What does the term in the right mean?












1












$begingroup$


Reading an article about clustering, I came across this equality:



enter image description here



It's basically a sum over $ mu_{i,j}(1-cos(x_i, p_j))$ (where $ cos(x_i, p_j) $ is the cosinus between two vectors in $mathbb{R}^n$). And then suddenly some magic happens and the formula in the right appears. Can somebody explain?



Note: I understand that the parenthesis was expanded (each term got multiplied by $ mu_{i,j}$). But what is that huge dot product and why does it have this form?










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$endgroup$








  • 1




    $begingroup$
    Refer to the inner products on $mathbb R^n$. Use the linearity w.r.t. each entry.
    $endgroup$
    – xbh
    Nov 24 '18 at 17:00












  • $begingroup$
    @xbh I changed R to $mathbb{R}$. I don't understand what you mentioned about the linearity. Oh, ok I'm stupid, you were actually answering my question, not referring to formating. I suck at math.
    $endgroup$
    – Hello Lili
    Nov 24 '18 at 17:03








  • 1




    $begingroup$
    $(lambda a+mu b|c) = lambda (a|c) + mu(b|c)$, where $(a|b) = langle a,b rangle$ [and this is just a preference of notations about $(|)$ and $langle,rangle$].
    $endgroup$
    – xbh
    Nov 24 '18 at 17:05


















1












$begingroup$


Reading an article about clustering, I came across this equality:



enter image description here



It's basically a sum over $ mu_{i,j}(1-cos(x_i, p_j))$ (where $ cos(x_i, p_j) $ is the cosinus between two vectors in $mathbb{R}^n$). And then suddenly some magic happens and the formula in the right appears. Can somebody explain?



Note: I understand that the parenthesis was expanded (each term got multiplied by $ mu_{i,j}$). But what is that huge dot product and why does it have this form?










share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    Refer to the inner products on $mathbb R^n$. Use the linearity w.r.t. each entry.
    $endgroup$
    – xbh
    Nov 24 '18 at 17:00












  • $begingroup$
    @xbh I changed R to $mathbb{R}$. I don't understand what you mentioned about the linearity. Oh, ok I'm stupid, you were actually answering my question, not referring to formating. I suck at math.
    $endgroup$
    – Hello Lili
    Nov 24 '18 at 17:03








  • 1




    $begingroup$
    $(lambda a+mu b|c) = lambda (a|c) + mu(b|c)$, where $(a|b) = langle a,b rangle$ [and this is just a preference of notations about $(|)$ and $langle,rangle$].
    $endgroup$
    – xbh
    Nov 24 '18 at 17:05
















1












1








1





$begingroup$


Reading an article about clustering, I came across this equality:



enter image description here



It's basically a sum over $ mu_{i,j}(1-cos(x_i, p_j))$ (where $ cos(x_i, p_j) $ is the cosinus between two vectors in $mathbb{R}^n$). And then suddenly some magic happens and the formula in the right appears. Can somebody explain?



Note: I understand that the parenthesis was expanded (each term got multiplied by $ mu_{i,j}$). But what is that huge dot product and why does it have this form?










share|cite|improve this question











$endgroup$




Reading an article about clustering, I came across this equality:



enter image description here



It's basically a sum over $ mu_{i,j}(1-cos(x_i, p_j))$ (where $ cos(x_i, p_j) $ is the cosinus between two vectors in $mathbb{R}^n$). And then suddenly some magic happens and the formula in the right appears. Can somebody explain?



Note: I understand that the parenthesis was expanded (each term got multiplied by $ mu_{i,j}$). But what is that huge dot product and why does it have this form?







vectors products






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 24 '18 at 17:02







Hello Lili

















asked Nov 24 '18 at 16:56









Hello LiliHello Lili

1104




1104








  • 1




    $begingroup$
    Refer to the inner products on $mathbb R^n$. Use the linearity w.r.t. each entry.
    $endgroup$
    – xbh
    Nov 24 '18 at 17:00












  • $begingroup$
    @xbh I changed R to $mathbb{R}$. I don't understand what you mentioned about the linearity. Oh, ok I'm stupid, you were actually answering my question, not referring to formating. I suck at math.
    $endgroup$
    – Hello Lili
    Nov 24 '18 at 17:03








