Generalized Singular Values
summary
Defines singular values from the geometry of two dual norms and an inner product by plotting the Pareto subfrontier (X vs Y) and reading off piecewise linear slopes; vectors are tight when frontier equals subfrontier and then admit a slope decomposition that recovers spectral, nuclear and Frobenius data. Shows concrete tensors: matrix-multiplication tensor has spectral norm 1, multiplicity N^3 and nuclear norm N^3; permutation-sign tensor has spectral 1 and nuclear norm n!; some tensors are not tight. Applies to ECG and total-variation image denoising.
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