Clarification on the weight parameter in PersistenceImage

[caki35]

First of all, thank you for providing this excellent library — it has been very useful in my work.

I am a bit confused about the role of the weight parameter in the PersistenceImage constructor.

According to the documentation:

“weight (function) – weight function for the persistence diagram points (default constant function, i.e. lambda x: 1). This function must be defined on 2D points, i.e. lists or numpy arrays of the form [p_x, p_y].”

From this, I understood that the input points are given as (birth, death), so a natural choice for weighting by persistence would be weight = lambda x: x[1] - x[0].

However, in the tutorial it is stated:

“A persistence image is obtained by rotating by -π/4, centering Gaussian functions on all diagram points (usually weighted by a parameter function — here we consider the squared distance to the diagonal) and summing all these Gaussians.”

In that example, the weight function is weight = lambda x: x[1]**2, which seems to imply that x[1] corresponds to the vertical axis after rotation (i.e., persistence) rather than the raw death coordinate from the input diagram.

Could you clarify this apparent inconsistency? Specifically: Does the weight function receive points in (birth, death) or in (birth, persistence) form?

Thank you for your time.

[hschreiber]

Hm, I agree that the documentation is not clear on this one. I am not an expert of this pipeline, so I will let @MathieuCarriere confirm or correct me if I am wrong, but from what I see, the input type for PersistenceImage.fit/transform is the usual (birth,death) diagram, but this diagram is then called on BirthPersistenceTransform which changes the format to (birth,death-birth). So the input of the weight function is the latter one.
We should clarify this in the description of weight in the documentation.

[MathieuCarriere]

Thank you for answering @hschreiber
You are right, the weight function expects points in the (birth, persistence) format