# What is a "residual mapping"?

A recent paper by He et al. (Deep Residual Learning for Image Recognition, Microsoft Research, 2015) claims that they use up to 4096 layers (not neurons!).

I am trying to understand the paper, but I stumble about the word "residual".

Could somebody please give me an explanation / definition what residual means in this case?

## Examples

We explicitly reformulate the layers as learning residual functions with reference to the layer inputs, instead of learning unreferenced functions.

[...]

Instead of hoping each few stacked layers directly fit a desired underlying mapping, we explicitly let these layers fit a residual mapping. Formally, denoting the desired underlying mapping as $\mathcal{H}(x)$, we let the stacked nonlinear layers fit another mapping of $\mathcal{F}(x) := \mathcal{H}(x)−x$. The original mapping is recast into $\mathcal{F}(x)+x$. We hypothesize that it is easier to optimize the residual mapping than to optimize the original, unreferenced mapping

• This might be a language problem. If you know the German translation of "residual" in this context, I would be happy about it, too. Jan 24, 2016 at 16:49

It's $F(x)$; the difference between the mapping $H(x)$ and its input $x$. It's a common term in mathematics (DE).