  One of the main reasons for topological persistence being useful in data
analysis is that it is backed up by a stability (isometry) property:
persistence diagrams of $1$-parameter persistence modules are stable in the
sense that the bottleneck distance between two diagrams equals the interleaving
distance between their generating modules. However, in multi-parameter setting
this property breaks down in general. A simple special case of persistence
modules called rectangle decomposable modules is known to admit a weaker
stability property. Using this fact, we derive a stability-like property for
$2$-parameter persistence modules. For this, first we consider interval
decomposable modules and their optimal approximations with rectangle
decomposable modules with respect to the bottleneck distance. We provide a
polynomial time algorithm to exactly compute this optimal approximation which,
together with the polynomial-time computable bottleneck distance among interval
decomposable modules, provides a lower bound on the interleaving distance.
Next, we leverage this result to derive a polynomial-time computable distance
for general multi-parameter persistence modules which enjoys similar
stability-like property. This distance can be viewed as a generalization of the
matching distance defined in the literature.
