What is the meaning of stacked integers in linear algebra notation?
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I was recently reading a stack exchange post on solving a polynomial regression problem with gradient descent, and one contributor identified the objective function as:
$$text{argmin}_{beta} | mathbf{y - Xbeta}|_2^2$$
They identified this as the "squared loss" i.e. L2 Loss function, which makes sense.
The notation in the equation, however, is new to me. Why are the 2's stacked atop each other in this equation? What do those 2's signify? What is the difference between the formulation above and:
$$text{argmin}_{beta} | mathbf{y - Xbeta}|^2$$
I'd be very grateful for any help others can offer with this question.
linear-algebra optimization notation
add a comment |
up vote
1
down vote
favorite
I was recently reading a stack exchange post on solving a polynomial regression problem with gradient descent, and one contributor identified the objective function as:
$$text{argmin}_{beta} | mathbf{y - Xbeta}|_2^2$$
They identified this as the "squared loss" i.e. L2 Loss function, which makes sense.
The notation in the equation, however, is new to me. Why are the 2's stacked atop each other in this equation? What do those 2's signify? What is the difference between the formulation above and:
$$text{argmin}_{beta} | mathbf{y - Xbeta}|^2$$
I'd be very grateful for any help others can offer with this question.
linear-algebra optimization notation
It's just the square of the $ell^2$ norm.
– anomaly
Nov 16 at 1:35
One is a subscript (indicating the particular norm), one is a square.
– Randall
Nov 16 at 1:36
Hmm, I'm still not following. What is the subscript for? Why is the subscript a 2? Sorry I'm pretty daft (I was an English major so have to learn all this on my own)
– duhaime
Nov 16 at 1:38
1
See: en.wikipedia.org/wiki/List_of_Banach_spaces
– Randall
Nov 16 at 1:42
Thanks @Randall, that link and zahbaz's note below cleared this up for me
– duhaime
Nov 16 at 1:50
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I was recently reading a stack exchange post on solving a polynomial regression problem with gradient descent, and one contributor identified the objective function as:
$$text{argmin}_{beta} | mathbf{y - Xbeta}|_2^2$$
They identified this as the "squared loss" i.e. L2 Loss function, which makes sense.
The notation in the equation, however, is new to me. Why are the 2's stacked atop each other in this equation? What do those 2's signify? What is the difference between the formulation above and:
$$text{argmin}_{beta} | mathbf{y - Xbeta}|^2$$
I'd be very grateful for any help others can offer with this question.
linear-algebra optimization notation
I was recently reading a stack exchange post on solving a polynomial regression problem with gradient descent, and one contributor identified the objective function as:
$$text{argmin}_{beta} | mathbf{y - Xbeta}|_2^2$$
They identified this as the "squared loss" i.e. L2 Loss function, which makes sense.
The notation in the equation, however, is new to me. Why are the 2's stacked atop each other in this equation? What do those 2's signify? What is the difference between the formulation above and:
$$text{argmin}_{beta} | mathbf{y - Xbeta}|^2$$
I'd be very grateful for any help others can offer with this question.
linear-algebra optimization notation
linear-algebra optimization notation
asked Nov 16 at 1:33
duhaime
1427
1427
It's just the square of the $ell^2$ norm.
– anomaly
Nov 16 at 1:35
One is a subscript (indicating the particular norm), one is a square.
– Randall
Nov 16 at 1:36
Hmm, I'm still not following. What is the subscript for? Why is the subscript a 2? Sorry I'm pretty daft (I was an English major so have to learn all this on my own)
– duhaime
Nov 16 at 1:38
1
See: en.wikipedia.org/wiki/List_of_Banach_spaces
– Randall
Nov 16 at 1:42
Thanks @Randall, that link and zahbaz's note below cleared this up for me
– duhaime
Nov 16 at 1:50
add a comment |
It's just the square of the $ell^2$ norm.
– anomaly
Nov 16 at 1:35
One is a subscript (indicating the particular norm), one is a square.
– Randall
Nov 16 at 1:36
Hmm, I'm still not following. What is the subscript for? Why is the subscript a 2? Sorry I'm pretty daft (I was an English major so have to learn all this on my own)
– duhaime
Nov 16 at 1:38
1
See: en.wikipedia.org/wiki/List_of_Banach_spaces
– Randall
Nov 16 at 1:42
Thanks @Randall, that link and zahbaz's note below cleared this up for me
– duhaime
Nov 16 at 1:50
It's just the square of the $ell^2$ norm.
– anomaly
Nov 16 at 1:35
It's just the square of the $ell^2$ norm.
– anomaly
Nov 16 at 1:35
One is a subscript (indicating the particular norm), one is a square.
– Randall
Nov 16 at 1:36
One is a subscript (indicating the particular norm), one is a square.
