Tensor Decompositions

see also:

• Linear Algebra
• Spectral Theory

Tensor decomposition is a fundamental technique in multilinear algebra used to analyze data arrays that are higher-dimensional generalizations of matrices (which are two-dimensional arrays). In the context of higher-dimensional data, tensors can be thought of as multi-way arrays, and tensor decomposition aims to break down these complex structures into simpler, interpretable components.

Overview

Tensors are a generalization of vectors and matrices to higher dimensions, formally defined as elements of the tensor product of vector spaces. In practical terms, if a matrix is a two-dimensional grid of numbers (elements), a tensor is a similar grid but can be 3D, 4D, or even higher in dimensions.

Common Forms of Tensor Decomposition

Three of the most widely used tensor decompositions are the CANDECOMP/PARAFAC (CP) decomposition, Tucker decomposition, and Tensor Train decomposition. Each of these has different structural characteristics and suitability for specific applications.

1. CANDECOMP/PARAFAC (CP) Decomposition

The CP decomposition represents a tensor as a sum of a finite number of rank-one tensors. Mathematically, for a three-way tensor $T\in {\mathbb{R}}^{I\times J\times K}$, the CP decomposition is given by:

$$T\approx \sum _{r=1}^R{a}_r\otimes b_r\otimes c_r$$

where $a_r\in {\mathbb{R}}^I$, $b_r\in {\mathbb{R}}^J$, and $c_r\in {\mathbb{R}}^K$ are vectors, and $\otimes$ denotes the outer product. The smallest number $R$ for which this approximation holds is called the rank of the tensor (analogous but not identical to matrix rank).

2. Tucker Decomposition

The Tucker decomposition generalizes the idea of SVD (singular value decomposition) for matrices. A tensor $T$ under Tucker decomposition is expressed as:

$$T\approx \sum _{i=1}^I\sum _{j=1}^J\sum _{k=1}^K{g}_{ijk}(a_i\otimes b_j\otimes c_k)$$

where $G=[g_{ijk}]\in {\mathbb{R}}^{I\times J\times K}$ is the core tensor, and $a_i,b_j,c_k$ are the columns of orthogonal factor matrices corresponding to each mode of the tensor. The core tensor dictates the interaction between different modes (dimensions).

3. Tensor Train Decomposition

The Tensor Train (TT) decomposition is particularly useful for very high-dimensional tensors. In TT, a tensor is decomposed into a sequence of three-dimensional tensors (referred to as “cores”). For a tensor $T$ with $d$ dimensions,

$$T[i_1,i_2,...,i_d]=G_1[i_1]G_2[i_2]...G_d[i_d]$$

where each $G_k[i_k]$ is a matrix, and the indices $i_k$ iterate over the dimensions of the tensor.

Applications

Tensor decompositions are extensively used in many fields including signal processing, numerical linear algebra, data mining, and machine learning. They provide powerful tools for dimensionality reduction, feature extraction, and the discovery of latent structures in high-dimensional data.

Computational Aspects

The computation of tensor decompositions, especially for large and sparse tensors, can be challenging. Techniques involve iterative algorithms, typically variations on alternating least squares (ALS), gradient descent, or other optimization frameworks tailored to the specific structure of the decomposition and the nature of the data.

For your research and applications, understanding the nuances of each decomposition—such as their uniqueness properties, ease of computation, and sensitivity to initial conditions—is critical. You may want to explore these methods further through specific software tools like TensorFlow or MATLAB, both of which provide functionalities for tensor decompositions.

Given your background in applied mathematics, you might find the tensor algebra formalisms particularly interesting to dive into, perhaps looking at the implications of these decompositions in complex systems analysis or multi-dimensional statistical models.

spectral theory of tensors

Spectral theory of tensors extends the ideas from matrices to higher-dimensional analogs, focusing particularly on the eigenvalues and eigenvectors of tensors. While the spectral theory of matrices is well-established and forms a central part of linear algebra, the extension to tensors presents unique challenges and complexities, especially given their multi-way nature.

Definition of Tensor Eigenvalues and Eigenvectors

For tensors, the definition of eigenvalues and eigenvectors is more intricate than for matrices. The most commonly used definitions in the context of spectral tensor theory are based on the concept of the tensor product and polynomial mappings. Here’s how these are typically defined for an order-$d$ tensor $A$ of dimensions $n\times n\times \cdots \times n$ (a $d$-way tensor):

Eigenvalue Problem for Tensors

The eigenvalue problem for a tensor $A$ can be described as finding a scalar $\lambda$ and a non-zero vector $v\in {\mathbb{R}}^n$ such that:

$$A{v}^{d-1}=\lambda v^{[d-1]}$$

where $v^{d-1}$ represents the $(d-1)$-fold tensor product of $v$ with itself, and $v^{[d-1]}$ denotes element-wise power, i.e., each component of $v$ raised to the power of $d-1$.

