6
votes
Accepted
Relationship between KL, chi-squared, and Hellinger
This inequality is false in general.
E.g., let $f$ and $g$ be pdf's on $[0,1]$ given by the formulas $f=1$ and $g=f+t\,1_{(0,1/2)}-t\,1_{(1/2.1)}$ for $t\in(0,1)$. Then the left- and right-hand sides ...
- 80.3k
4
votes
Inverse quadratic norms
The result follows from operator monotonicity of inverse over symmetric positive definite matrices, i.e., $A \succeq B \Rightarrow A^{-1} \preceq B^{-1}$, which is a special case of the Löwner-Heinz ...
- 1,293
4
votes
Accepted
Inverse quadratic norms
The proposed result holds true.
I am assuming throughout that $spd$ means symmetric positive definite and that the matrices $A$ and $B$ are $n$-by-$n$ matrices over $\mathbb{R}$ for some $n > 0$.
...
- 6,453
3
votes
Relationship between KL, chi-squared, and Hellinger
The inequality does not hold for even simple densities which fulfill the conditions given in Section 29.3 of the paper shared by the OP.
Here is a piecewise constant counterexample,
$$ f(x) = \frac{3}...
- 4,875
3
votes
Accepted
A bounded sequence $a_k=\frac{2 k(k-1)}{\Gamma[k-1]} \int_0^\infty \frac{e^{-r^2} r^{2k-3}}{2+4 r^2+r^4} dr$
$\newcommand\Ga\Gamma$We have
$$a_k=\frac{2 k(k-1)}{\Ga(k-1)} J_k,\quad J_k:=\int_0^\infty e^{-r^2} r^{2k-3}g(r)\,dr,$$
$$g(r):=\frac1{2+4 r^2+r^4}.$$
Next (for $r>0$),
$$g(r)=h(r)-\frac{8 \left(70 ...
- 80.3k
2
votes
Accepted
How do I integrate this inequality that appears in a paper of Rabinowitz?
For any unit vector $u$ and real $t>0$, let
\begin{equation}
h(t):=H(tu).
\end{equation}
Then
\begin{equation}
h'(t)=H'(tu)\cdot u=\frac{H'(tu)\cdot(tu)}t\ge\frac{\mu H(tu)}t=\frac{h(t)}t.
...
- 80.3k
2
votes
Accepted
Coarea formula for measure of epsilon neighbourhood
Your formula is true (up to appropriate constants) if you take $\nu$ to be the Minkowski content, assuming that the volume of the tubular neighborhood is locally Lipschitz as a function of the tube ...
- 1,966
1
vote
Accepted
For which value of $C(f)$ would the following inequality hold?
The question is not well posed, as it is not quite clear in what terms you want $C(f)$ to be expressed.
If one only uses the terms you did mention -- the Lipschitz constant and the size of the domain, ...
- 80.3k
1
vote
Strengthened version of Isoperimetric inequality with n-polygon
The keyword you are looking for is "quantitative isoperimetric inequalities". The case of polygons was solved in the following paper: https://arxiv.org/abs/1402.4460
The "quantitative&...
- 1,855
1
vote
Accepted
Lower bound for KL divergence of bounded densities and $L_{2}$ metric
$\newcommand{\ep}{\varepsilon}$As in your post and comments, suppose that $f$ and $f_0$ are supported on a compact set $S$, and
\begin{equation*}
a\le f\le b,\quad a\le f_0\le b
\end{equation*}
on ...
- 80.3k
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