  We introduce a linear dimensionality reduction technique preserving
topological features via persistent homology. The method is designed to find
linear projection $L$ which preserves the persistent diagram of a point cloud
$\mathbb{X}$ via simulated annealing. The projection $L$ induces a set of
canonical simplicial maps from the Rips (or \v{C}ech) filtration of
$\mathbb{X}$ to that of $L\mathbb{X}$. In addition to the distance between
persistent diagrams, the projection induces a map between filtrations, called
filtration homomorphism. Using the filtration homomorphism, one can measure the
difference between shapes of two filtrations directly comparing simplicial
complexes with respect to quasi-isomorphism $\mu_{\operatorname{quasi-iso}}$ or
strong homotopy equivalence $\mu_{\operatorname{equiv}}$. These
$\mu_{\operatorname{quasi-iso}}$ and $\mu_{\operatorname{equiv}}$ measures how
much portion of corresponding simplicial complexes is quasi-isomorphic or
homotopy equivalence respectively. We validate the effectiveness of our
framework with simple examples.

Machine Learning;Machine Learning