An exemplary real-valued prediction function for classification with the dashed black line being the decision boundary which separates the blue from the green dots.

<p>The blue dots are labeled negatively, the green dots are labeled positively. Left: Local gradient of the classification function at the prediction point. Right: Taylor approximation relative to a root point on the decision boundary. This figure depicts the intuition that a gradient at a prediction point <i>x</i>—here indicated by a square—does not necessarily point to a close point on the decision boundary. Instead it may point to a local optimum or to a far away point on the decision boundary. In this example the explanation vector from the local gradient at the prediction point <i>x</i> has a too large contribution in an irrelevant direction. The closest neighbors of the other class can be found at a very different angle. Thus, the local gradient at the prediction point <i>x</i> may not be a good explanation for the contributions of single dimensions to the function value <i>f</i>(<i>x</i>). Local gradients at the prediction point in the left image and the Taylor root point in the right image are indicated by black arrows. The nearest root point <i>x</i>₀ is shown as a triangle on the decision boundary. The red arrow in the right image visualizes the approximation of <i>f</i>(<i>x</i>) by Taylor expansion around the nearest root point <i>x</i>₀. The approximation is given as a vector representing the dimension-wise product between <i>Df</i>(<i>x</i>₀) (the black arrow in the right panel) and <i>x</i> − <i>x</i>₀ (the dashed red line in the right panel) which is equivalent to the diagonal of the outer product between <i>Df</i>(<i>x</i>₀) and <i>x</i> − <i>x</i>₀.