Getting closer to $k$ min-entropy using summation
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A distribution $D$ over $Lambda$ has $k$ min-entropy if the largest probability mass given to any element in $Lambda$ is $2^{-k}$ (i.e., for all $ainLambda$, $D(a)leq 2^{-k}$, and for some $a$ it is $2^{-k}$). We denote it by $H_infty(D)=k$.
Let $X$ and $Y$ be two independent distributions over ${0, 1}^n$ such that both $X$ is $epsilon$-close in statistical distance to a distribution with $k$ min-entropy and $Y$ is $epsilon$-close in statistical distance to a distribution with $k$ min-entropy.
Let $Z=X+Y$ denote the distribution over ${0, 1}^n$ obtained by sampling $xsim X$ and $ysim Y$ and outputting $x+y$, where the sum is addition modulo $2$ coordinate wise.
Prove that $Z$ is $epsilon^2$-close in statistical distance to a distribution with min-entropy at least $k$.
It is clear that $H_infty(Z)leqmin{H_infty(X),H_infty(Y)}$, but I don't see how that helps.
Note that by statistical distance I mean TV distance.
probability-theory entropy
asked Dec 24 '18 at 21:08
Don FanucciDon Fanucci
1,325521
$endgroup$
add a comment |
$begingroup$
A distribution $D$ over $Lambda$ has $k$ min-entropy if the largest probability mass given to any element in $Lambda$ is $2^{-k}$ (i.e., for all $ainLambda$, $D(a)leq 2^{-k}$, and for some $a$ it is $2^{-k}$). We denote it by $H_infty(D)=k$.
Let $X$ and $Y$ be two independent distributions over ${0, 1}^n$ such that both $X$ is $epsilon$-close in statistical distance to a distribution with $k$ min-entropy and $Y$ is $epsilon$-close in statistical distance to a distribution with $k$ min-entropy.
Let $Z=X+Y$ denote the distribution over ${0, 1}^n$ obtained by sampling $xsim X$ and $ysim Y$ and outputting $x+y$, where the sum is addition modulo $2$ coordinate wise.
Prove that $Z$ is $epsilon^2$-close in statistical distance to a distribution with min-entropy at least $k$.
It is clear that $H_infty(Z)leqmin{H_infty(X),H_infty(Y)}$, but I don't see how that helps.
Note that by statistical distance I mean TV distance.
probability-theory entropy
asked Dec 24 '18 at 21:08
Don FanucciDon Fanucci
1,325521
$endgroup$
1
$begingroup$
Just to be sure, by statistical distance you mean total variation, yes? In any case, this can't be true at the level of generality of your formulation - suppose $X ,Y$ are uniform. It is easy to see that $Z$ is also uniform. So $Z$ cannot get closer to any other distribution than $X$ and $Y$. In particular, if I pick $epsilon$ to be the minimum TV distance that a distribution with min-entropy $k$ is from the uniform, then $Z$ is also at a distance $ge epsilon$ from every $k$ min-entropy distribution. Are there missing hypotheses?
$endgroup$
– stochasticboy321
Dec 26 '18 at 1:16
$begingroup$
@stochasticboy321 I have edited it. Indeed by the statistical distance I mean TV distance. Also I have changed the claim to be proven.
$endgroup$
– Don Fanucci
Dec 26 '18 at 4:46
$begingroup$
I'm curious, did you ever find an answer to this? If yes, would you mind giving me a high-level overview of the argument?
$endgroup$
– stochasticboy321
Mar 8 at 2:00
add a comment |
$begingroup$
A distribution $D$ over $Lambda$ has $k$ min-entropy if the largest probability mass given to any element in $Lambda$ is $2^{-k}$ (i.e., for all $ainLambda$, $D(a)leq 2^{-k}$, and for some $a$ it is $2^{-k}$). We denote it by $H_infty(D)=k$.
Let $X$ and $Y$ be two independent distributions over ${0, 1}^n$ such that both $X$ is $epsilon$-close in statistical distance to a distribution with $k$ min-entropy and $Y$ is $epsilon$-close in statistical distance to a distribution with $k$ min-entropy.
Let $Z=X+Y$ denote the distribution over ${0, 1}^n$ obtained by sampling $xsim X$ and $ysim Y$ and outputting $x+y$, where the sum is addition modulo $2$ coordinate wise.
Prove that $Z$ is $epsilon^2$-close in statistical distance to a distribution with min-entropy at least $k$.
It is clear that $H_infty(Z)leqmin{H_infty(X),H_infty(Y)}$, but I don't see how that helps.
Note that by statistical distance I mean TV distance.
probability-theory entropy
asked Dec 24 '18 at 21:08
Don FanucciDon Fanucci
1,325521
$endgroup$
A distribution $D$ over $Lambda$ has $k$ min-entropy if the largest probability mass given to any element in $Lambda$ is $2^{-k}$ (i.e., for all $ainLambda$, $D(a)leq 2^{-k}$, and for some $a$ it is $2^{-k}$). We denote it by $H_infty(D)=k$.
Let $X$ and $Y$ be two independent distributions over ${0, 1}^n$ such that both $X$ is $epsilon$-close in statistical distance to a distribution with $k$ min-entropy and $Y$ is $epsilon$-close in statistical distance to a distribution with $k$ min-entropy.
Let $Z=X+Y$ denote the distribution over ${0, 1}^n$ obtained by sampling $xsim X$ and $ysim Y$ and outputting $x+y$, where the sum is addition modulo $2$ coordinate wise.
Prove that $Z$ is $epsilon^2$-close in statistical distance to a distribution with min-entropy at least $k$.
It is clear that $H_infty(Z)leqmin{H_infty(X),H_infty(Y)}$, but I don't see how that helps.
