6
votes
Accepted
How is the max function differentiable wrt multiple arguments?
Not sure wether this question is suitable here or should rather be on the math SE but here goes.
First restrict to two arguments, the general case is similar. Let $f(x,y):=max(x,y)$. This is a ...
- 281
2
votes
Accepted
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 ...
- 1,211
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.
- 1,751
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 ...
- 1,106
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).
- 1,211
1
vote
What does it mean "having Lipschitz continuous derivatives"?
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 ...
- 1,211
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, ...
- 1,211
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 ...
- 39.7k
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