  • 1




    $begingroup$
    $(lambda a+mu b|c) = lambda (a|c) + mu(b|c)$, where $(a|b) = langle a,b rangle$ [and this is just a preference of notations about $(|)$ and $langle,rangle$].
    $endgroup$
    – xbh
    Nov 24 '18 at 17:05
















  • 1




    $begingroup$
    Refer to the inner products on $mathbb R^n$. Use the linearity w.r.t. each entry.
    $endgroup$
    – xbh
    Nov 24 '18 at 17:00












  • $begingroup$
    @xbh I changed R to $mathbb{R}$. I don't understand what you mentioned about the linearity. Oh, ok I'm stupid, you were actually answering my question, not referring to formating. I suck at math.
    $endgroup$
    – Hello Lili
    Nov 24 '18 at 17:03








  • 1




    $begingroup$
    $(lambda a+mu b|c) = lambda (a|c) + mu(b|c)$, where $(a|b) = langle a,b rangle$ [and this is just a preference of notations about $(|)$ and $langle,rangle$].
    $endgroup$
    – xbh
    Nov 24 '18 at 17:05










1




1




$begingroup$
Refer to the inner products on $mathbb R^n$. Use the linearity w.r.t. each entry.
$endgroup$
– xbh
Nov 24 '18 at 17:00






$begingroup$
Refer to the inner products on $mathbb R^n$. Use the linearity w.r.t. each entry.
$endgroup$
– xbh
Nov 24 '18 at 17:00














$begingroup$
@xbh I changed R to $mathbb{R}$. I don't understand what you mentioned about the linearity. Oh, ok I'm stupid, you were actually answering my question, not referring to formating. I suck at math.
$endgroup$
– Hello Lili
Nov 24 '18 at 17:03






$begingroup$
@xbh I changed R to $mathbb{R}$. I don't understand what you mentioned about the linearity. Oh, ok I'm stupid, you were actually answering my question, not referring to formating. I suck at math.
$endgroup$
– Hello Lili
Nov 24 '18 at 17:03






1




1




$begingroup$
$(lambda a+mu b|c) = lambda (a|c) + mu(b|c)$, where $(a|b) = langle a,b rangle$ [and this is just a preference of notations about $(|)$ and $langle,rangle$].
$endgroup$
– xbh
Nov 24 '18 at 17:05






$begingroup$
$(lambda a+mu b|c) = lambda (a|c) + mu(b|c)$, where $(a|b) = langle a,b rangle$ [and this is just a preference of notations about $(|)$ and $langle,rangle$].
$endgroup$
– xbh
Nov 24 '18 at 17:05












1 Answer
1






active

oldest

votes


















1












$begingroup$

Remember that the inner product is bilinear: for three vectors $u,v,w$ and scalars $alpha,beta$,
$$
langle alpha u, beta vrangle = alphabetalangle u, vrangle, qquad langle u, v+wrangle = langle u, vrangle+langle u, wrangle
$$

What happened above is exactly what follows when applying these rules: for every $i,j$
$$
frac{langle x_i, p_jrangle}{lVert x_irVertlVert p_jrVert} = leftlangle frac{x_i}{lVert x_irVert}, frac{p_j}{lVert p_jrVert}rightrangle tag{1}
$$

since $lVert x_irVert,lVert p_jrVertinmathbb{R}$ are scalars (the brackets became "bigger" just for style considerations, like parentheses would be drawn bigger when what's inside looks "large").



Then, each $mu_{ij}$ is also a scalar, and you have a sum, so
$$
sum_{i} mu_{ij}leftlangle frac{x_i}{lVert x_irVert}, frac{p_j}{lVert p_jrVert}rightrangle
= sum_{i} leftlangle mu_{ij}frac{x_i}{lVert x_irVert}, frac{p_j}{lVert p_jrVert}rightrangle
= leftlangle sum_{i}mu_{ij}frac{x_i}{lVert x_irVert}, frac{p_j}{lVert p_jrVert}rightrangle tag{2}
$$

Note that it does make sense: $sum_{i}mu_{ij}frac{x_i}{lVert x_irVert}$ is a vector, and so is $frac{p_j}{lVert p_jrVert}$. So it's indeed the inner product between two vectors.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    This answers my question. Thank you! I will accept this answer in 3 minutes.
    $endgroup$
    – Hello Lili
    Nov 24 '18 at 17:07










