What is the loss function defined by Mnih and Hinton in their paper “Learning to Label Aerial Images from Noisy Data”?
Section 3.3 simply gives the equation for the negative log-likelihood.
They say that it takes the form of cross-entropy (because it just looks like a cross-entropy equation, perhaps?), but mathematically it seems to come from the fact that they define the model in equation 2 to follow a Bernoulli distribution, which can be 0 or 1 with probabilities p or q respectively:

$$
\Pr(X=1)=p=1-\Pr(X=0)=1-q
$$

The probability mass function of the Bernoulli looks like this:

$$
f(x;p)=px+(1-p)(1-x)\!\quad {\text{for }}x\in \{0,1\}
$$

and the maximum likelihood for the Bernoulli looks like this:

$$
L(p) = \prod_{i+1}^n p^{x_1} (1 - p)^{1 - x_1}
$$

Taking the log (to get the log-likelihood that they mention), gives something of the form we see in equation 3:

$$
\log{L(p)} = \log{p}\sum_{i=1}^n x_i + \log{(1-p)}\sum_{i=1}^n (1-x_i)
$$

Then you can takes the sum out the front as it is over the image patches:

$$
\sum_{i}^{w^2_m} (\tilde{m}_i ln\hat{m}_i + (1-\tilde{m}_i)ln(1- \hat{m}_i))
$$
So here 
This sums over all the pixel-wise output units in each image patch (map: m), hence why the summation goes to $w_m^2$; the width of the image patch squared. $\tilde{m}$ is a realisation (possible outcome) of $\hat{m}$, just as $x$ is a realisation of distribution $p$ in the above equations.

This looks like the binary cross-entropy loss function, which is why I think they say:


  For the model given in Equation 2 the negative log likelihood
  takes the form of a cross entropy between the
  patch $\tilde{m}$ derived from the given map and the predicted
  patch $\hat{m}$