Timeline for Quantization of normal distribution
Current License: CC BY-SA 4.0
8 events
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| Apr 12, 2021 at 19:09 | comment | added | Iosif Pinelis | Previous comment continued: Here, the reduction of the dimension from $2n-1$ to $n$, as well as the special structure of $S_*(x)$, will help a lot. | |
| Apr 12, 2021 at 19:09 | comment | added | Iosif Pinelis | @Steve : No, I did not mean finding numerically zeroes of derivatives of $S_*$. What I meant is this: According to the interval arithmetic method, partition $\mathbb R$ into, say, $k$ subintervals and then accordingly partition $\mathbb R^n$ into $k^n$ $n$-dimensional boxes. Using the partial derivatives (possibly of higher orders), bound $S_*$ from below on each box. The lower bounds for most of the boxes will be greater then the quasi-minimum of $S_*$ found by Mathematica (say). Work similarly with each of the remaining boxes, etc. | |
| Apr 12, 2021 at 17:54 | comment | added | Steve | Thanks for the explanation. I definitely see the benefit of the reformulation. Do I understand correctly that to obtain the concrete values for $n=10$ you posted (so, what mathematica is doing), one calculates numerically where the derivative of $S_*$ is zero? I am wondering whether one can show that $S_*$ really has a unique global minimum. $S_*$ is not convex as far as I can tell? | |
| Apr 12, 2021 at 17:26 | comment | added | Iosif Pinelis | Previous comment continued: If, instead of $\varphi$, you have some other density, then the calculations could be more difficult, depending of course on the density. | |
| Apr 12, 2021 at 17:26 | comment | added | Iosif Pinelis | @Steve : I think the main benefit of this answer is the reduction of the number of variables from $2n-1$ to $n$. As for the structure of $S_*(x)$, note that its partial derivatives in $x_j$ depend only on at most three variables: $x_ j,x_{j-1},x_{j+1}$. So, these partial derivatives can be bounded comparatively easily using the interval arithmetic. This should help quite a bit. I have not done such specific calculations -- which is of course a lot of work, but which I think can be done. | |
| Apr 12, 2021 at 6:52 | comment | added | Steve | Thanks for the answer! I don't quite understand how the interval arithmetic method is used to solve problem (3). As I understand it, additional to the interval arithmetic method there has to be some optimization procedure or optimality criteria that is applied? I don't yet see what the important structure of $x \mapsto S_*(x)$ is which is utilized to solve (3). (For instance, would any other density $\varphi$ be valid as well?). Could you perhaps elaborate on that? | |
| Apr 11, 2021 at 19:31 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
added 185 characters in body
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| Apr 11, 2021 at 19:14 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |