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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.

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Yes. The fact that the variance, for example the scale or the orientation of an object in an image do not affect the NN results, makes the NN invariate to scale or orientation.

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