Common-subspace volume power means hit a sharp threshold: one factor is reachable in polynomial time, anything larger is NP-hard

A single low-dimensional subspace can now preserve volume from several observations with a guarantee that does not grow with the surrounding dimension: for each fixed subspace dimension and fixed rational power, a randomized polynomial-time algorithm reaches a definite factor. Earlier practice optimized on the subspace manifold with local iterations that guarantee only stationary points, while relaxing the projector to a trace-one positive semidefinite matrix with a linear trace score can lose an arbitrarily large factor even for one positive definite observation. For each fixed power, any constant approximation larger than that factor is proved NP-hard, so the factor is optimal at fixed parameters and independent of ambient dimension.

Above a critical power the approximation factor has a closed form, and at or below that power the problem becomes inapproximable. Earlier scalar factors covered only parts of the power range and did not identify a constant-factor threshold. Above the critical power the factor is an explicit Gamma-function product; at and below it every positive constant-factor approximation is NP-hard, separately for equally weighted rational positive semidefinite observations and for positive definite ones.

The algorithm relaxes the subspace to a trace-one positive semidefinite matrix whose relaxed score agrees with the volume score at every subspace, then draws independent Gaussian columns and keeps their span. The earlier trace relaxation did not equal the determinant volume at subspaces, so rounding loss could not be controlled directly; the new score is built from elementary symmetric polynomials of eigenvalues and stays concave. The rounding analysis writes each Gaussian determinant as a sum of squared minors via Cauchy–Binet and bounds it with Bartlett-type determinant moments; at finite rank the factor is strictly larger than the final threshold, and the surplus pays for optimization and output precision.

With only two observations of rank k, the relaxation is exact and no rounding loss occurs. In the general case returning from a positive semidefinite matrix to one subspace costs an approximation factor, and in the two-observation case that loss disappears. Every trace-one positive semidefinite matrix can be replaced by a subspace that decreases neither determinant score; the construction is deterministic, uses principal vectors and square roots of nonnegative scalars, and yields a closed-form global optimum for the Martin mean of two subspaces.

The results cover fixed subspace dimension and fixed rational power over the real and complex fields, with rational positive semidefinite observations or rank-k projections and positive rational weights; two-observation exactness requires both observations to have rank k. The additive Martin guarantee concerns projection observations and can run in the complementary dimension.

Whether hardness already holds for three or more observations is left unresolved; at power one the threshold requires a tensor construction, and near power one the treatment of rational output carries an additional qualification.