Matrix Completion and Sums of Squares
summary
Characterizes exactly when nonnegative quadratic forms on a variety X equal sums of squares: equivalently the dual slice of the PSD cone has only rank‑1 extreme rays, and those X are unions of basic varieties of minimal degree glued along coordinate subspaces. Uses convex duality, Hankel spectra, classical algebraic geometry and SDP geometry. For matrix completion the necessary principal‑minor condition is sufficient iff the pattern graph G is chordal, and the smallest new extreme rank p satisfies cycle length = p+2.
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