Deep learning ... notion of depth I'm reading from http://www.iro.umontreal.ca/~bengioy/papers/ftml.pdf around page 8.
Here is a quote that I'm not understanding:
"If we include affine operations and their possible composition with sigmoids 
 in the set of computational elements, linear regression and logistic regression 
have depth 1, i.e., have a single level."

Depth or Depth of Architecture refers "to the depth of that graph, i.e. the longest path from an input node to an output node."
I can see why the graph represented in the image has depth of 4. I'm not sure how to think about, say, linear regression in these terms.
 A: Linear regression is of the form $w^T x + b$. "If we include affine operations...in the set of computational elements," then $f(x) = w^T x + b$ is an affine operation, and so linear regression only applies a single "computational element" to the input, and the graph is of depth one.
Similarly, logistic regression is of the form $g(x) = \sigma(w^T x + b)$, where $\sigma(x) = 1 / (1 + e^{-x})$. This as an affine operation composed with a sigmoid, which we're including in the set of computational elements, so by definition $g$ is only applying a single operation to the input $x$ and so it's also of depth 1.
A: Once a linear regresion model is fitted, the output is simply a linear combination of the inputs (the coefficients are the fitted parameters of the regression model, $\hat{\beta}$.)  In particular, the model prediction is $y=\hat{\beta}^{T}x$, or if a binary output is desired, we can apply a threshhold:
$y=\left\{
\begin{array}{ll}
1 & \hat{\beta}x \geq c \\
0 & \hat{\beta}x < c
\end{array}
\right.
$
In logtistic regression, the prediction model is only slightly more complicated.  The predicted probability is:  
$y=\mbox{logit}^{-1}(\hat{\beta}^{T}x)$  
In many cases we make a binary yes or no prediction by applying a threshhold to the above formula.  Since $\mbox{logit}^{-1}$ is monotone, this is just a threshhold on $\hat{\beta}x$:
$y=\left\{
\begin{array}{ll}
1 & \hat{\beta}x \geq c \\
0 & \hat{\beta}x < c
\end{array}
\right.
$
A classical perceptron node computes $\hat{\beta}x$ and applies a threshhold, so a single perceptron node can be used to implement either a linear regression or logistic regression classifier.    
