Persistence diagrams (PDs) are now routinely used to summarize the underlying topology of sophisticated dataencountered in challenging learning problems . Despite several appealing properties, integrating PDs in learning pipelines can be challenging because their natural geometry is not Hilbertian. In particular, algorithms to average a family of PDs have only been considered recently and are known to be computationally prohibitive. We propose in this article a tractable framework to carry out fundamental tasks on PDs , namely evaluating distances, computing barycenters and carrying out clustering. This framework builds upon a formulation of PD metrics as optimal transport (OT) problems , for which recent computational advances , in particular entropic regularization and its convolutional formulation on regular grids, can all be leveraged to provide efficient and (GPU) scalable computations . We demonstrate the efficiency of our approach by carrying out clustering on PDs at scales never seen before in the literature.
Persistence diagrams (PDs) are now routinely used to summarize the underlying topology of complex data . Despite several appealing properties, incorporating PDs in learning pipelines can be challenging because their natural geometry is not Hilbertian. Indeed, this was recently exemplified in a string of papers which show that the simple task of averaging a few PDs can be computationally prohibitive. We propose in this article a tractable framework to carry out standard tasks on PDs at scale, notably evaluating distances, estimating barycenters and performing clustering. This framework builds upon a reformulation of PD metrics as optimal transport (OT) problems . Doing so, we can exploit recent computational advances : the OT problem on a planar grid, when regularized with entropy, is convex can be solved in linear time using the Sinkhorn algorithm and convolutions. This results in scalable computations that can stream on GPUs . We demonstrate the efficiency of our approach by carrying out clustering with diagrams metrics on several thousands of PDs, a scale never seen before in the literature.
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sophisticated dataencountered in challenging learning problems
complex data
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integrating
incorporating
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In particular, algorithms to average a family of PDs have only been considered recently and are known to
Indeed, this was recently exemplified in a string of papers which show that the simple task of averaging a few PDs can
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fundamental
standard
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, namely
at scale, notably
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computing barycenters and carrying out
estimating barycenters and performing
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formulation
reformulation
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, for which
. Doing so, we can exploit
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, in particular entropic regularization and its convolutional formulation on regular grids, can all be leveraged to provide efficient and (GPU) scalable computations
: the OT problem on a planar grid, when regularized with entropy, is convex can be solved in linear time using the Sinkhorn algorithm and convolutions. This results in scalable computations that can stream on GPUs
R
on PDs at scales
with diagrams metrics on several thousands of PDs, a scale