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### Reason for relaxing limit in derivative in this context?

The equation $$f(x + \epsilon) \approx f(x) + \epsilon f'(x)$$ is justified from taylor series. It is not derived from the limit definition of a derivative. Let $f'(a)$ and $f''(a)$ exist for $a$ in ...
• 1,131
1 vote

### Reason for relaxing limit in derivative in this context?

It is just an assumption. If we assume $\epsilon$ is small enough (depending on the function $f$), you can remove the limit $\lim_{\epsilon \to 0}$ for the approximation.
• 1,723
1 vote

### Why is my derivation of the back-propagation equations inconsistent with Andrew Ng's slides from Coursera?

TL;DR: This has to do with the way A. Ng has defined back propagation for the course. Left Column This is only with respect to one input example and so the $\frac{1}{m}$ factor reduces to 1 and can ...
• 1,076
1 vote
Accepted

### What is the correct partial derivative of $Y^c$ with respect to $A_{ij}^{kc}$?

After some reflection I noticed that the actual final expression should contain derivatives of the alphas w.r.t. $A_{ij}^k$ too, because the alphas cannot be constants that do not depend on $A_{ij}^k$....
• 141
1 vote

### What is the rigorous and formal definition for the direction pointed by a gradient?

If $u$ is a vector, the direction pointed by the vector is defined as $\dfrac{u}{\lVert {u}\rVert}$ where $\lVert \cdot \rVert$ is the 2 norm (euclidean norm).
• 1,131
1 vote

Consider a function $f(x) : \mathcal{R}^m\rightarrow\mathcal{R}^n$ defined for $x \in X$. If $f$ is Lipschitz continuous, it has three main properties: $f(x)$ is continuous for all $x \in X$ $\frac{d ... • 1,131 1 vote ### Why does critical points and stationary points are used interchangeably? From reading the text, it's clear that the authors are using critical point to mean the same thing as stationary point, so they are not using the proper mathematical definition. More generally, ... • 1,131 1 vote ### Why does critical points and stationary points are used interchangeably? A critical point of a function$f$can be a stationary point (i.e.$f'(x) = 0$), or a point where the derivative is undefined (for example, in the case of the absolute value function$f(x)$,$x=0\$ is ...
• 37k

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