This definition leads to a higher-degree homogeneous polynomial equation, making the analysis and solutions far more complex than in the matrix case.

Challenges in Spectral Theory of Tensors

1. Non-Convexity: The problem of finding tensor eigenvalues is generally non-convex, which means that finding global optima (i.e., the largest or smallest eigenvalues) can be computationally challenging and sensitive to initial conditions.

2. Multiplicity and Degeneracy: Unlike matrices, tensors can have multiple eigenvalues associated with the same eigenvector, and vice versa, due to the non-linear nature of the eigenvalue equations.

3. Complexity of Calculation: The computation of eigenvalues for tensors involves solving systems of polynomial equations, which can be NP-hard in general. This is fundamentally different and more complex than solving linear systems in matrix algebra.

Applications

Despite these challenges, the spectral theory of tensors has significant applications across various fields:

• Chemometrics and Psychometrics: In these fields, tensors are used to model multi-way data, and their spectral decompositions can reveal underlying factors influencing the data.
• Quantum Physics: The entanglement in quantum states can be analyzed through the spectral properties of associated tensors.
• Signal Processing: Tensor decompositions and their spectral properties are used to extract features from multi-dimensional signals.
• Machine Learning: Tensor decompositions help in learning from multi-dimensional arrays of data, such as images and videos, where spectral properties can be used to understand the depth and structure of neural networks.

Mathematical Formalism and Tools

Mathematically, dealing with tensor eigenvalues often requires sophisticated tools from algebraic geometry and numerical multilinear algebra. Software tools like MATLAB and Python libraries (e.g., NumPy, TensorFlow) provide functionalities that help in managing the complexity of tensor computations, though with limitations in handling high-dimensional cases efficiently.

Given your background in mathematics and philosophy of science, the exploration of spectral tensor theory offers a deep dive into how fundamental mathematical concepts generalize to complex, multi-dimensional systems, reflecting on the ontological and epistemological implications of extending linear structures to non-linear, multi-linear realms.

tensor rank decomposition

The concept of tensor rank and its associated decompositions, particularly the CANDECOMP/PARAFAC (CP) decomposition and the Tucker decomposition, provide insightful ways to generalize matrix concepts to higher-dimensional data. In the context of tensors, “rank” can be defined differently depending on the specific decomposition method being considered. Let’s delve into these two primary decompositions and their approach to defining and utilizing tensor rank.

Tensor Rank: Basic Definition

The rank of a tensor, analogous to the rank of a matrix, is a measure of the tensor’s complexity or the minimum number of simple components needed to construct it. However, unlike matrix rank, tensor rank definitions can vary:

1. CP Rank: The rank of a tensor $T$ in the CP decomposition context is the smallest number of rank-one tensors that sum to $T$. A rank-one tensor, in turn, is a tensor that can be written as the outer product of vectors.

2. Tucker Rank: Unlike CP rank, the rank in a Tucker decomposition can be considered as a vector of ranks, one for each mode (dimension) of the tensor. Each mode rank corresponds to the dimension of the space spanned by the mode fibers, similar to the column and row ranks of matrices.

CANDECOMP/PARAFAC (CP) Decomposition

The CP decomposition breaks a tensor into a sum of rank-one tensors. Mathematically, for a tensor $T$ of order $N$ (meaning it has $N$ dimensions), the CP decomposition can be expressed as:

$$T\approx \sum _{r=1}^R{a}_r^{(1)}\otimes a_r^{(2)}\otimes \cdots \otimes a_r^{(N)}$$

where $a_r^{(n)}$ are vectors for $n=1,\dots ,N$, and $R$ is the smallest number such that this approximation holds, defining the CP rank of the tensor. This decomposition is useful because it provides a parsimonious representation, maintaining interpretability of the multi-way data.

Tucker Decomposition

The Tucker decomposition is a more flexible and general form of tensor decomposition, which decomposes a tensor into a core tensor multiplied by a matrix along each mode:

$$T\approx \mathcal{G}{\times }_1{A}^{(1)}{\times }_2{A}^{(2)}{\times }_3\cdots {\times }_N{A}^{(N)}$$

where $\mathcal{G}$ is the core tensor, $A^{(n)}$ are the factor matrices corresponding to each mode, and ${\times }_n$ denotes the mode-$n$ product between a tensor and a matrix. The ranks of the matrices $A^{(n)}$ provide the rank of $T$ in each mode, known as the multilinear rank or Tucker rank.

