# [scikit-learn] Why is subset invariance necessary for transfom()?

Charles Pehlivanian pehlivaniancharles at gmail.com
Tue Jan 21 21:19:45 EST 2020

```    This is what I thought we usually do. It looks like you said we are
doing a greedy transform.
I'm not sure I follow that. In particular for spectral embedding for
example there is a pretty way to describe
the transform and that's what we're doing. You could also look at doing
transductive learning but that's
not really the standard formulation, is it?

Batch transform becomes greedy if one does:

for x_i in X:
X_new_i = self.transform(x_i)

I said that LLE uses greedy algorithm. The algorithm implemented is
pointwise. It may be that that's the only approach (in which case it's not
greedy), but I don't think so - looks like all of the spectral embedding,
lle, mds transforms have batch versions. So I probably shouldn't call it
greedy. Taking a *true* batch transform and enclosing it in a loop like
that - I'm calling that greedy. I'm honestly not sure if the LLE qualifies.

Spectral embedding - agree, the method you refer to is implemented in
fit_transform(). How to apply to oos points?

Non-distributable, non-subset-invariant, optimal batch transform
Can you give an example of that?

Most of the manifold learners can be expressed as solutions to
eigenvalue/vector problems. For MDS batch transform, form a new constrained
double-centered distance matrix and solve a constrained least-squares
problem that mimics the SVD solution to the eigenvalue problem.  They're
all like this - least-squares estimates for some constrained eigenvalue
problem. The question is whether you want to solve the full problem, or
solve on each point, adding one row and optimzing each time, ... that would
be subset-invariant though.

For this offline/batch approach to an oos transform, the only way I see to
make it pass tests is to enclose it in a loop as above. That's what I see
at least.

On Tue, Jan 21, 2020 at 8:35 PM Andreas Mueller <t3kcit at gmail.com> wrote:

>
>
> On 1/21/20 8:23 PM, Charles Pehlivanian wrote:
>
> I understand - I'm kind of conflating the idea of data sample with test set, my view assumes there are a sample space of samples, might require rethinking the cross-validation setup...
>
> I also think that part of it relies on the notion of online vs. offline algorithm. For offline fits, a batch transform (non-subset invariant) is preferred. For a transformer that can only be used in an online sense, or is primarily used that way, keep the invariant.
>
>
> I see 3 options here - all I can say is that I don't vote for the first
>
> + No transform method on the manifold learners, so no cross-validation
>
> This is what I thought we usually do. It looks like you said we are doing
> a greedy transform.
> I'm not sure I follow that. In particular for spectral embedding for
> example there is a pretty way to describe
> the transform and that's what we're doing. You could also look at doing
> transductive learning but that's
> not really the standard formulation, is it?
>
> + Pointwise, distributable, subset-invariant, suboptimal greedy transform
>
> + Non-distributable, non-subset-invariant, optimal batch transform
>
> Can you give an example of that?
>
> -Charles
>
> On Mon., Jan. 20, 21:24:52 2020 <joel.nothman at gmail.com <scikit-learn%40python.org?Subject=Re%3A%20%5Bscikit-learn%5D%20Why%20is%20subset%20invariance%20necessary%20for%0A%20transfom%28%29%3F&In-Reply-To=%3CCAAkaFLWfWyu%2BDdQ3RX5tBays6jLX6A3W_QpqAcWn_RAxbRz5cQ%40mail.gmail.com%3E>> wrote
>
> I think allowing subset invariance to not hold is making stronger
>
> assumptions than we usually do about what it means to have a "test set".
> Having a transformation like this that relies on test set statistics
> implies that the test set is more than just selected samples, but rather
> that a large collection of samples is available at one time, and that it is
> in some sense sufficient or complete (no more samples are available that
> would give a better fit). So in a predictive modelling context you might
> have to set up your cross validation splits with this in mind.
>
> In terms of API, the subset invariance constraint allows us to assume that
> the transformation can be distributed or parallelized over samples. I'm not
> sure whether we have exploited that assumption within scikit-learn or
> whether related projects do so.
>
> I see the benefit of using such transformations in a prediction Pipeline,
> and really appreciate this challenge to our assumptions of what "transform"
> means.
>
> Joel
>
> On Tue., 21 Jan. 2020, 11:50 am Charles Pehlivanian, <pehlivaniancharles at gmail.com <https://mail.python.org/mailman/listinfo/scikit-learn>> wrote:
>
> >* Not all data transformers have a transform method. For those that do,
> *>* subset invariance is assumed as expressed
> *>* in check_methods_subset_invariance(). It must be the case that
> *>* T.transform(X)[i] == T.transform(X[i:i+1]), e.g. This is true for classic
> *>* projections - PCA, kernel PCA, etc., but not for some manifold learning
> *>* transformers - MDS, SpectralEmbedding, etc. For those, an optimal placement
> *>* of the data in space is a constrained optimization, may take into account
> *>* the centroid of the dataset etc.
> *>>* The manifold learners have "batch" oos transform() methods that aren't
> *>* implemented, and wouldn't pass that test. Instead, those that do -
> *>* LocallyLinearEmbedding - use a pointwise version, essentially replacing a
> *>* batch fit with a suboptimal greedy one [for LocallyLinearEmbedding]:
> *>>*     for i in range(X.shape[0]):
> *>*         X_new[i] = np.dot(self.embedding_[ind[i]].T, weights[i])
> *>>* Where to implement the batch transform() methods for MDS,
> *>* SpectralEmbedding, LocallyLinearEmbedding, etc?
> *>>* Another verb? Both batch and pointwise versions? The latter is easy to
> *>* implement once the batch version exists. Relax the test conditions?
> *>* transform() is necessary for oos testing, so necessary for cross
> *>* validation. The batch versions should be preferred, although as it stands,
> *>* the pointwise versions are.
> *>>* Thanks
> *>* Charles Pehlivanian
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