A gentle introduction to deep learning in medical image processing

Network training

Having gained basic insights into neural networks and their basic topology, we still need to discuss how its parameters θ are actually determined. The answer is fairly easy: gradient descent. In order to compute a gradient, we need to define a function that measures the quality of our parameter set θ, the so-called loss function L(θ). In the following, we will work with simple examples for loss functions to introduce the concept of back-propagation, which is the algorithm that

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Figure 3. A decision tree allows to describe any partition of space and can thus model any decision boundary. Mapping the tree into a one-layer network is possible. Yet, there still is significant residual error in the resulting function. In the center example, ‹ 0.7. In order to reduce this error further, a higher number of neurons would be required. If we construct a network with one node for every inner node in the first layer and one node for every leaf node in the second layer, we are able to construct a network that results in ‹ = 0.

is commonly used to efficiently compute gradients for neural network training.
We can represent a single-layer fully connected network with linear activations simply as yˆ = f^ˆ (x) = Wx, i.e., a
where η is the so-called learning rate and j is used to index the iteration number.
Now, let us consider a slightly more complicated net- work structure with three layers yˆ = f^ˆ ₃(f^ˆ ₂(f^ˆ ₁(x))) =

matrix multiplication. Note that the network’s output is now

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multidimensional with yˆ, y R^m. Using an L2-loss, we end up with the following objective function:



₂ 2 ₂ 2
L(θ) = ¹ ||f^ˆ (x) − y||² = ¹ ||Wx − y||². (4)
In order to update the parameters θ = W in this example, we need to compute