I'd be happy to have the other implementation as an alternative, and a parameter to set is as the algorithm.

related to #4485 ?

What's the relation between PCA and PCoA? They seem quite similar but I'm not really familiar with MDS and have never heard the term PCoA

What's the relation between PCA and PCoA? They seem quite similar but I'm not really familiar with MDS and have never heard the term PCoA

PCoA is essentially a PCA, but starting for a distance matrix (which doesn't need to be euclidean).
It seems indeed, that it is mostly what #4485 is implementing.

The scikit-bio implementation is quite nice: _principal_coordinate_analysis.py

It seems indeed, that is mostly what #4485 is implementing.

@maxibor You would be very welcome to continue that PR.

Is anyone working on that? If not, I can take it.

Seems like it's available, feel free to take it @panpiort8

Ok, I'll take it (then I suppose 'help wanted' is misleading)

Could someone review above PR?

We're busy mostly with the release at the moment. We should be able to focus on this more if you ping in 3 weeks ;)

Ok, I fully understand. I'll ping soon :)

gentle ping (;

Could someone review above PR? (gentle ping no. 2)

Could someone review above PR? (gentle ping no. 2)

Has this been finally implemented? I read the documentation, but there is nothing about SVD yet :(

It's almost implemented (PR #16067), but still needs pretty much work to be merged. Feel free to continue this PR.

Has there been any progress on this recently? Looking forward to decreasing my reliance on scikit-bio for PCoA.

@jolespin Current progress is in PR #22330 which provides and SVD implementation to the metric MDS context.

There is some confusion in the discussion above.

PCoA is another name for "classical MDS". It takes a matrix of pairwise distances, performs some linear algebra and eigendecomposition, and gets the embedding. The loss function that is minimized is sometimes called "strain". If the input pairwise distances are Euclidean distances between some vectors, then this procedure is equivalent to PCA of those vectors. In contrast, "metric MDS" optimizes a DIFFERENT loss function called "stress", and can only be solved iteratively. These are two different optimization problems with two different solutions. There is also "non-metric MDS" which is a yet another different thing.

I fully agree that classical MDS aka PCoA should be implemented in sklearn. My question is: what is the best API for that?

Ideally I would prefer algorithm={"metric", "classical", "non-metric"}. But currently there is metric={True, False} switch. So the least invasive option (that would not require a deprecation cycle) would be to add a separate parameter classical={True, False}.

What do people think?

@amueller @adrinjalali What do you think about the optimal API (see my comment above)? Does classical={True, False} look like a reasonable solution? Or should we go with algorithm={"metric", "classical", "non-metric"} and gradually deprecate metric={True, False}?

I think the algorithm path is the nicest API FAICT. To move this forward we need reviewers for #22330

I would like to understand the specific stress function currently implemented in scikit-learn for both metric and non-metric MDS. Does the implementation follow the Kruskal stress function as described in the referenced papers in the documentation? If not, could you kindly clarify the formula used for each and any key differences?
I previously raised this #30240 but was advised to add my thoughts here. Thank you for your patience!

I would like to understand the specific stress function currently implemented in scikit-learn for both metric and non-metric MDS.

I replied in more detail in your closed issue, but I think Wikipedia has a good overview:

• https://en.wikipedia.org/wiki/Multidimensional_scaling#Metric_multidimensional_scaling_(mMDS)
• https://en.wikipedia.org/wiki/Multidimensional_scaling#Non-metric_multidimensional_scaling_(NMDS).

I made a new PR to implement classical MDS aka PCoA: #31322.