Only in a desperate sales pitch or a desparate research grants. There are of course some situations were certain measurements are untrustworthy, but to claim that is the common case is very snake oily.
When certain measurements become untrustworthy, that it does so only because of some unknown smooth transformation, is not very frequent (this is what purely topological methods will deal with). Random noise will also do that for you.
Not disputing the fact that sometimes metrics cannot be trusted entirely, but to go to a topological approach seems extreme. One should use as much of the relevant non-topological information as possible.
As the hackneyed example goes a topological methods would not be able to distinguish between a cup and a donut. For that you would need to trust non-topological features such as distances and angles. Deep learning methods can indeed differentiate between cop-nip and coffee mugs.
BTW I am completely on-board with the idea that data often looks as if it has been sampled from an unknown, potentially smooth, possibly non-Euclidean manifold and then corrupted by noise.
In such cases recovering that manifold from noisy data is a very worthy cause.
In fact that is what most of your blogpost is about. But that's differential geometry and manifolds, they have structure far richer than a topology. For example they may have tangent planes, a Reimann metric or a symplectic form etc. A topological method would throw all of that away and focus on topology.
Only in a desperate sales pitch or a desparate research grants. There are of course some situations were certain measurements are untrustworthy, but to claim that is the common case is very snake oily.
When certain measurements become untrustworthy, that it does so only because of some unknown smooth transformation, is not very frequent (this is what purely topological methods will deal with). Random noise will also do that for you.
Not disputing the fact that sometimes metrics cannot be trusted entirely, but to go to a topological approach seems extreme. One should use as much of the relevant non-topological information as possible.
As the hackneyed example goes a topological methods would not be able to distinguish between a cup and a donut. For that you would need to trust non-topological features such as distances and angles. Deep learning methods can indeed differentiate between cop-nip and coffee mugs.
BTW I am completely on-board with the idea that data often looks as if it has been sampled from an unknown, potentially smooth, possibly non-Euclidean manifold and then corrupted by noise. In such cases recovering that manifold from noisy data is a very worthy cause.
In fact that is what most of your blogpost is about. But that's differential geometry and manifolds, they have structure far richer than a topology. For example they may have tangent planes, a Reimann metric or a symplectic form etc. A topological method would throw all of that away and focus on topology.