deletions | additions       diff --git a/subsection_Evaluation_Scheme_label_sec__.tex b/subsection_Evaluation_Scheme_label_sec__.tex  index bc35455..104ba56 100644  --- a/subsection_Evaluation_Scheme_label_sec__.tex  +++ b/subsection_Evaluation_Scheme_label_sec__.tex 
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\begin{align}  \ell(x, y) &= x - y\\  d_a(h_i, h_j) &= \text{cosine\_similarity}(h_i, \text{CS}(h_i,  h_j)\\ d_e(h_i, h_j) &= f_X(pos_i \mid \mu, \Sigma) = f_X(x_i, y_i \mid \mu, \Sigma)\\  \end{align} 
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\end{bmatrix}  \end{align}  The cosine similarity (CS) is defined as:  \begin{align}  CS(h_i, h_j) = \frac{h_i^Th_j}{||h_i||||h_j||}  \end{align}  The cosine similarity has the convenient property that its values are bounded between $-1$ and $1$. In the present case, since the elements of $h_i$ and $h_j$ are non-negative, it is even bounded between $0$ and $1$. The  function $f_x$ describes the non-normalized  probability density function of the normal distribution. distribution: $f_x = e^{- \frac{(x - \mu)^2}{2 \sigma ^ 2}}$. This function is also bounded between $0$ and $1$, which makes the functions $f_X$ and $CS$ easily comparable.  The idea behind the global loss function $L$ is that histograms in closeby areas should be similar and the similarity should decrease the further away  two positions are. This is modeled as a 2-dimensional Gaussian with zero  covariance. The variance is depended on the