  • $begingroup$
    @HelloLili Glad this helped!
    $endgroup$
    – Clement C.
    Nov 24 '18 at 17:08











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1 Answer
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1 Answer
1






active

oldest

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active

oldest

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active

oldest

votes









1












$begingroup$

Remember that the inner product is bilinear: for three vectors $u,v,w$ and scalars $alpha,beta$,
$$
langle alpha u, beta vrangle = alphabetalangle u, vrangle, qquad langle u, v+wrangle = langle u, vrangle+langle u, wrangle
$$

What happened above is exactly what follows when applying these rules: for every $i,j$
$$
frac{langle x_i, p_jrangle}{lVert x_irVertlVert p_jrVert} = leftlangle frac{x_i}{lVert x_irVert}, frac{p_j}{lVert p_jrVert}rightrangle tag{1}
$$

since $lVert x_irVert,lVert p_jrVertinmathbb{R}$ are scalars (the brackets became "bigger" just for style considerations, like parentheses would be drawn bigger when what's inside looks "large").



Then, each $mu_{ij}$ is also a scalar, and you have a sum, so
$$
sum_{i} mu_{ij}leftlangle frac{x_i}{lVert x_irVert}, frac{p_j}{lVert p_jrVert}rightrangle
= sum_{i} leftlangle mu_{ij}frac{x_i}{lVert x_irVert}, frac{p_j}{lVert p_jrVert}rightrangle
= leftlangle sum_{i}mu_{ij}frac{x_i}{lVert x_irVert}, frac{p_j}{lVert p_jrVert}rightrangle tag{2}
$$

Note that it does make sense: $sum_{i}mu_{ij}frac{x_i}{lVert x_irVert}$ is a vector, and so is $frac{p_j}{lVert p_jrVert}$. So it's indeed the inner product between two vectors.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    This answers my question. Thank you! I will accept this answer in 3 minutes.
    $endgroup$
    – Hello Lili
    Nov 24 '18 at 17:07










  • $begingroup$
    @HelloLili Glad this helped!
    $endgroup$
    – Clement C.
    Nov 24 '18 at 17:08
















1












$begingroup$

Remember that the inner product is bilinear: for three vectors $u,v,w$ and scalars $alpha,beta$,
$$
langle alpha u, beta vrangle = alphabetalangle u, vrangle, qquad langle u, v+wrangle = langle u, vrangle+langle u, wrangle
$$

What happened above is exactly what follows when applying these rules: for every $i,j$
$$
frac{langle x_i, p_jrangle}{lVert x_irVertlVert p_jrVert} = leftlangle frac{x_i}{lVert x_irVert}, frac{p_j}{lVert p_jrVert}rightrangle tag{1}
$$

since $lVert x_irVert,lVert p_jrVertinmathbb{R}$ are scalars (the brackets became "bigger" just for style considerations, like parentheses would be drawn bigger when what's inside looks "large").



Then, each $mu_{ij}$ is also a scalar, and you have a sum, so
$$
sum_{i} mu_{ij}leftlangle frac{x_i}{lVert x_irVert}, frac{p_j}{lVert p_jrVert}rightrangle
= sum_{i} leftlangle mu_{ij}frac{x_i}{lVert x_irVert}, frac{p_j}{lVert p_jrVert}rightrangle
= leftlangle sum_{i}mu_{ij}frac{x_i}{lVert x_irVert}, frac{p_j}{lVert p_jrVert}rightrangle tag{2}
$$

Note that it does make sense: $sum_{i}mu_{ij}frac{x_i}{lVert x_irVert}$ is a vector, and so is $frac{p_j}{lVert p_jrVert}$. So it's indeed the inner product between two vectors.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    This answers my question. Thank you! I will accept this answer in 3 minutes.
    $endgroup$
    – Hello Lili
    Nov 24 '18 at 17:07










  • $begingroup$
    @HelloLili Glad this helped!
    $endgroup$
    – Clement C.
    Nov 24 '18 at 17:08














1












1








1





$begingroup$

Remember that the inner product is bilinear: for three vectors $u,v,w$ and scalars $alpha,beta$,
$$
langle alpha u, beta vrangle = alphabetalangle u, vrangle, qquad langle u, v+wrangle = langle u, vrangle+langle u, wrangle
$$

What happened above is exactly what follows when applying these rules: for every $i,j$
$$
frac{langle x_i, p_jrangle}{lVert x_irVertlVert p_jrVert} = leftlangle frac{x_i}{lVert x_irVert}, frac{p_j}{lVert p_jrVert}rightrangle tag{1}
$$

since $lVert x_irVert,lVert p_jrVertinmathbb{R}$ are scalars (the brackets became "bigger" just for style considerations, like parentheses would be drawn bigger when what's inside looks "large").



