Table 1

Mathematical notations

X0 is the input tensor given to the network, and X refers to any input, sampled from the set X.
y is a vector of target classes corresponding to the input.
f is a network of L layers. The first layer is the closest to the input; the last layer is the closest to the output. A layer is a function.
g is a transparent function which aims at reproducing the behavior of f.
w and b are the weights and the bias associated to a linear function (e.g., in a fully connected layer).
u and v are locations (set of coordinates) corresponding to a node in a feature map. They belong respectively to the set U and V.
Ak(l)(u) is the value of the feature map computed by layer l, of K channels at channel k, at position u.
Rk(l)(u) is the value of a property back-propagated through the l + 1, of K channels at channel k, at position u. R(l) and A(l) have the same number of channels.
oc is the output node of interest (in a classification framework, it corresponds to the node of the class c).
Sc is an attribution map corresponding to the output node oc.
m is a mask of perturbations. It can be applied to X to compute its perturbed version Xm.
Φ is a function producing a perturbed version of an input X.
Γc is the function computing the attribution map Sc from the black-box function f and an input X0.

From: Chapter 22, Interpretability of Machine Learning Methods Applied to Neuroimaging

Cover of Machine Learning for Brain Disorders
Machine Learning for Brain Disorders [Internet].
Colliot O, editor.
New York, NY: Humana; 2023.
Copyright 2023, The Author(s)

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