Structured statistical methods are promising for recovering or completing information from noisy and incomplete data with high fidelity. In particular, matrix completion exploits underlying structural properties such as rank or sparsity. Our objective is to employ matrix completion to reduce computational effort associated with the calculation of multiple quantum chemical Hessians, which are necessary for identification of temperature-dependent free energy maxima under canonical variational transition state theory (VTST). We demonstrate proof-of-principle of an algebraic variety-based matrix completion method for recovering missing elements in a matrix of transverse Hessian eigenvalues constituting the minimum energy path (MEP) of a reaction. The algorithm, named harmonic variety-based matrix completion (HVMC), utilizes the fact that the points lying on the MEP of a reaction step constitute an algebraic variety in the reaction path Hamiltonian representation. We demonstrate that, with as low as 30% random sampling of matrix elements for the largest system in our test set (46 atoms), the complete matrix of eigenvalues can be recovered. We further establish algorithm performance for VTST rate calculations by quantifying errors in zero-point energies and vibrational free energies. Motivated by this success, we outline next steps toward developing a practical HVMC algorithm, which utilizes a gradient-based sampling protocol for low-cost VTST rate computations.