# Dimensionality reduction with b-splines - conceptual question

In this R code:

# data is say 1000 x 50 data matrix
require(fda)
n = 1000  # no of observations
k = 5    # reduced dimensionality
x = seq(0, 1, length.out = n)
splinebasis_B = create.bspline.basis(c(0, 1), k)
base_B = eval.basis(x, splinebasis_B)
data_reduced = data %*% base_B


as I understand, the last operation does not project (or otherwise reducingly transform) data just yet, but rather it gives us data_reduced on which we can run some fitting algos (regressions, trees etc.).

For this reason, we cannot claim that data_reduced is compressed version of data.

Is this correct? Am I missing anything? What's the right way to think about data_reduced? It's clearly mapped to a lower-dimensional space, and there is loss as data's variance got collapsed during the multiplication with fewer columns... but I feel like somehow data_reduced is not "optimal". It's not the best representation of data in k-dimensional space... What is it then?

For background, I'm talking specifically about using splines for dimensionality reduction, in particular as applied to functional data (screenshot)

Thus, if you have a set of points $$(x,y)$$, you can approximate the non-linear relation between $$x$$ and $$y$$ by first expanding x into a B-spline basis and then fitting the expanded basis to $$y$$ using a standard linear model such as linear regression.
So now if you wanted to reduce the dimensionality of several sets of $$(x,y)$$ pairs, you first do a basis expansion of all the $$x$$'s and then for each set, fit the B-spline expansion to the paired $$y$$'s. Then, the fitted coefficients from each $$(x,y)$$ set can be seen as a reduced rank approximation of the original $$(x,y)$$ set.
Note that it is vitally important that each $$x$$ be expanded at the same knot locations with the same degrees of freedom, otherwise comparing fitted coefficients becomes an apples to oranges comparison.
• thanks a lot, Cliff. Any change you could elaborate. What exactly is the operation for 'basis expansion of all the $x$'s? Is data_reduced matrix from the snippet such expanded version of $\boldsymbol X$? – Tim Jul 7 at 17:54