Then, each $mu_{ij}$ is also a scalar, and you have a sum, so
$$
sum_{i} mu_{ij}leftlangle frac{x_i}{lVert x_irVert}, frac{p_j}{lVert p_jrVert}rightrangle
= sum_{i} leftlangle mu_{ij}frac{x_i}{lVert x_irVert}, frac{p_j}{lVert p_jrVert}rightrangle
= leftlangle sum_{i}mu_{ij}frac{x_i}{lVert x_irVert}, frac{p_j}{lVert p_jrVert}rightrangle tag{2}
$$

Note that it does make sense: $sum_{i}mu_{ij}frac{x_i}{lVert x_irVert}$ is a vector, and so is $frac{p_j}{lVert p_jrVert}$. So it's indeed the inner product between two vectors.






share|cite|improve this answer









$endgroup$



Remember that the inner product is bilinear: for three vectors $u,v,w$ and scalars $alpha,beta$,
$$
langle alpha u, beta vrangle = alphabetalangle u, vrangle, qquad langle u, v+wrangle = langle u, vrangle+langle u, wrangle
$$

What happened above is exactly what follows when applying these rules: for every $i,j$
$$
frac{langle x_i, p_jrangle}{lVert x_irVertlVert p_jrVert} = leftlangle frac{x_i}{lVert x_irVert}, frac{p_j}{lVert p_jrVert}rightrangle tag{1}
$$

since $lVert x_irVert,lVert p_jrVertinmathbb{R}$ are scalars (the brackets became "bigger" just for style considerations, like parentheses would be drawn bigger when what's inside looks "large").



Then, each $mu_{ij}$ is also a scalar, and you have a sum, so
$$
sum_{i} mu_{ij}leftlangle frac{x_i}{lVert x_irVert}, frac{p_j}{lVert p_jrVert}rightrangle
= sum_{i} leftlangle mu_{ij}frac{x_i}{lVert x_irVert}, frac{p_j}{lVert p_jrVert}rightrangle
= leftlangle sum_{i}mu_{ij}frac{x_i}{lVert x_irVert}, frac{p_j}{lVert p_jrVert}rightrangle tag{2}
$$

Note that it does make sense: $sum_{i}mu_{ij}frac{x_i}{lVert x_irVert}$ is a vector, and so is $frac{p_j}{lVert p_jrVert}$. So it's indeed the inner product between two vectors.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Nov 24 '18 at 17:04









Clement C.Clement C.

49.8k33886




49.8k33886












  • $begingroup$
    This answers my question. Thank you! I will accept this answer in 3 minutes.
    $endgroup$
    – Hello Lili
    Nov 24 '18 at 17:07










  • $begingroup$
    @HelloLili Glad this helped!
    $endgroup$
    – Clement C.
    Nov 24 '18 at 17:08


















  • $begingroup$
    This answers my question. Thank you! I will accept this answer in 3 minutes.
    $endgroup$
    – Hello Lili
    Nov 24 '18 at 17:07










  • $begingroup$
    @HelloLili Glad this helped!
    $endgroup$
    – Clement C.
    Nov 24 '18 at 17:08
















$begingroup$
This answers my question. Thank you! I will accept this answer in 3 minutes.
$endgroup$
– Hello Lili
Nov 24 '18 at 17:07




$begingroup$
This answers my question. Thank you! I will accept this answer in 3 minutes.
$endgroup$
– Hello Lili
Nov 24 '18 at 17:07












$begingroup$
@HelloLili Glad this helped!
$endgroup$
– Clement C.
Nov 24 '18 at 17:08




$begingroup$
@HelloLili Glad this helped!
$endgroup$
– Clement C.
Nov 24 '18 at 17:08


















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