A group-algebraic tensor framework delivers Eckart-Young optimal equivariant approximations and recovers physical selection rules from data alone via a Lean-formalized star_G algebra.
Sufficient and Necessary Conditions for Eckart-Young like Result for Tubal Tensors
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abstract
A valuable feature of the tubal tensor framework is that many familiar constructions from matrix algebra carry over to tensors, including SVD and notions of rank. Importantly, it has been shown that for a specific family of tubal products, an Eckart-Young type theorem holds, i.e., the best low-rank approximation of a tensor under the Frobenius norm is obtained by truncating its tubal SVD. In this paper, we provide a complete characterization of the family of tubal products that yield an Eckart-Young type result. We demonstrate the practical implications of our theoretical findings by conducting experiments with video data and data-driven dynamical systems.
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Copositive matrices with nondecreasing off-diagonal entries admit a PSD plus nonnegative decomposition, which implies exactness of a natural relaxation for separable quadratic optimization over the simplex.
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Group-Algebraic Tensors: Provably-optimal Equivariant Learning and Physical Symmetry Discovery
A group-algebraic tensor framework delivers Eckart-Young optimal equivariant approximations and recovers physical selection rules from data alone via a Lean-formalized star_G algebra.
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Copositive Matrices with Ordered Off-Diagonal Entries
Copositive matrices with nondecreasing off-diagonal entries admit a PSD plus nonnegative decomposition, which implies exactness of a natural relaxation for separable quadratic optimization over the simplex.