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1

$begingroup$

I have two summations: $y = sum_i=1^s w_i x_i$, and $y' = sum_i=1^s' w'_i x'_i$.

$s$ and $s'$ are the number of terms in each summation, and both are big numbers.

$w_i$ and $w'_i$ are weights of the terms $x$ and $x'$, so we have: $sum_i=1^s w_i = sum_i=1^s' w'_i = 1$. We can therefore treat $x$ and $x'$ as terms in a probability distribution.

Now, I want to approximate the multiplication of the two summations: $y cdot y'$. The accurate multiplication could easily be expressed as a sum of $s cdot s'$ terms, but we want to reduce the number of terms; so we do approximation.

To this end, I draw a sample of $M$ pairs of indices $(k_r, l_r)^M_r=1$, where the indices are sampled independently from the two probability distributions $w$ and $w'$.

The next step is to compute $z_r = x_k_r cdot x'_l_r$ for each $r$.

Next, we need to assign new weights $q$ to each $z_r$. Apparently, we can use the frequency of the sampled indices as new weights. We will then get another sum $sum q_i z_i$ as the final product.

I am trying to implement the above steps in a programming language. Due to the nature of the program I write, I can't sample exact indices.

To give an example, in the above, when I sample an index of 3 from a total of 5 elements, I am in effect generating a one-hot vector $[0, 0, 0, 1, 0]$. This vector could next be dot-multiplied with the original sum to get the 3rd element in the sum.

In the program I am writing, I am generating approximate one-hot vectors that look like $[0.05, 0.05, 0.1, 0.75, 0.05]$, where the largest value approximately represents index of the sampled element.

If we had an exact one-hot vector, it is easy to get the frequencies and new weights. However, it is not so obvious how to do so when the one-hot vectors themselves are approximate. Can anyone help me get an expression of the new weights with these approximate one-hot vectors?

probability-distributions sampling

share|cite|improve this question

asked Mar 21 at 12:35

Su XuanSu Xuan

1112

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Could you loop through each approximate vector and set all of the elements to zero except for the biggest one to get an exact vector?
  $endgroup$
  – CyborgOctopus
  Mar 21 at 13:04
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Nope, this is not possible (and not very clean). I want a formula to compute the weights that explicitly involves those approximate vectors.
  $endgroup$
  – Su Xuan
  Mar 21 at 13:10
  

  
  
  
  

add a comment | 

1

$begingroup$

I have two summations: $y = sum_i=1^s w_i x_i$, and $y' = sum_i=1^s' w'_i x'_i$.

$s$ and $s'$ are the number of terms in each summation, and both are big numbers.

$w_i$ and $w'_i$ are weights of the terms $x$ and $x'$, so we have: $sum_i=1^s w_i = sum_i=1^s' w'_i = 1$. We can therefore treat $x$ and $x'$ as terms in a probability distribution.

Now, I want to approximate the multiplication of the two summations: $y cdot y'$. The accurate multiplication could easily be expressed as a sum of $s cdot s'$ terms, but we want to reduce the number of terms; so we do approximation.

To this end, I draw a sample of $M$ pairs of indices $(k_r, l_r)^M_r=1$, where the indices are sampled independently from the two probability distributions $w$ and $w'$.

The next step is to compute $z_r = x_k_r cdot x'_l_r$ for each $r$.

Next, we need to assign new weights $q$ to each $z_r$. Apparently, we can use the frequency of the sampled indices as new weights. We will then get another sum $sum q_i z_i$ as the final product.

I am trying to implement the above steps in a programming language. Due to the nature of the program I write, I can't sample exact indices.

To give an example, in the above, when I sample an index of 3 from a total of 5 elements, I am in effect generating a one-hot vector $[0, 0, 0, 1, 0]$. This vector could next be dot-multiplied with the original sum to get the 3rd element in the sum.

In the program I am writing, I am generating approximate one-hot vectors that look like $[0.05, 0.05, 0.1, 0.75, 0.05]$, where the largest value approximately represents index of the sampled element.

If we had an exact one-hot vector, it is easy to get the frequencies and new weights. However, it is not so obvious how to do so when the one-hot vectors themselves are approximate. Can anyone help me get an expression of the new weights with these approximate one-hot vectors?

probability-distributions sampling

share|cite|improve this question

asked Mar 21 at 12:35

Su XuanSu Xuan

1112

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Could you loop through each approximate vector and set all of the elements to zero except for the biggest one to get an exact vector?
  $endgroup$
  – CyborgOctopus
  Mar 21 at 13:04
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Nope, this is not possible (and not very clean). I want a formula to compute the weights that explicitly involves those approximate vectors.
  $endgroup$
  – Su Xuan
  Mar 21 at 13:10
  

  
  
  
  

add a comment | 

$begingroup$

I have two summations: $y = sum_i=1^s w_i x_i$, and $y' = sum_i=1^s' w'_i x'_i$.

$s$ and $s'$ are the number of terms in each summation, and both are big numbers.

$w_i$ and $w'_i$ are weights of the terms $x$ and $x'$, so we have: $sum_i=1^s w_i = sum_i=1^s' w'_i = 1$. We can therefore treat $x$ and $x'$ as terms in a probability distribution.

Now, I want to approximate the multiplication of the two summations: $y cdot y'$. The accurate multiplication could easily be expressed as a sum of $s cdot s'$ terms, but we want to reduce the number of terms; so we do approximation.

To this end, I draw a sample of $M$ pairs of indices $(k_r, l_r)^M_r=1$, where the indices are sampled independently from the two probability distributions $w$ and $w'$.

The next step is to compute $z_r = x_k_r cdot x'_l_r$ for each $r$.

Next, we need to assign new weights $q$ to each $z_r$. Apparently, we can use the frequency of the sampled indices as new weights. We will then get another sum $sum q_i z_i$ as the final product.

I am trying to implement the above steps in a programming language. Due to the nature of the program I write, I can't sample exact indices.

To give an example, in the above, when I sample an index of 3 from a total of 5 elements, I am in effect generating a one-hot vector $[0, 0, 0, 1, 0]$. This vector could next be dot-multiplied with the original sum to get the 3rd element in the sum.

In the program I am writing, I am generating approximate one-hot vectors that look like $[0.05, 0.05, 0.1, 0.75, 0.05]$, where the largest value approximately represents index of the sampled element.

If we had an exact one-hot vector, it is easy to get the frequencies and new weights. However, it is not so obvious how to do so when the one-hot vectors themselves are approximate. Can anyone help me get an expression of the new weights with these approximate one-hot vectors?

probability-distributions sampling

share|cite|improve this question

asked Mar 21 at 12:35

Su XuanSu Xuan

1112

$endgroup$

share|cite|improve this question

asked Mar 21 at 12:35

Su XuanSu Xuan

1112

share|cite|improve this question

asked Mar 21 at 12:35

Su XuanSu Xuan

1112

asked Mar 21 at 12:35

Su XuanSu Xuan

1112

asked Mar 21 at 12:35

Su XuanSu Xuan

1112