Note that by statistical distance I mean TV distance.
probability-theory entropy
probability-theory entropy
asked Dec 24 '18 at 21:08
Don FanucciDon Fanucci
1,325521
asked Dec 24 '18 at 21:08
Don FanucciDon Fanucci
1,325521
edited Dec 26 '18 at 5:59
Don Fanucci
asked Dec 24 '18 at 21:08
Don FanucciDon Fanucci
1,325521
asked Dec 24 '18 at 21:08
Don FanucciDon Fanucci
1,325521
asked Dec 24 '18 at 21:08
Don FanucciDon Fanucci
1,325521
1,325521
1
$begingroup$
Just to be sure, by statistical distance you mean total variation, yes? In any case, this can't be true at the level of generality of your formulation - suppose $X ,Y$ are uniform. It is easy to see that $Z$ is also uniform. So $Z$ cannot get closer to any other distribution than $X$ and $Y$. In particular, if I pick $epsilon$ to be the minimum TV distance that a distribution with min-entropy $k$ is from the uniform, then $Z$ is also at a distance $ge epsilon$ from every $k$ min-entropy distribution. Are there missing hypotheses?
$endgroup$
– stochasticboy321
Dec 26 '18 at 1:16
$begingroup$
@stochasticboy321 I have edited it. Indeed by the statistical distance I mean TV distance. Also I have changed the claim to be proven.
$endgroup$
– Don Fanucci
Dec 26 '18 at 4:46
$begingroup$
I'm curious, did you ever find an answer to this? If yes, would you mind giving me a high-level overview of the argument?
$endgroup$
– stochasticboy321
Mar 8 at 2:00
add a comment |
1
$begingroup$
Just to be sure, by statistical distance you mean total variation, yes? In any case, this can't be true at the level of generality of your formulation - suppose $X ,Y$ are uniform. It is easy to see that $Z$ is also uniform. So $Z$ cannot get closer to any other distribution than $X$ and $Y$. In particular, if I pick $epsilon$ to be the minimum TV distance that a distribution with min-entropy $k$ is from the uniform, then $Z$ is also at a distance $ge epsilon$ from every $k$ min-entropy distribution. Are there missing hypotheses?
$endgroup$
– stochasticboy321
Dec 26 '18 at 1:16
$begingroup$
@stochasticboy321 I have edited it. Indeed by the statistical distance I mean TV distance. Also I have changed the claim to be proven.
$endgroup$
– Don Fanucci
Dec 26 '18 at 4:46
$begingroup$
I'm curious, did you ever find an answer to this? If yes, would you mind giving me a high-level overview of the argument?
$endgroup$
– stochasticboy321
Mar 8 at 2:00
1
1
$begingroup$
Just to be sure, by statistical distance you mean total variation, yes? In any case, this can't be true at the level of generality of your formulation - suppose $X ,Y$ are uniform. It is easy to see that $Z$ is also uniform. So $Z$ cannot get closer to any other distribution than $X$ and $Y$. In particular, if I pick $epsilon$ to be the minimum TV distance that a distribution with min-entropy $k$ is from the uniform, then $Z$ is also at a distance $ge epsilon$ from every $k$ min-entropy distribution. Are there missing hypotheses?
$endgroup$
– stochasticboy321
Dec 26 '18 at 1:16
$begingroup$
Just to be sure, by statistical distance you mean total variation, yes? In any case, this can't be true at the level of generality of your formulation - suppose $X ,Y$ are uniform. It is easy to see that $Z$ is also uniform. So $Z$ cannot get closer to any other distribution than $X$ and $Y$. In particular, if I pick $epsilon$ to be the minimum TV distance that a distribution with min-entropy $k$ is from the uniform, then $Z$ is also at a distance $ge epsilon$ from every $k$ min-entropy distribution. Are there missing hypotheses?
$endgroup$
– stochasticboy321
Dec 26 '18 at 1:16
$begingroup$
@stochasticboy321 I have edited it. Indeed by the statistical distance I mean TV distance. Also I have changed the claim to be proven.
$endgroup$
– Don Fanucci
Dec 26 '18 at 4:46
$begingroup$
@stochasticboy321 I have edited it. Indeed by the statistical distance I mean TV distance. Also I have changed the claim to be proven.
$endgroup$
– Don Fanucci
Dec 26 '18 at 4:46
$begingroup$
I'm curious, did you ever find an answer to this? If yes, would you mind giving me a high-level overview of the argument?
$endgroup$
– stochasticboy321
Mar 8 at 2:00
$begingroup$
I'm curious, did you ever find an answer to this? If yes, would you mind giving me a high-level overview of the argument?
$endgroup$
– stochasticboy321
Mar 8 at 2:00
add a comment |
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$begingroup$
Just to be sure, by statistical distance you mean total variation, yes? In any case, this can't be true at the level of generality of your formulation - suppose $X ,Y$ are uniform. It is easy to see that $Z$ is also uniform. So $Z$ cannot get closer to any other distribution than $X$ and $Y$. In particular, if I pick $epsilon$ to be the minimum TV distance that a distribution with min-entropy $k$ is from the uniform, then $Z$ is also at a distance $ge epsilon$ from every $k$ min-entropy distribution. Are there missing hypotheses?
$endgroup$
– stochasticboy321
Dec 26 '18 at 1:16
$begingroup$
@stochasticboy321 I have edited it. Indeed by the statistical distance I mean TV distance. Also I have changed the claim to be proven.
$endgroup$
– Don Fanucci
Dec 26 '18 at 4:46
$begingroup$
I'm curious, did you ever find an answer to this? If yes, would you mind giving me a high-level overview of the argument?
$endgroup$
– stochasticboy321
Mar 8 at 2:00