumptions.

In conclusion, the paper meticulously defines the landscape of tensor decompositions, aligning them with machine learning challenges and advancing the field’s methodology armamentarium. It effectively bridges theoretical constructs with algorithmic implementations, providing a scaffold upon which further advancements can be built, potentially impacting domains that demand nuanced interpretability of complex, high-dimensional data.