Why is the property of neural networks being robust to variances in the input referred to as invariance?
Is it that the neural network's output is invariant, regardless of a variance in the input?
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The term "invariance" or "invariant" in this context is not directly related to the statistical meaning of the term "variance" - it is using the basic English meaning of the words variant/varying/etc: invariant as in not varying.
In the context of biological and artificial neural networks it is usually referring to a network that gives the same/similar response to an input that is transformed in some way; for vision, common transformations are of scale, position, orientation/rotation, color, etc.