– Randall
Nov 16 at 1:36
Hmm, I'm still not following. What is the subscript for? Why is the subscript a 2? Sorry I'm pretty daft (I was an English major so have to learn all this on my own)
– duhaime
Nov 16 at 1:38
Hmm, I'm still not following. What is the subscript for? Why is the subscript a 2? Sorry I'm pretty daft (I was an English major so have to learn all this on my own)
– duhaime
Nov 16 at 1:38
1
1
See: en.wikipedia.org/wiki/List_of_Banach_spaces
– Randall
Nov 16 at 1:42
See: en.wikipedia.org/wiki/List_of_Banach_spaces
– Randall
Nov 16 at 1:42
Thanks @Randall, that link and zahbaz's note below cleared this up for me
– duhaime
Nov 16 at 1:50
Thanks @Randall, that link and zahbaz's note below cleared this up for me
– duhaime
Nov 16 at 1:50
add a comment |
2 Answers
2
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1
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accepted
The subscript is indicating the type of norm. For instance, the finite dimensional Lp norms are given by
$$||vec{x}||_p = left(sum_{i=1}^n |x_i|^p right)^{1/p}$$
The superscript is simply the square. So, the L2 norm is then
$$||vec{x}||_2 = left(sum_{i=1}^n |x_i|^2 right)^{1/2}$$
$$||vec{x}||_2^2 = sum_{i=1}^n x_i^2.$$
The main idea here is that there are many types of norms, and we often want to specify which one it is that we're using.
add a comment |
up vote
0
down vote
It just ensures you are not doing something like taking the Lp norm and then squaring it. That would have a subscript p and exponent 2. In this case, you get extra clarity that is L2 as usual. That is just an example showing how the subscript provides extra (usually unnecessary) clarification.
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
The subscript is indicating the type of norm. For instance, the finite dimensional Lp norms are given by
$$||vec{x}||_p = left(sum_{i=1}^n |x_i|^p right)^{1/p}$$
The superscript is simply the square. So, the L2 norm is then
$$||vec{x}||_2 = left(sum_{i=1}^n |x_i|^2 right)^{1/2}$$
$$||vec{x}||_2^2 = sum_{i=1}^n x_i^2.$$
The main idea here is that there are many types of norms, and we often want to specify which one it is that we're using.
add a comment |
up vote
1
down vote
accepted
The subscript is indicating the type of norm. For instance, the finite dimensional Lp norms are given by
$$||vec{x}||_p = left(sum_{i=1}^n |x_i|^p right)^{1/p}$$
The superscript is simply the square. So, the L2 norm is then
$$||vec{x}||_2 = left(sum_{i=1}^n |x_i|^2 right)^{1/2}$$
$$||vec{x}||_2^2 = sum_{i=1}^n x_i^2.$$
The main idea here is that there are many types of norms, and we often want to specify which one it is that we're using.
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
The subscript is indicating the type of norm. For instance, the finite dimensional Lp norms are given by
$$||vec{x}||_p = left(sum_{i=1}^n |x_i|^p right)^{1/p}$$
The superscript is simply the square. So, the L2 norm is then
$$||vec{x}||_2 = left(sum_{i=1}^n |x_i|^2 right)^{1/2}$$
$$||vec{x}||_2^2 = sum_{i=1}^n x_i^2.$$
The main idea here is that there are many types of norms, and we often want to specify which one it is that we're using.
The subscript is indicating the type of norm. For instance, the finite dimensional Lp norms are given by
$$||vec{x}||_p = left(sum_{i=1}^n |x_i|^p right)^{1/p}$$
The superscript is simply the square. So, the L2 norm is then
$$||vec{x}||_2 = left(sum_{i=1}^n |x_i|^2 right)^{1/2}$$
$$||vec{x}||_2^2 = sum_{i=1}^n x_i^2.$$
The main idea here is that there are many types of norms, and we often want to specify which one it is that we're using.
answered Nov 16 at 1:45
zahbaz
8,14921937
8,14921937
add a comment |
add a comment |
up vote
0
down vote
It just ensures you are not doing something like taking the Lp norm and then squaring it. That would have a subscript p and exponent 2. In this case, you get extra clarity that is L2 as usual. That is just an example showing how the subscript provides extra (usually unnecessary) clarification.
add a comment |
up vote
0
down vote
It just ensures you are not doing something like taking the Lp norm and then squaring it. That would have a subscript p and exponent 2. In this case, you get extra clarity that is L2 as usual. That is just an example showing how the subscript provides extra (usually unnecessary) clarification.
add a comment |
up vote
0
down vote
up vote
0
down vote
It just ensures you are not doing something like taking the Lp norm and then squaring it. That would have a subscript p and exponent 2. In this case, you get extra clarity that is L2 as usual. That is just an example showing how the subscript provides extra (usually unnecessary) clarification.
It just ensures you are not doing something like taking the Lp norm and then squaring it. That would have a subscript p and exponent 2. In this case, you get extra clarity that is L2 as usual. That is just an example showing how the subscript provides extra (usually unnecessary) clarification.
answered Nov 16 at 1:39
AHusain
2,7602816
2,7602816
add a comment |
add a comment |
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It's just the square of the $ell^2$ norm.
– anomaly
Nov 16 at 1:35
One is a subscript (indicating the particular norm), one is a square.
– Randall
Nov 16 at 1:36
Hmm, I'm still not following. What is the subscript for? Why is the subscript a 2? Sorry I'm pretty daft (I was an English major so have to learn all this on my own)
– duhaime
Nov 16 at 1:38
1
See: en.wikipedia.org/wiki/List_of_Banach_spaces
– Randall
Nov 16 at 1:42
Thanks @Randall, that link and zahbaz's note below cleared this up for me
– duhaime
Nov 16 at 1:50