Prove $(-1)^dbig(operatornameTr(KH^-1)-1big)H_ij>0$
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I came across an interesting inequality in studying the stability of attractors.
Given an i.i.d random vector $mathbfx$ (e.g. $x_msim N(0,1)$), construct following matrices $H$ and $K$ with $a_i,a_j>0$ and $a_ineq a_j$
$$H_ij = langle h(a_imathbfx),h(a_jmathbfx)rangle qquad K_ij = langle a_imathbfx,h(a_jmathbfx)rangle $$
where $h(a_imathbfx)$ is a vector, such that the $m$th term is $tanh(a_ix_m)$. Thus $H$ is symmetric and $K_ij>H_ij>0$.
Assuming $H$ is invertable and $d =dim(mathbfa)< dim(mathbfx)$, then $forall mathbfa$
$$(-1)^dbig(operatornameTr(KH^-1)-1big)<0$$
Note that $Kto H$ when $a_i, a_j to 0$, but $operatornameTr(KH^-1)to1$ for any $d$. When $d=1$, since sgn$(x)=$ sgn$(h(x))$ and $|ax_m|>|h(ax_m)|$ for each m, we have
$$fracsum_m ax_m h(ax_m)sum_mh(ax_m)^2>1$$
Can anyone come up with a rigorous proof for arbitrary dimensionality? Although the inequality holds when $h(x)=tanh(x)$, what are the necessary properties of the nonlinear function $h(x)$?
linear-algebra matrices inequality
asked Aug 21 at 8:52
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up vote
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down vote
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I came across an interesting inequality in studying the stability of attractors.
Given an i.i.d random vector $mathbfx$ (e.g. $x_msim N(0,1)$), construct following matrices $H$ and $K$ with $a_i,a_j>0$ and $a_ineq a_j$
$$H_ij = langle h(a_imathbfx),h(a_jmathbfx)rangle qquad K_ij = langle a_imathbfx,h(a_jmathbfx)rangle $$
where $h(a_imathbfx)$ is a vector, such that the $m$th term is $tanh(a_ix_m)$. Thus $H$ is symmetric and $K_ij>H_ij>0$.
Assuming $H$ is invertable and $d =dim(mathbfa)< dim(mathbfx)$, then $forall mathbfa$
$$(-1)^dbig(operatornameTr(KH^-1)-1big)<0$$
Note that $Kto H$ when $a_i, a_j to 0$, but $operatornameTr(KH^-1)to1$ for any $d$. When $d=1$, since sgn$(x)=$ sgn$(h(x))$ and $|ax_m|>|h(ax_m)|$ for each m, we have
$$fracsum_m ax_m h(ax_m)sum_mh(ax_m)^2>1$$
Can anyone come up with a rigorous proof for arbitrary dimensionality? Although the inequality holds when $h(x)=tanh(x)$, what are the necessary properties of the nonlinear function $h(x)$?
linear-algebra matrices inequality
asked Aug 21 at 8:52
pear
134
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I came across an interesting inequality in studying the stability of attractors.
Given an i.i.d random vector $mathbfx$ (e.g. $x_msim N(0,1)$), construct following matrices $H$ and $K$ with $a_i,a_j>0$ and $a_ineq a_j$
$$H_ij = langle h(a_imathbfx),h(a_jmathbfx)rangle qquad K_ij = langle a_imathbfx,h(a_jmathbfx)rangle $$
where $h(a_imathbfx)$ is a vector, such that the $m$th term is $tanh(a_ix_m)$. Thus $H$ is symmetric and $K_ij>H_ij>0$.
Assuming $H$ is invertable and $d =dim(mathbfa)< dim(mathbfx)$, then $forall mathbfa$
$$(-1)^dbig(operatornameTr(KH^-1)-1big)<0$$
Note that $Kto H$ when $a_i, a_j to 0$, but $operatornameTr(KH^-1)to1$ for any $d$. When $d=1$, since sgn$(x)=$ sgn$(h(x))$ and $|ax_m|>|h(ax_m)|$ for each m, we have
$$fracsum_m ax_m h(ax_m)sum_mh(ax_m)^2>1$$
Can anyone come up with a rigorous proof for arbitrary dimensionality? Although the inequality holds when $h(x)=tanh(x)$, what are the necessary properties of the nonlinear function $h(x)$?
linear-algebra matrices inequality
asked Aug 21 at 8:52
pear
134
I came across an interesting inequality in studying the stability of attractors.
Given an i.i.d random vector $mathbfx$ (e.g. $x_msim N(0,1)$), construct following matrices $H$ and $K$ with $a_i,a_j>0$ and $a_ineq a_j$
$$H_ij = langle h(a_imathbfx),h(a_jmathbfx)rangle qquad K_ij = langle a_imathbfx,h(a_jmathbfx)rangle $$
where $h(a_imathbfx)$ is a vector, such that the $m$th term is $tanh(a_ix_m)$. Thus $H$ is symmetric and $K_ij>H_ij>0$.
Assuming $H$ is invertable and $d =dim(mathbfa)< dim(mathbfx)$, then $forall mathbfa$
$$(-1)^dbig(operatornameTr(KH^-1)-1big)<0$$
Note that $Kto H$ when $a_i, a_j to 0$, but $operatornameTr(KH^-1)to1$ for any $d$. When $d=1$, since sgn$(x)=$ sgn$(h(x))$ and $|ax_m|>|h(ax_m)|$ for each m, we have
$$fracsum_m ax_m h(ax_m)sum_mh(ax_m)^2>1$$
Can anyone come up with a rigorous proof for arbitrary dimensionality? Although the inequality holds when $h(x)=tanh(x)$, what are the necessary properties of the nonlinear function $h(x)$?
linear-algebra matrices inequality
asked Aug 21 at 8:52
pear
134
edited Aug 21 at 12:20
asked Aug 21 at 8:52
pear
134
asked Aug 21 at 8:52
pear
134
asked Aug 21 at 8:52
pear
134
134
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