Tensor-tensor algebra for optimal representation and compression of multiway data.

Area of Science:

• Data Science
• Machine Learning
• Numerical Analysis

Background:

• Matrix Singular Value Decomposition (SVD) is the standard for data dimensionality reduction.
• Current methods require data to be in matrix format, limiting efficiency for high-dimensional datasets.

Purpose of the Study:

• Demonstrate superior compressibility of high-dimensional datasets using tensor representations.
• Investigate the efficiency of tensor Singular Value Decomposition (SVD) methods.
• Compare tensor-based SVDs with matrix-based SVD.

Main Methods:

• Proving Eckart-Young optimality for tensor-SVDs under different truncation strategies.
• Developing tensor-tensor product constructs and generalizations.
• Theoretically and empirically analyzing truncated tensor SVD against high-order SVD and tensor-train SVD.

Main Results:

• Tensor representations show higher compressibility for high-dimensional data than matrix representations.
• Tensor-tensor representations can be superior to matrix counterparts for equal dimensional spanning spaces.
• Optimality results for tensor-SVDs are established.

Conclusions:

• Tensors provide a more efficient framework for representing and compressing large, high-dimensional datasets.
• Tensor SVD generalizes matrix SVD, offering enhanced performance in machine learning applications.
• The study validates the theoretical and empirical advantages of tensor-based dimensionality reduction techniques.