  Topological Data Analysis (TDA) provides a pipeline to extract quantitative
topological descriptors from structured objects. This enables the definition of
topological loss functions, which assert to what extent a given object exhibits
some topological properties. These losses can then be used to perform
topological optimizationvia gradient descent routines. While theoretically
sounded, topological optimization faces an important challenge: gradients tend
to be extremely sparse, in the sense that the loss function typically depends
on only very few coordinates of the input object, yielding dramatically slow
optimization schemes in practice.Focusing on the central case of topological
optimization for point clouds, we propose in this work to overcome this
limitation using diffeomorphic interpolation, turning sparse gradients into
smooth vector fields defined on the whole space, with quantifiable Lipschitz
constants. In particular, we show that our approach combines efficiently with
subsampling techniques routinely used in TDA, as the diffeomorphism derived
from the gradient computed on a subsample can be used to update the coordinates
of the full input object, allowing us to perform topological optimization on
point clouds at an unprecedented scale. Finally, we also showcase the relevance
of our approach for black-box autoencoder (AE) regularization, where we aim at
enforcing topological priors on the latent spaces associated to fixed,
pre-trained, black-box AE models, and where we show thatlearning a
diffeomorphic flow can be done once and then re-applied to new data in linear
time (while vanilla topological optimization has to be re-run from scratch).
Moreover, reverting the flow allows us to generate data by sampling the
topologically-optimized latent space directly, yielding better interpretability
of the model.
