User anton sukhinov - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T03:56:19Z http://mathoverflow.net/feeds/user/21661 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/101589/convexified-threshold-of-a-function Convexified threshold of a function Anton Sukhinov 2012-07-07T15:53:10Z 2012-07-08T08:40:26Z <p><strong>Upd.</strong> The question in a nutshell: <em>find convex set on plane which is «closest» to a given non-convex set</em>.</p> <p>It is given integrable function $0\leq f(x,y)\leq 1$ with bounded support: $f(x,y)=0$ when $x^2+y^2\geq R$. Needed to find function $t(x,y)$ which takes only two values («binary» function): <code>$t(x,y)\in\{0,1\}$</code> such as <code>$$\iint\limits_{x^2+y^2&lt;R}\left|f(x,y)-t(x,y)\right|dxdy\to\min.$$</code> We will call this task «threshold of a function». The solution is trivial: <code>$$t(x,y)=\left\{\begin{array}{cl}0&amp;\text{if }f(x,y)&lt;1/2;\\1&amp;\text{if }f(x,y)\geq1/2.\end{array}\right.$$</code> But what to do if I need to find threshold having convex support? I.e. I need to find a function <code>$c(x,y)\in\{0,1\}$</code> that have convex support and satisfies minimization criteria stated above among all binary functions with convex support. I can use any norm which will be convenient.</p> <p>I also interested in discrete analog of the problem: given set of points <code>$p_i=\{x_i,y_i\}\in\mathbb{R}^2$</code>, $i=1,...,N$, and values $0\leq v(p_i)\leq 1$. Find convex polygon $P$, such as <code>$$\sum_{p_i\in P}(1-v(p_i))+\sum_{p_i\notin P}v(p_i)\to\min.$$</code> I tried greedy algorithm to solve last problem: take convex hull of all points which have value $>1/2$ and then reduce convex hull point-by-point while target function decreases, but this fails for structures where 0-valued points are surrounded by 1-valued points.</p> http://mathoverflow.net/questions/100386/information-amount-of-fuzzy-data-transfer Information amount of fuzzy data transfer Anton Sukhinov 2012-06-22T19:01:45Z 2012-06-22T19:01:45Z <p>Suppose we have binary channel from which we are able to receive zeroes and ones. We also know apriory probability $p$ of receiving "1". Then we can calculate information amount of each digit $q$ we receive:</p> <p><em><strong>q=0</em></strong>: $I=-\log_2(1-p)$ bits;</p> <p><em><strong>q=1</em></strong>: $I=-\log_2(p)$ bits.</p> <p>Now imagine that the channel is "fussy": instead of receiving exact digits we receiving probability $q$ that transferred digit is "1". Previous example of "unfussy" channel is when $q$ can take only two values: <code>$q\in\{0,1\}$</code>.</p> <p>What will be the amount of information of receiving probability $q$ given probability $p$?</p> <p>Thanks!</p> http://mathoverflow.net/questions/89302/scale-random-variables-in-a-way-they-have-equal-probabilities-of-being-minimal Scale random variables in a way they have equal probabilities of being minimal Anton Sukhinov 2012-02-23T17:03:58Z 2012-02-23T21:11:25Z <p>I have several <em>positive</em> random variables $x_i,\ i=1,...,N$ taken from different unknown distributions (these distributions can be closely approximated by log-normal if needed). I can sample these variables as many times as needed; all variables are sampled simultaneously.</p> <p>From each sample my software selects the minimal variable $x_j$ and outputs its index $j$.</p> <p>The problem is: all values of output index $1 \leq j \leq N$ should have equal probabalities. So, random variables should be somehow adjusted to be "equal" for the minumum operation.</p> <p>I decided to multiply each random variable by some constant $k_i$. First idea was that $k_i=1/E[x_i]$, so that all adjusted variables will have equal expected values. The output indices indeed became much more uniform, but I pretty sure that my solution is wrong.</p> <p>The software should also dynamically adapt to gradually changing distributions of random variables (noise shape of sensors depend on temperature).</p> <p>Do you have any advice?</p> http://mathoverflow.net/questions/100386/information-amount-of-fuzzy-data-transfer Comment by Anton Sukhinov Anton Sukhinov 2012-06-23T18:41:54Z 2012-06-23T18:41:54Z I have not found this question in the FAQ http://mathoverflow.net/questions/89302/scale-random-variables-in-a-way-they-have-equal-probabilities-of-being-minimal/89332#89332 Comment by Anton Sukhinov Anton Sukhinov 2012-02-25T06:15:41Z 2012-02-25T06:15:41Z That completely solves my problem. Thank you!