# Data transformation - pointwise or batch?

A data transformer performs a pre-preprocessing step ("transformation") before an estimator can fit or classify the data. The transformation step is a projection (any idempotent map: $$T^2 = T$$), usually to a lower dimensional space.

Another property satisfied by some transformers I'll call the "pointwise" property: for a set $$X = \{x_0, x_1, ..., x_n\}$$, $$T(X)\vert{x_i} = T(\{x_i\})$$

that is, the restriction of T fitted on $$X$$ to the point $$x_i$$ is equal to the transform of the singleton $$\{x_i\}$$.

Some transformers satisfy the pointwise property - PCA, kernel PCA, some don't - spectral embedding, MDS, or any projection that includes special handling of outliers, as those are dependent on the set in question. Those are still idempotent.

How important is the pointwise property? Do any pipeline steps require it? Cross validation does not, it just requires an out-of-sample transform method applied to the test sample. It would be applied batchwise on the entire sample, not pointwise, nothing seems to be violated. Is any desirable transformer property missing if the pointwise property is not satisfied?