OPERATOR NORM INEQUALITIES BETWEEN TENSOR UNFOLDINGS ON THE PARTITION LATTICE

Area of Science:

• Multilinear algebra
• Data science
• Applied mathematics

Background:

• Higher-order tensors are crucial in data-intensive fields like image processing and feature extraction.
• Tensor unfolding (flattening) into matrices is a common but poorly understood technique.
• Existing research primarily focuses on specific tensor matricizations.

Purpose of the Study:

• To analyze the impact of all possible tensor unfoldings on functional properties.
• To derive inequalities between the Lp-norms of arbitrary tensor unfoldings.
• To establish bounds for spectral and Frobenius norms and investigate norm invariance.

Main Methods:

• Consideration of all possible unfoldings of an order-k tensor, corresponding to partitions of {1, ..., k}.
• Derivation of general inequalities between Lp-norms of tensor unfoldings.
• Analysis of spectral norm bounds and the ratio of Frobenius to spectral norms.
• Investigation of norm invariance for orthogonally decomposable tensors.

Main Results:

• General inequalities are derived for the Lp-norms of arbitrary tensor unfoldings.
• The spectral norm of a tensor is shown to be bounded by its unfoldings.
• An improved upper bound is obtained for the ratio of Frobenius to spectral norms.
• For specific tensors, the spectral norm is invariant under certain unfolding operations.

Conclusions:

• Tensor unfolding significantly impacts tensor properties, contrary to common assumptions.
• The derived inequalities and bounds offer new insights into tensor norm behavior.
• Understanding unfolding effects is crucial for developing robust tensor-based algorithms.