Why do we use PCA to speed up learning algorithms when we could just reduce the number of features? In a machine learning course, I learned that one common use of PCA (Principal Component Analysis) is to speed up other machine learning algorithms. For example, imagine you are training a logistic regression model. If you have a training set $(x^{(i)},y^{(i)})$ for i from 1 to n and it turns out the dimension of your vector x is very large (let's say a dimensions), you can use PCA to get a smaller dimension (let's say k dimensions) feature vector z. Then you can train your logistic regression model on the training set $(z^{(i)},y^{(i)})$ for i from 1 to n. Training this model will be faster because your feature vector has less dimensions.
However, I don't understand why you can't just reduce the dimension of your feature vector to k dimensions by just choosing k of your features at random and eliminating the rest. 
The z vectors are linear combinations of your a feature vectors. Since the z vectors are confined to a k-dimensional surface, you can write the a-k eliminated feature values as a linear function of the k remaining feature values, and thus all the z's can be formed by linear combinations of your k features. So shouldn't a model trained on an training set with eliminated features have the same power as a model trained on a training set whose dimension was reduced by PCA? Does it just depend on the type of model and whether it relies on some sort of linear combination?
 A: PCA reduces features while preserving the variance/information in the original data. This helps with enabling computation while not losing the data's resemblance of reality. 
A: Let's say you initially have $p$ features but this is too many so you want to actually fit your model on $d < p$ features. You could choose $d$ of your features and drop the rest. If $X$ is our feature matrix, this corresponds to using $XD$ where $D \in \{0,1\}^{p \times d}$ picks out exactly the columns of $X$ that we want to include. But this ignores all information in the other columns, so why not consider a more general dimension reduction $XV$ where $V \in \mathbb R^{p \times d}$? This is exactly what PCA does: we find the matrix $V$ such that $XV$ contains as much of the information in $X$ as possible. Not all linear combinations are created equally. Unless our $X$ matrix is so low rank that a random set of $d$ columns can (with high probability) span the column space of all $p$ columns we will certainly not be able to do just as well as with all $p$ features. Some information will be lost, and so it behooves us to lose as little information as possible. With PCA, the "information" that we're trying to avoid losing is the variation in the data.
As for why we restrict ourselves to linear transformations of the predictors, the whole point in this use-case is computation time. If we could do fancy non-linear dimension reduction on $X$ we could probably just fit the model on all of $X$ too. So PCA sits perfectly at the intersection of fast-to-compute and effective.
A: PCA solution
First, beware when using PCA for this purpose. As I wrote in response to a related question PCA does not necessarily lead to selection of features that are informative for the regression you intend to do (see also Jolliffe 1982). 
OP proposed solution
Now consider the proposed alternative mechanism: reduce the dimension of your feature vector to k dimensions by just choosing k of your features at random and eliminating the rest. Now in the problem statement we were asked to suppose that dimension of your vector x is very large. Let's call this dimension $p$
There are $pCk$ ways to choose $k$ predictors from a group of $p$. To give an example if $p=1000$ and we choose $k=5$ predictors from the dataset there would be $\approx 8.25 \times 10^{12}$ different models we would have to fit. And that's supposing we knew that $k=5$, and not $k=6$ etc etc. Put simply, it's not a problem you'd want to brute force in a large $p$ setting. 
Suggested solution
To cope with regressions where $p$ is large a number of penalised regression strategies have been proposed. In particular the LASSO method will do dimension reduction while constructing a regression model by zeroing out the contribution from predictors that do not contribute enough to the model. There is a very clever algroithm (LARS) to fit the model efficiently.