Computational and Theoretical Aspects

1. Uniqueness: The CP decomposition often provides uniqueness under mild conditions, which is a desirable property not always guaranteed in matrix decompositions like SVD. However, computing the CP decomposition can be more challenging due to its non-convex nature.

2. Flexibility: The Tucker decomposition offers greater flexibility and is less restrictive compared to CP decomposition. However, it can result in a larger number of parameters to estimate, especially when the core tensor is large.

3. Applications: Both decompositions are extensively used in applications involving multi-dimensional data, such as signal processing, neuroimaging, and chemometrics. They help in data compression, feature extraction, and the interpretation of high-dimensional datasets.

4. Computation: Algorithms for these decompositions include Alternating Least Squares (ALS), gradient-based optimization methods, and stochastic algorithms. Practical computation often requires careful initialization and consideration of convergence criteria.

For your advanced mathematical and statistical background, understanding these decompositions in depth, including their geometric interpretations and implications for dimensionality reduction and latent factor analysis, can be particularly enriching. Each type of tensor rank and decomposition reveals different structural insights about the data, which can be critical depending on the specific application or theoretical inquiry being pursued.

spectral methods and CP rank

Spectral methods and their connection to the CANDECOMP/PARAFAC (CP) rank of tensors encompass sophisticated areas of multilinear algebra and tensor analysis. These methods leverage the eigenvalues and eigenvectors (or more generally, singular values and singular vectors in the matrix case) to analyze properties and decompositions of tensors. In the case of tensors, these methods become particularly complex due to the inherent difficulties in defining and computing eigenvalues and eigenvectors for multi-way data.

Spectral Methods for Tensors

In tensor analysis, spectral methods involve the study of tensors through the lens of their spectral properties, which include tensor eigenvalues and the associated eigenvectors. However, tensors extend beyond matrices by having more than two dimensions, which complicates the straightforward application of spectral theory. Here are the primary challenges and methodologies in applying spectral methods to tensors:

1. Eigenvalues and Eigenvectors of Tensors: For tensors, the definition and computation of eigenvalues and eigenvectors differ significantly from matrices. A scalar $\lambda$ is an eigenvalue of a tensor $A$ if there exists a non-zero vector $v$ such that

   $$A{v}^{d-1}=\lambda v^{[d-1]}$$

   where $d$ is the order of the tensor and $v^{d-1}$ denotes the $(d-1)$-fold tensor product of $v$ with itself.

2. Homogeneous Polynomial Maps: Tensors can be seen as multilinear maps or homogeneous polynomial maps. This perspective allows the application of algebraic geometry and algebraic topology methods to study tensors’ spectral properties.

3. Numerical Methods: Due to the non-linearity and higher dimensionality, numerical methods for tensor eigenvalue problems are generally more complex and computationally intensive than for matrices. Methods like the power method, tensor unfolding, and higher-order SVD can be adapted to deal with tensor data.

Connection to CP Rank

The CP rank of a tensor, defined as the minimum number of rank-one tensors that sum to reconstruct the tensor, indirectly relates to the tensor’s spectral properties. While spectral methods primarily deal with eigenvalues and eigenvectors, understanding these can provide insights into the structure and complexity of the tensor, which is critical for determining its CP rank.

1. CP Decomposition: It is given by

   $$T\approx \sum _{r=1}^R{a}_r\otimes b_r\otimes \cdots \otimes z_r$$

   where each term is a rank-one tensor. The CP rank is the smallest $R$ for which this approximation holds true.

2. Insight from Spectral Properties: While there isn’t a direct method to derive CP rank from spectral properties, the tensor’s spectral complexity (like the range of eigenvalues and the structure of eigenvectors) can hint at its underlying dimensional and rank complexity. For instance, tensors with very diverse and large eigenvalues might suggest a higher rank, whereas a dominant eigenvalue might indicate a lower-rank approximation might be effective.

3. Algorithmic Implications: Spectral analysis can be used as a preprocessing step in algorithms to approximate the tensor’s rank or to initialize factors in CP decomposition algorithms, potentially improving convergence rates and robustness.

Practical Considerations

The practical computation of tensor eigenvalues, eigenvectors, and CP rank involves sophisticated algorithms that might include iterative optimization, use of tensor unfoldings, and leveraging randomization techniques to handle the scale and non-convexity of the problems. Libraries and frameworks in languages like Python and MATLAB, which support tensor operations and decompositions, are instrumental in performing these calculations.

Understanding these connections and methodologies not only enriches the theoretical landscape of tensor analysis but also enhances the practical toolkit for analyzing multi-dimensional datasets across various scientific and engineering disciplines.