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you are asking for the condition I would say. If you skim through the formulae, the german explanation is more detailed.

In particular, the absolute condition is defined as $\kappa_{\text{abs}}:= \lim \sup _{\tilde{x} \rightarrow x} \frac{||f(x) - f(\tilde{x})||}{||x - \tilde{x}||}$, which means $\kappa_{\text{abs}} \geq 0 $ is the smallest number such that there is a $\delta >0 $ so that all $\tilde{x}$ with $||\tilde{x}-x|| < \delta $ it holds that $|| f(\tilde{x})-f(x) || \leq \kappa_\text{abs} || \tilde{x} - x ||.$

The relative condition $\kappa_{\text{rel}} \geq 0 $ is the smallest number such that there is a $\delta > 0 $ so that all $\tilde{x}$ with $||\tilde{x}-x|| < \delta $ satisfy: $ \frac{|| f(\tilde x)-f(x) ||}{||f(x)||} \leq \kappa_\text{rel} \frac{|| \tilde{x} - x ||}{||x||}.$

The difference is thus, that the $\kappa_{\text{rel}}$ compares the relative change of the output with the relative change of the input.

In general, let $\tilde{f}$ denote an approximation of the real function $f$, $\tilde{x}$ the input with some noise, then there are 4 categories in numerics:

• condition: Analyze $||f(x)-f(\tilde{x})||$
• stability: Analyze $||\tilde{f}(x)-\tilde{f}(\tilde{x})||$
• consistence: Analyze $||\tilde{f}(x)-f(x)||$
• convergence: Analyze $||\tilde{f}(\tilde{x})-f(x)||$.

Your question is also related to adverserial attacks. You can have a look into the literature.

you are asking for the condition I would say. If you skim through the formulae, the german explanation is more detailed.

In particular, the absolute condition at $x$ is defined as $\kappa_{\text{abs}}:= \lim \sup _{\tilde{x} \rightarrow x} \frac{||f(x) - f(\tilde{x})||}{||x - \tilde{x}||}$, which means $\kappa_{\text{abs}} \geq 0 $ is the smallest number such that there is a $\delta >0 $ so that all $\tilde{x}$ with $||\tilde{x}-x|| < \delta $ it holds that $|| f(\tilde{x})-f(x) || \leq \kappa_\text{abs} || \tilde{x} - x ||.$

The relative condition $\kappa_{\text{rel}} \geq 0 $ at $x$ is the smallest number such that there is a $\delta > 0 $ so that all $\tilde{x}$ with $||\tilde{x}-x|| < \delta $ satisfy: $ \frac{|| f(\tilde x)-f(x) ||}{||f(x)||} \leq \kappa_\text{rel} \frac{|| \tilde{x} - x ||}{||x||}.$

The difference is thus, that the $\kappa_{\text{rel}}$ compares the relative change of the output with the relative change of the input.

In general, let $\tilde{f}$ denote an approximation of the real function $f$, $\tilde{x}$ the input with some noise, then there are 4 categories in numerics:

• condition: Analyze $||f(x)-f(\tilde{x})||$
• stability: Analyze $||\tilde{f}(x)-\tilde{f}(\tilde{x})||$
• consistence: Analyze $||\tilde{f}(x)-f(x)||$
• convergence: Analyze $||\tilde{f}(\tilde{x})-f(x)||$.

Your question is also related to adverserial attacks. You can have a look into the literature.

Note also that $\kappa_{\text{rel}} = \frac{||Df(x)||||x||}{||f(x)||}$, so that you could compute the condition number for any given neural network.

If you have a look at this definitions, at the parts "Kondition = Condition", "Stabilität = Stabilty", where $\tilde{x}$ means that there is noise on the input and $\tilde{f}$ means that the algorithm is an approximation of the correct function $f$, you see the different consideratins. If you compare this to the definiton on the english wikipedia page, the definition on the english page seems to be wrong. Anyhow you are asking for the absolute condition I would say, so $\kappa_{\text{abs}}:= \lim \sup _{\tilde{x} \rightarrow x} \frac{||f(x) - f(\tilde{x})||}{||x - \tilde{x}||}$, which means $\kappa_{\text{abs}} > 0 $ is the smallest number such that there is a $\delta >0 $ so that all $x$ with $||\tilde{x}-x|| < \delta $ it holds that $|| f(\tilde x)-f(x) || \leq \kappa_\text{abs} || \tilde x - x ||.$

Anyhow, it is connected to the adverserial attacks. You can have a look into the literature.

you are asking for the condition I would say. If you skim through the formulae, the german explanation is more detailed.

In particular, the absolute condition is defined as $\kappa_{\text{abs}}:= \lim \sup _{\tilde{x} \rightarrow x} \frac{||f(x) - f(\tilde{x})||}{||x - \tilde{x}||}$, which means $\kappa_{\text{abs}} \geq 0 $ is the smallest number such that there is a $\delta >0 $ so that all $\tilde{x}$ with $||\tilde{x}-x|| < \delta $ it holds that $|| f(\tilde{x})-f(x) || \leq \kappa_\text{abs} || \tilde{x} - x ||.$

The relative condition $\kappa_{\text{rel}} \geq 0 $ is the smallest number such that there is a $\delta > 0 $ so that all $\tilde{x}$ with $||\tilde{x}-x|| < \delta $ satisfy: $ \frac{|| f(\tilde x)-f(x) ||}{||f(x)||} \leq \kappa_\text{rel} \frac{|| \tilde{x} - x ||}{||x||}.$

The difference is thus, that the $\kappa_{\text{rel}}$ compares the relative change of the output with the relative change of the input.

In general, let $\tilde{f}$ denote an approximation of the real function $f$, $\tilde{x}$ the input with some noise, then there are 4 categories in numerics:

• condition: Analyze $||f(x)-f(\tilde{x})||$
• stability: Analyze $||\tilde{f}(x)-\tilde{f}(\tilde{x})||$
• consistence: Analyze $||\tilde{f}(x)-f(x)||$
• convergence: Analyze $||\tilde{f}(\tilde{x})-f(x)||$.

Your question is also related to adverserial attacks. You can have a look into the literature.

If you have a look at this definitions, you see "Kondition = Condition", "Stabilität = Stabilty", where $\tilde{x}$ means that there is noise on the input and $\tilde{f}$ means that the algorithm is an approximation of the correct function $f$. If you compare this to the definiton on the english wikipedia page, the definition on the english page seems to be wrong.

Anyhow, it is connected to the adverserial attacks. You can have a look into the literature.

If you have a look at this definitions, at the parts "Kondition = Condition", "Stabilität = Stabilty", where $\tilde{x}$ means that there is noise on the input and $\tilde{f}$ means that the algorithm is an approximation of the correct function $f$, you see the different consideratins. If you compare this to the definiton on the english wikipedia page, the definition on the english page seems to be wrong. Anyhow you are asking for the absolute condition I would say, so $\kappa_{\text{abs}}:= \lim \sup _{\tilde{x} \rightarrow x} \frac{||f(x) - f(\tilde{x})||}{||x - \tilde{x}||}$, which means $\kappa_{\text{abs}} > 0 $ is the smallest number such that there is a $\delta >0 $ so that all $x$ with $||\tilde{x}-x|| < \delta $ it holds that $|| f(\tilde x)-f(x) || \leq \kappa_\text{abs} || \tilde x - x ||.$

Anyhow, it is connected to the adverserial attacks. You can have a look into the literature.