We suggest a three step strategy to find a good basis (dictionary) for nonlinear $m$-term approximation. The first step consists of solving an optimization problem of finding a near best basis for a given function class $F$, when we optimize over a collection $\Bbb D$ of bases (dictionaries). The second step is devoted to finding a universal basis (dictionary) $\D_u\in \Bbb D$ for a given pair $(\Cal F,\Bbb D)$ of collections: $\Cal F$ of function classes and $\Bbb D$ of bases (dictionaries). This means that $\D_u$ provides near optimal approximation for each class $F$ from a collection $\Cal F$. The third step deals with constructing a theoretical algorithm that realizes near best $m$-term approximation with regard to $\D_u$ for function classes from $\Cal F$.

In the talk we will discuss this strategy in the model case of anisotropic multivariate function classes and the set of orthogonal bases. We will show that a natural tensor-product-wavelet type basis has a property of universality. Moreover, we will point out that greedy type algorithm realizes near best $m$-term approximation with regard to this universal basis for all anisotropic function classes.