Bregman divergence and its property
$begingroup$
Suppose, $S$ is a convex set in $mathbb{R^n}$and $ri(S)$ denotes the relative interior set of $S$.For, $x_1in S$ and $x_2,x_3 in ri(S)$ the following property holds
begin{equation}
d_{phi}(x_1,x_3)=d_{phi}(x_1,x_2)+d_{phi}(x_2,x_3)-langle(x_1-x_2)(nabla phi(x_3)-nabla phi(x_2))rangle.
end{equation}
where, $d_{phi}(.,.)$ denotes the Bregman divergence with the convex function $phi$.
Show that, when $x_1,x_2,x_3$ are such that $x_1 in S'$, $S'$ being convex subset of $S$ and $x_2$ is given by
$x_2=rm{argmin} ~{d_{phi}(x,x_3):x in S'}$
then the third term of the previous equation is negative.
Proof: when $x_2=rm{argmin} ~{d_{phi}(x,x_3):x in S'}$ then,
$d_{phi}(x_2,x_3)<d_{phi}(x_1,x_3)$
Now, $phi$ is a convex function,hence $(nabla phi(x_3)-nabla phi(x_2))>0$ whenever $x_3>x_2$. On the other hand,
$(x_1-x_2)>0$ if $x_1>x_2$.Therefore the third term is negative when both $x_1,x_3>x_2$.
But how can I prove it in general??Give me some hint please.
self-learning
asked Dec 22 '18 at 5:43
Shefali royShefali roy
93
$endgroup$
add a comment |
$begingroup$
Suppose, $S$ is a convex set in $mathbb{R^n}$and $ri(S)$ denotes the relative interior set of $S$.For, $x_1in S$ and $x_2,x_3 in ri(S)$ the following property holds
begin{equation}
d_{phi}(x_1,x_3)=d_{phi}(x_1,x_2)+d_{phi}(x_2,x_3)-langle(x_1-x_2)(nabla phi(x_3)-nabla phi(x_2))rangle.
end{equation}
where, $d_{phi}(.,.)$ denotes the Bregman divergence with the convex function $phi$.
Show that, when $x_1,x_2,x_3$ are such that $x_1 in S'$, $S'$ being convex subset of $S$ and $x_2$ is given by
$x_2=rm{argmin} ~{d_{phi}(x,x_3):x in S'}$
then the third term of the previous equation is negative.
Proof: when $x_2=rm{argmin} ~{d_{phi}(x,x_3):x in S'}$ then,
$d_{phi}(x_2,x_3)<d_{phi}(x_1,x_3)$
Now, $phi$ is a convex function,hence $(nabla phi(x_3)-nabla phi(x_2))>0$ whenever $x_3>x_2$. On the other hand,
$(x_1-x_2)>0$ if $x_1>x_2$.Therefore the third term is negative when both $x_1,x_3>x_2$.
But how can I prove it in general??Give me some hint please.
self-learning
asked Dec 22 '18 at 5:43
Shefali royShefali roy
93
$endgroup$
add a comment |
$begingroup$
Suppose, $S$ is a convex set in $mathbb{R^n}$and $ri(S)$ denotes the relative interior set of $S$.For, $x_1in S$ and $x_2,x_3 in ri(S)$ the following property holds
begin{equation}
d_{phi}(x_1,x_3)=d_{phi}(x_1,x_2)+d_{phi}(x_2,x_3)-langle(x_1-x_2)(nabla phi(x_3)-nabla phi(x_2))rangle.
end{equation}
where, $d_{phi}(.,.)$ denotes the Bregman divergence with the convex function $phi$.
Show that, when $x_1,x_2,x_3$ are such that $x_1 in S'$, $S'$ being convex subset of $S$ and $x_2$ is given by
$x_2=rm{argmin} ~{d_{phi}(x,x_3):x in S'}$
then the third term of the previous equation is negative.
Proof: when $x_2=rm{argmin} ~{d_{phi}(x,x_3):x in S'}$ then,
$d_{phi}(x_2,x_3)<d_{phi}(x_1,x_3)$
Now, $phi$ is a convex function,hence $(nabla phi(x_3)-nabla phi(x_2))>0$ whenever $x_3>x_2$. On the other hand,
$(x_1-x_2)>0$ if $x_1>x_2$.Therefore the third term is negative when both $x_1,x_3>x_2$.
But how can I prove it in general??Give me some hint please.
self-learning
asked Dec 22 '18 at 5:43
Shefali royShefali roy
93
$endgroup$
Suppose, $S$ is a convex set in $mathbb{R^n}$and $ri(S)$ denotes the relative interior set of $S$.For, $x_1in S$ and $x_2,x_3 in ri(S)$ the following property holds
begin{equation}
d_{phi}(x_1,x_3)=d_{phi}(x_1,x_2)+d_{phi}(x_2,x_3)-langle(x_1-x_2)(nabla phi(x_3)-nabla phi(x_2))rangle.
end{equation}
where, $d_{phi}(.,.)$ denotes the Bregman divergence with the convex function $phi$.
Show that, when $x_1,x_2,x_3$ are such that $x_1 in S'$, $S'$ being convex subset of $S$ and $x_2$ is given by
$x_2=rm{argmin} ~{d_{phi}(x,x_3):x in S'}$
then the third term of the previous equation is negative.
Proof: when $x_2=rm{argmin} ~{d_{phi}(x,x_3):x in S'}$ then,
$d_{phi}(x_2,x_3)<d_{phi}(x_1,x_3)$
Now, $phi$ is a convex function,hence $(nabla phi(x_3)-nabla phi(x_2))>0$ whenever $x_3>x_2$. On the other hand,
$(x_1-x_2)>0$ if $x_1>x_2$.Therefore the third term is negative when both $x_1,x_3>x_2$.
But how can I prove it in general??Give me some hint please.
self-learning
self-learning
asked Dec 22 '18 at 5:43
Shefali royShefali roy
93
asked Dec 22 '18 at 5:43
Shefali royShefali roy
93
asked Dec 22 '18 at 5:43
Shefali royShefali roy
93
asked Dec 22 '18 at 5:43
Shefali royShefali roy
93
asked Dec 22 '18 at 5:43
Shefali royShefali roy
93
93
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