Timeline for How to learn a continuous function?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Jul 5, 2020 at 14:49 | comment | added | Rajesh D | Agree, the finite arithmetic precision to represent data points $x_i$ is problematic. Even if you have a large number of data points, it won't make up for it. | |
| Jul 5, 2020 at 11:05 | comment | added | Ivan Meir | However if you are allowed to sample at any scale then you can compensate for the limited precision of each sample by making more samples over different scales. In general precision and data size are intimately connected and make up to the total number of bits required to store your sample data. This is what you generally want to limit in the real world. | |
| Jul 5, 2020 at 10:58 | comment | added | Ivan Meir | Yes the precision required in general will depend on $f$ as clearly if you required a large number of data points to get your sampling of $\Omega$ within $\delta$ then you need enough precision to prevent sample data points merging which would not then allow you to pick up the fine details of the function - it could oscillate with arbitrarily small period. If you take only functions with variation bounded by a constant value you can do it I think. | |
| Jul 5, 2020 at 10:15 | comment | added | Rajesh D | @AryehKontorovich : Yes bound on precision. Not the number of data points. Precision means arithmetic precision of floating point number. Suppose the function value at a data point $x$ is $f(x)$, the number of bits used to represent the real number $f(x)$ is what I meant by arithmetic precision. Like we use double precison floating point numbers and single precision floating point numbers in matlab. | |
| Jul 5, 2020 at 10:03 | comment | added | Aryeh Kontorovich | Your conjecture specifically states "one can derive a bound on required precision p that depends only on Ω and epsilon and is independent of f". | |
| Jul 5, 2020 at 9:40 | comment | added | Rajesh D | If you read my question, " at sufficiently large but finite number of data points", so the number of data points required need not be bounded by a bound independent of $f$. They just have to be finite for a given $f$. The bound I am conjecturing is on required precision, and that be independent of $f$. So your answer is valid for all continuous functions, but the problem I am worried is that you may need to draw sample points infinite times to get an $delta$-even distribution? I hope I am wrong. | |
| Jul 5, 2020 at 9:40 | comment | added | Aryeh Kontorovich | So you could do this for an equicontinuous family of functions on a compact domain. | |
| Jul 5, 2020 at 9:28 | history | answered | Ivan Meir | CC BY-SA 4.0 |