Using NMF for guitar pitch detection using fixed pitch template
The goal of non-negative matrix factorization (NMF) is to factorize (as the name says) a matrix, denoted by $\mathbf{X}\in\mathbb{R}^{(N\times K)}$ in the reference you mention, into the product of two matrices:
$$
\mathbf{W}\mathbf{H}\approx\mathbf{X},
$$
with $\mathbf{W}\in\mathbb{R}^{(N\times M)}$ the template, dictionary, or basis matrix and $\mathbf{H}\in\mathbb{R}^{(M\times K)}$ the activation matrix. The approximation sign ($\approx$) in the equation indicates that in many cases it is not possible to actually find a true factorization but rather an approximation.

Inspecting the above equation reveals that each column of $\mathbf{X}$ is approximated by a linear combination of columns of $\mathbf{W}$, with the coefficients of linear combination in $\mathbf{H}$. (Experimenting with some simple examples, e.g., with $N=3,\,M=3, K=5$ and $\mathbf{X}_{ij}\in\left\{0,\,1\right\}$, will surely help gaining insight into this fundamental behaviour).

In many cases it is desired, given a matrix $\mathbf{X}$, to determine both the dictionary matrix $\mathbf{W}$ and the activation matrix $\mathbf{H}$. Optionally, of course, only one can be estimated if the other is given.

Your goal, if I understand correctly, is to keep $\mathbf{W}$ fixed, estimate $\mathbf{H}$ and use the resulting activations to determine when which note has been played. That should work as a first attempt, I think.

After this coarse overview now for your questions:


You are right: to create the $\mathbf{W}$ matrix put an estimate of each note's power spectral density estimate (e.g., obtained with Welch's method) into a column of $\mathbf{W}$ (I'm not sure what you mean by "do the FFT for each, take its spectrum".).
Yes, the matrix $\mathbf{X}$ in the notation used in this answer ($V^{(k)}$ in the paper that you mention) contains the STDFT magnitude spectrogram of the full input recording.
Silence in the beginning of the recording will obviously have an influence on the first columns of $\mathbf{X}$ (or, respectively, $v_j^{(k)}$ for $j&lt;\text{some number}$). However, what do you guess will happen with $\mathbf{H}$ if the first columns of $\mathbf{X}$ contain zeros or only very small values? Correct: the first columns of $\mathbf{H}$ will also receive zeros or very small values.


I hope this answer helps in clarifying the basic functionality of the NMF algorithm. A rather intuitive description of the NMF idea is also given in this famous paper.