Tensor Contraction and Dimensionality Reduction

Tensor contraction is a nuanced and potent mathematical operation employed extensively across various scientific disciplines, including physics, engineering, and computer science. I intend to deep dive on Tensor contraction and provide an understanding on dimensionality reduction. The learning from this blog also offers an analytical tool that is necessary for working with complex tensor operations.

Basic Concepts and Notation

Tensors are multi-dimensional arrays of numerical values that generalize scalars, vectors, and matrices. They can be represented as:

• Scalar (0th-order tensor): A single number.
• Vector (1st-order tensor): An array of numbers.
• Matrix (2nd-order tensor): A two-dimensional array of numbers.
• Higher-order tensor: An array with three or more indices.

Tensors can have components that transform according to certain rules under coordinate transformations, classified as covariant (lower indices) or contravariant (upper indices).

Defining Tensor Contraction

Tensor contraction is the process of reducing a tensor’s order by summing over pairs of matching indices. This operation can be visualized as generalizing the trace of a matrix to higher-dimensional arrays.

Mathematical Formulation

Contraction of a Second-Order Tensor (Matrix):

For a matrix Aij​, the trace, which is a form of contraction, sums over the diagonal elements where the indices are equal:

General Tensor Contraction:

For a tensor T_i1 i2​…ip​​ of order p, contracting over the indices ik​ and il​ involves summing over these indices:

Comprehensive Examples

Example 1: Contraction of a Fourth-Order Tensor

Consider a tensor T_ijkl​. To contract this tensor over indices i and k:

This contraction results in a second-order tensor (or matrix) Cjl​, where the dimensions depend on the remaining indices j and l.

Here, the summation is performed across all values that i and k can assume, resulting in a matrix where each element represents the sum across two dimensions of the original tensor.

Example 2: Double Contraction of a Fourth-Order Tensor

If we extend the contraction to another pair of indices, for instance, j and l in Cjl​:

This results in a scalar D, which is the sum of all elements in the original fourth-order tensor T_ijkl​.

Advanced Tensor Operations Involving Contraction

Contracting Tensor Products:

Consider two tensors, Aij​ and Bkl​, and their tensor product Cijkl​=Aij​ ⊗ Bkl​. Contracting over i and k, and j and l, we get:

​Here, δik​ and δjl​ are Kronecker deltas, which are 1 when the indices are equal and 0 otherwise. This contraction essentially multiplies A and B element-wise and then sums all products.

Properties and Implications of Tensor Contraction

1. Symmetry Considerations: If a tensor is symmetric or antisymmetric with respect to any of the contracting indices, certain properties or simplifications can emerge. For example, contraction over symmetric indices in a symmetric tensor will not alter the resultant tensor’s symmetry properties.
2. Dimensional Analysis: The dimensions of the resulting tensor after contraction depend on the dimensions corresponding to the uncontracted indices. This property is crucial when applying tensor contraction to solve physical problems where dimensions represent physical quantities.

Practical Applications of Tensor Contraction

• Physics: Calculation of physical quantities like energy and momentum often involves contracting tensors that describe stress or electromagnetic fields.
• Physics: In the formulation of Einstein’s field equations in general relativity, tensor contraction is used to derive the Ricci tensor from the Riemann curvature tensor.
• Machine Learning: In neural networks, particularly in operations involving convolutional layers, tensor contraction simplifies the computation by reducing the dimensions of data arrays.

Tensor contraction offers a powerful method for manipulating and simplifying tensors, crucial in many areas of mathematics and science. By mastering these operations, students and researchers can tackle complex multidimensional problems more effectively, leading to deeper insights and innovative solutions in their respective fields.