How to interpret the eigenvalues of spectral embedding plot

[newtonharry]

Hi,

I'm trying to understand this plot coming from usually using PCA plots to understand how well the DR performed. PCA tells you how much variance of the data is captured in each component. What do the values between 0 and 1 for each component mean for an eigenvalue in laplacian eigenmaps? How do I know how well the DR is performing aside from performing downstream analysis?

Thanks!

[kaizhang]

The eigenvalues in a Laplacian eigenmap provide a ranked measure of the components' importance for preserving the data's local neighborhood structure during dimensionality reduction. They help in identifying the most significant dimensions for embedding, revealing the data's underlying manifold structure, and understanding the clustering or geometric distribution of the dataset in a lower-dimensional space.

[newtonharry]

Ok that makes sense.

So in a hypothetical scenario where I was able to influence the features in such a way prior to dimensional reduction, that each components importance is greater than in that plot, would that mean its preserving the local neighborhood more accurately? Lets say component 2 goes from 0.3 to 0.5. Or should I only be interpreting it in terms of rank, as you mentioned?

[kaizhang]

So in a hypothetical scenario where I was able to influence the features in such a way prior to dimensional reduction, that each components importance is greater than in that plot, would that mean its preserving the local neighborhood more accurately?

I'm not sure if you can say that. But I think comparing the values from two different runs may not be meaningful.

Lets say component 2 goes from 0.3 to 0.5. Or should I only be interpreting it in terms of rank, as you mentioned?

The magnitude also matters, similar to the values in PCA.

[newtonharry]

I'm not sure if you can say that. But I think comparing the values from two different runs may not be meaningful.

So interestingly, I did some experimentation on the bin sizes ranging from 50bp up to 2000bp and saw the eigenvalues after the first component increased/decreased steadily. The lower I went from 500bp, it increased noticeably, apart from some components where 50bp results in a significantly lower eigenvalue. The higher I went from 500bp it decreased noticeably. I think one would have to perform some type of correlation analysis to ensure that the eigenvalues are capturing the correct aspect of the data.

[TingTingShao]

Hi,

Regarding to the number of components, I think there is some information from the API reference (https://kzhang.org/SnapATAC2/api/_autosummary/snapatac2.tl.spectral.html):

By default, this function adopts a strategy where all eigenvectors are weighted according to the square root of their corresponding eigenvalues, rather than implementing a strict cutoff threshold.

From what I understood, the cutoff is not that important if you set weighted_by_sd=True.

So I didn't put much efforts to optimize the number of components to choose, simply used the default value.

Plz correct me if I am wrong.

Thanks,
tingting

[newtonharry]

Hi,

I understand that an optimal number of components can be chosen, I just want to understand how to interpret what they meant in Laplacian eigenmaps. Thanks for your response though, I'll be mindful of that :)