Sample projected onto principal component Working through digit recognition on the MNIST data set, one of the example problems reduces the dimensionality of the feature set, which is 768 samples, down to 18 using PCA. For illustration purposes, we project the input examples onto the principal components.

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*What exactly does this projection mean, how should it be interpreted intuitively?

*What is the distinction between a principal component and the projection of the input sample onto a principal component?


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*Projection means that we use the axis proposed by the principal component to map the data to this new space. Mathematically this is usually called a vector projection.

*Principal component (PC) is the axis. The projection is the point on the axis. So for example, we have a principal component $\text{PC}_1$ that takes our 768-long vector $x_i$ representing a written number and we make the relevant "vector projection" as this is defined by  $\text{PC}_1$ to get the  $\text{PC}_1$ score for that $x_i$ (i.e. the projection along the axis proposed by the PC).

In case you haven't seen it already, check the CV.SE thread on: Making sense of principal component analysis, eigenvectors & eigenvalues, it is very good.
