Gram Determinants of Real Binary Tensors

Area of Science:

• Multilinear algebra
• Numerical analysis
• Tensor decomposition

Background:

• Binary tensors are defined as hypercubes with 2^n entries.
• Flattening a tensor into a matrix is a common operation in tensor analysis.
• Gram determinants are crucial in understanding matrix properties and relationships.

Purpose of the Study:

• To characterize the image of a map that associates a tensor with its n-tuple of Gram determinants.
• To address a question posed by Hackbusch and Uschmajew regarding higher-order singular values of tensors.
• To provide a semi-algebraic description for the set of possible Gram determinant tuples.

Main Methods:

• Definition of binary tensors and their flattening into matrices.
• Calculation of the Gram matrix determinant for each flattened matrix.
• Development of a map from tensors to n-tuples of Gram determinants.
• Proposing a semi-algebraic characterization for the image of this map.

Main Results:

• A novel map is introduced that transforms a binary tensor into an n-tuple of Gram determinants.
• A semi-algebraic characterization for the image of this tensor-to-determinant map is proposed.
• The findings offer insights into the structure of higher-order singular values of tensors.

Conclusions:

• The proposed semi-algebraic characterization provides a new way to understand tensor properties.
• This work contributes to the ongoing research on tensor decomposition and analysis.
• The results directly answer a specific question about tensor singular values raised in prior literature.