Classical interlacing status of Q_k polynomials when U has singular k×k submatrices

Determine whether the polynomial set Q_k := { det[x·I_d + (A − C_{:,W}(U_{S,W})^{†}R_{S,:})^T(A − C_{:,W}(U_{S,W})^{†}R_{S,:})] } over all k-subsets S ⊆ [n_R] and W ⊆ [d_C] forms a classical interlacing family in the case where the matrix U contains singular k×k submatrices; specifically, ascertain if these degree-d polynomials admit a common interlacing and thereby enable direct application of the classical interlacing polynomials method.

Background

In attempting to apply the classical interlacing polynomials method to the generalized CUR problem, a natural candidate family consists of the degree-d polynomials built from the residuals using pseudoinverses, denoted here as Q_k. This family would allow the direct use of the classical interlacing machinery provided it forms an interlacing family.

However, when U contains singular k×k submatrices, the authors note that it is not known whether Q_k satisfies the interlacing property. To overcome this obstruction, they introduce a different polynomial set P_k and the framework of generalized interlacing families, which accommodates varying polynomial degrees. Resolving the status of Q_k under singular submatrices would clarify whether the classical framework suffices without recourse to generalized interlacing.

References

However, it is not generally known or guaranteed that this polynomial set forms a classical interlacing family when U contains singular k × k submatrices. This uncertainty prevents the direct application of classical interlacing methods.

— Generalized Interlacing Families: New Error Bounds for CUR Matrix Decompositions (2512.07903 - Cai et al., 7 Dec 2025) in Remark in the subsubsection “Generalized CUR matrix decompositions” (following Theorem gcur-th2)