Switch to irlba to avoid Spectra dependencies.

[LTLA]

This uses the diagonal shift trick to convert the problem of "finding the smallest eigenvalues" into the largest ones.

[codecov-commenter]

Codecov Report

Merging #4 (068cc2e) into master (42a6412) will increase coverage by 0.20%.
The diff coverage is 95.23%.

@@            Coverage Diff             @@
##           master       #4      +/-   ##
==========================================
+ Coverage   95.75%   95.96%   +0.20%     
==========================================
  Files           9        9              
  Lines         448      446       -2     
==========================================
- Hits          429      428       -1     
+ Misses         19       18       -1     

Impacted Files	Coverage Δ
include/umappp/spectral_init.hpp	89.06% <95.23%> (+1.18%)	⬆️

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[jlmelville]

@LTLA was the motivation for this PR to remove Spectra as a dependency because of the weirdness reported at yixuan/spectra#126?

I am also toying with removing RSpectra as a dependency mainly because it causes PREPERRORs with rhub when using UBSAN/ASAN sanitizers and with valgrind (a pretty important safety net for me). The irlba approach seems good enough, although it is noticeably slower for some artificial cases like a 1D line embedded in 3D.

As an aside, the maximum eigenvalue that can be attained for the normalized laplacian is 2, so I think you could pre-shift your matrix without having to extract the largest eigenvalue first (the approach I used to test irlba is outlined here and almost the same as what you are currently doing).

[LTLA]

was the motivation for this PR to remove Spectra as a dependency because of the weirdness reported at yixuan/spectra#126?

That was the primary motivation. I also have some fairly stringent requirements re. compilation times and binary sizes for a non-R project I'm working on, so I've been trying to trim dependencies in all libraries involved - and I already was using irlba elsewhere so I figured I could just save myself a dependency and re-use it here.

As an aside, the maximum eigenvalue that can be attained for the normalized laplacian is 2, so I think you could pre-shift your matrix without having to extract the largest eigenvalue first

That would be neat. Is this an obvious/well-known result? I managed to convince myself that the normalized laplacian is PSD and that its smallest eigenvalue is zero, but I didn't get around to thinking through this property.

Edit: I have convinced myself. Needed to look it up but eventually worked it out.

[jlmelville]

Edit: I have convinced myself. Needed to look it up but eventually worked it out.

Great. For reference, I have seen this asserted in several places, but the canonical place including a proof seems to be Spectral Graph Theory. Part 5 of Lemma 1.7 in Section 1.3, 'Basic facts about the spectrum of a graph'.

[LTLA]

Just did the change for the max eigenvalue - seems to check out, I'll see how it goes with some of my downstream apps.