  We propose a method to simultaneously compute scalar basis functions with an
associated functional map for a given pair of triangle meshes. Unlike previous
techniques that put emphasis on smoothness with respect to the
Laplace--Beltrami operator and thus favor low-frequency eigenfunctions, we aim
for a spectrum that allows for better feature matching. This change of
perspective introduces many degrees of freedom into the problem which we
exploit to improve the accuracy of our computed correspondences. To effectively
search in this high dimensional space of solutions, we incorporate into our
minimization state-of-the-art regularizers. We solve the resulting highly
non-linear and non-convex problem using an iterative scheme via the Alternating
Direction Method of Multipliers. At each step, our optimization involves simple
to solve linear or Sylvester-type equations. In practice, our method performs
well in terms of convergence, and we additionally show that it is similar to a
provably convergent problem. We show the advantages of our approach by
extensively testing it on multiple datasets in a few applications including
shape matching, consistent quadrangulation and scalar function transfer.
