Restricted Boltzmann machine uniqueness

[Jufa]

I am dealing with restricted Boltzmann machines to model distributuins in my final degree project and some question has come to my mind.
A restricted Boltzmann machine with v visible binary neurons and h hidden neurons models a distribution in the following manner:

$f_i= e^{ \sum_k b[k] \sigma^i[k] + \sum_s \log(c[ s ] + e^{\sum_k w[ s ][ k ] b[ k ] })}$
$Z = \sum_i f_i$
$p_i = f_i/Z$
Where b[ k ] and c[ s ] are, respectively, the k-th and the s-th bias of, again respectively, the visible and hidden layer.
w[ s ][k] is the component s, k of the weight matrix of the network.
"i" here refers to a certain binary vector with components $\sigma^i[k]$.
My question is:
Given a certain restricted Boltzmann machine (i.e. a certain set of biases and weights) that models a certain distribution $p_i$, is it possible to find another configuration (i.e. a different set of parameters and weights) such that it gives the same distribution?
Thanks in advance.

 

[Jufa]

If some moderator can explain why my text is strikethrough I would appreciate it.

 

[jedishrfu]

We are attempting to fix the problem. I got rid of the strike through text but now the latex Mathjax isn’t rendering. We moved to a new version of forum software with a new post editor and perhaps it has injected or requires some special way to enter math expressions.

 

[jedishrfu]

Testing mathjax $e=mc^2$ now And $2+2=4$ in base 10.

 

[fresh_42]

I think I have reset the corrected original version. Please refresh your browsers.

Hint: Do not use [B],[U],[S],[I]. If you do not "cheat" as I did here, the interpreter takes it for the command tag BEGIN bold / underline / strikeout / italic.

 

[Jufa]

Many thanks!

 

[TeethWhitener]

I am dealing with restricted Boltzmann machines to model distributuins in my final degree project and some question has come to my mind.
A restricted Boltzmann machine with v visible binary neurons and h hidden neurons models a distribution in the following manner:

$f_i= e^{ \sum_k b[k] \sigma^i[k] + \sum_s \log(c[ s ] + e^{\sum_k w[ s ][ k ] b[ k ] })}$
$Z = \sum_i f_i$
$p_i = f_i/Z$
Where b[ k ] and c[ s ] are, respectively, the k-th and the s-th bias of, again respectively, the visible and hidden layer.
w[ s ][k] is the component s, k of the weight matrix of the network.
"i" here refers to a certain binary vector with components $\sigma^i[k]$.
My question is:
Given a certain restricted Boltzmann machine (i.e. a certain set of biases and weights) that models a certain distribution $p_i$, is it possible to find another configuration (i.e. a different set of parameters and weights) such that it gives the same distribution?
Thanks in advance.

I’m not very familiar with the topic, but trivially, if you set $c[ s]=0$, then you get the same $f_i$ by exchanging the $w[ s][k]$ and the $\sigma^i[k]$. So for instance,
$$f_{i,1}=\exp{\left(\sum_k{b[k]\sigma_1^i[k]}+\sum_s{ \log{e^{\sum_k{w_1[ s][k]b[k]}}}}\right)}$$
$$f_{i,2}=\exp{\left(\sum_k{b[k]\sigma_2^i[k]}+\sum_s{ \log{e^{\sum_k{w_2[ s][k]b[k]}}}}\right)}$$
Then let $\sigma_1^i[k]=w_2[ s][k]$ and $\sigma_2^i[k]=w_1[ s][k]$.
That gives you the same probability distributions but I don’t know if that’s allowed with restricted Boltzmann machines.