Timeline for Is the Gaussian Kernel still a valid Kernel when taking the negative of the inner function?
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15 events
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| Jan 21, 2020 at 4:09 | comment | added | Danica | @Mathmath Yes. But we’re not trying to prove that a kernel is psd; we’re showing that it’s not by exhibiting a specific $\{x_1, x_2\}$ for which the kernel matrix is not psd; hence the kernel function is not psd. | |
| Jan 20, 2020 at 21:44 | comment | added | Mathmath | @Dougal, I think the answer you gave is correct, but to prove that a kernel is PSD, we need to show that for any set of points $\{x_1,...x_n\}$, the matrix $K=[k(x_i,x_j)]$ is PSD. | |
| Oct 27, 2016 at 13:23 | comment | added | Danica | @tomka Not quite; $\varphi(r) = r$ works, as given by an example there. It depends on every derivative. | |
| Oct 27, 2016 at 13:22 | comment | added | tomka | @Dougal so a completely monotonous function cannot go against infinity, basically. It is evaluated positively on $[0,\inf]$ but has negative slope. | |
| Oct 27, 2016 at 13:17 | comment | added | Danica | @tomka Yes, $\varphi^{(l)}$ represents the $l$th derivative. | |
| Oct 27, 2016 at 13:16 | comment | added | tomka | @Dougal What is $\phi'$ or $\phi^{(l)}$ in their notation? The derivative? | |
| Oct 27, 2016 at 13:08 | comment | added | Danica | @tomka "Completely monotone" doesn't just mean "monotone everywhere" or something like that; it's defined e.g. in Definition 2.5.1 of the linked book chapter. One of the requirements is that $\varphi'(r) \le 0$, which holds for $f(r) = \exp\left( - \alpha r \right)$ when $\alpha \ge 0$ but not for $\alpha < 0$. | |
| Oct 27, 2016 at 13:03 | comment | added | tomka | @Dougal but doesn't the logic of monotonicity in the answer apply also if $\alpha<0$ | |
| Oct 27, 2016 at 12:59 | comment | added | Danica | @tomka That question is about $\exp(- \alpha \lVert x - y \rVert )$ for $\alpha > 0$; it doesn't conflict with this answer. | |
| Oct 27, 2016 at 12:55 | comment | added | tomka | @Dougal I just found a similar discussion on mathematics SE which seems to suggest (as acceped answer) the opposite of what you claim. I cannot fully follow their argument. Can you check? math.stackexchange.com/questions/248976/… | |
| Oct 27, 2016 at 8:14 | vote | accept | tomka | ||
| Oct 27, 2016 at 0:02 | history | edited | Danica | CC BY-SA 3.0 |
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| Oct 26, 2016 at 23:49 | history | edited | Danica | CC BY-SA 3.0 |
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| Oct 26, 2016 at 23:46 | history | edited | Sycorax♦ | CC BY-SA 3.0 |
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| Oct 26, 2016 at 23:42 | history | answered | Danica | CC BY-SA 3.0 |