Random walks on symmetric spaces and inequalities for matrix spectra

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Eigenvalues, singular values, and Littlewood-Richardson coefficients (2003, background): “hese three sequences must satisfy the Horn inequalities (see Proposition 2.1). Conversely, those inequalities are suﬃcient for the existence of A, B, and C = A+B with the desired singular values; see [11] for the proof, and [7, Section 5] for a detailed exposition. Proposition 1.16. Let 2p ≤ n, and ”
Sharp Minima Can Generalize For Deep Nets (2017, methodology): “Since D α and ( ∇ 2 L )( θ ) are positive symmetric semi-deﬁnite matrices, we can apply the multiplicative Horn inequalities (Klyachko, 2000) on singular values of the product , since we have It means that there exists an observationally equivalent parameter with at least r − min k ≤ K ( n k )…”