Mixing
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Equivalent forms Optimization
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We were told to assume in class that the below optimization formulations are equivalent-
$$min_wmax_delta:||(X+delta)w-y||_2^2$$
$$min_w||Xw-y||_2^2+lambda||w||_2^2 $$
for appropriately chosen $lambda$.
$X,deltain R^mtimes n,~win R^ntimes1,~yin R^mtimes1$
Can someone please explain why this is true? A reference paper pointing this out would also be appreciated.
matrices optimization convex-optimization least-squares quadratic-programming
edited Jul 27 at 14:11
Michael Grant
14.6k11743
asked Jul 24 at 3:38
Amrit Prasad
727
 |Â
show 4 more comments
up vote
4
down vote
favorite
3
We were told to assume in class that the below optimization formulations are equivalent-
$$min_wmax_delta:||(X+delta)w-y||_2^2$$
$$min_w||Xw-y||_2^2+lambda||w||_2^2 $$
for appropriately chosen $lambda$.
$X,deltain R^mtimes n,~win R^ntimes1,~yin R^mtimes1$
Can someone please explain why this is true? A reference paper pointing this out would also be appreciated.
matrices optimization convex-optimization least-squares quadratic-programming
edited Jul 27 at 14:11
Michael Grant
14.6k11743
asked Jul 24 at 3:38
Amrit Prasad
727
What does $|delta|_2$ mean when $delta$ is a matrix?
– angryavian
Jul 24 at 4:04
4
proceedings.mlr.press/v28/yang13e.pdf
– LinAlg
Jul 24 at 13:59
1
There's a perfectly standard defenition of the 2-norm of a matrix that isn't the Frobenius norm. Wny not write $| delta |_F$, since that's what you mean?
– Brian Borchers
Jul 24 at 15:23
1
@MichaelGrant Thanks, I am familiar with both definitions. I was just trying to get OP to clarify, since I have seen some people use $|cdot|_2$ to refer to Frobenius norm, while [most] others use it to refer to operator norm.
– angryavian
Jul 27 at 17:05
1
Fair enough. I consider the Frobenius interpretation incorrect. Let’s fight the good fight!
– Michael Grant
Jul 27 at 17:06
 |Â
show 4 more comments
up vote
4
down vote
favorite
3
up vote
4
down vote
favorite
3
3
We were told to assume in class that the below optimization formulations are equivalent-
$$min_wmax_delta:||(X+delta)w-y||_2^2$$
$$min_w||Xw-y||_2^2+lambda||w||_2^2 $$
for appropriately chosen $lambda$.
$X,deltain R^mtimes n,~win R^ntimes1,~yin R^mtimes1$
Can someone please explain why this is true? A reference paper pointing this out would also be appreciated.
matrices optimization convex-optimization least-squares quadratic-programming
edited Jul 27 at 14:11
Michael Grant
14.6k11743
asked Jul 24 at 3:38
Amrit Prasad
727
We were told to assume in class that the below optimization formulations are equivalent-
$$min_wmax_delta:||(X+delta)w-y||_2^2$$
$$min_w||Xw-y||_2^2+lambda||w||_2^2 $$
for appropriately chosen $lambda$.
$X,deltain R^mtimes n,~win R^ntimes1,~yin R^mtimes1$
Can someone please explain why this is true? A reference paper pointing this out would also be appreciated.
matrices optimization convex-optimization least-squares quadratic-programming
edited Jul 27 at 14:11
Michael Grant
14.6k11743
asked Jul 24 at 3:38
Amrit Prasad
727
edited Jul 27 at 14:11
Michael Grant
14.6k11743
edited Jul 27 at 14:11
Michael Grant
14.6k11743
edited Jul 27 at 14:11
Michael Grant
14.6k11743
14.6k11743
asked Jul 24 at 3:38
Amrit Prasad
727
asked Jul 24 at 3:38
Amrit Prasad
727
asked Jul 24 at 3:38
Amrit Prasad
727
727
What does $|delta|_2$ mean when $delta$ is a matrix?
– angryavian
Jul 24 at 4:04
4
proceedings.mlr.press/v28/yang13e.pdf
– LinAlg
Jul 24 at 13:59
1
There's a perfectly standard defenition of the 2-norm of a matrix that isn't the Frobenius norm. Wny not write $| delta |_F$, since that's what you mean?
– Brian Borchers
Jul 24 at 15:23
1
@MichaelGrant Thanks, I am familiar with both definitions. I was just trying to get OP to clarify, since I have seen some people use $|cdot|_2$ to refer to Frobenius norm, while [most] others use it to refer to operator norm.
– angryavian
Jul 27 at 17:05
1
Fair enough. I consider the Frobenius interpretation incorrect. Let’s fight the good fight!
– Michael Grant
Jul 27 at 17:06
 |Â
show 4 more comments
What does $|delta|_2$ mean when $delta$ is a matrix?
– angryavian
Jul 24 at 4:04
4
proceedings.mlr.press/v28/yang13e.pdf
– LinAlg
Jul 24 at 13:59
1
There's a perfectly standard defenition of the 2-norm of a matrix that isn't the Frobenius norm. Wny not write $| delta |_F$, since that's what you mean?
– Brian Borchers
Jul 24 at 15:23
1
@MichaelGrant Thanks, I am familiar with both definitions. I was just trying to get OP to clarify, since I have seen some people use $|cdot|_2$ to refer to Frobenius norm, while [most] others use it to refer to operator norm.
– angryavian
Jul 27 at 17:05
1
Fair enough. I consider the Frobenius interpretation incorrect. Let’s fight the good fight!
– Michael Grant
Jul 27 at 17:06
What does $|delta|_2$ mean when $delta$ is a matrix?
– angryavian
Jul 24 at 4:04
What does $|delta|_2$ mean when $delta$ is a matrix?
– angryavian
Jul 24 at 4:04
4
4
proceedings.mlr.press/v28/yang13e.pdf
– LinAlg
Jul 24 at 13:59
proceedings.mlr.press/v28/yang13e.pdf
– LinAlg
Jul 24 at 13:59
1
1
There's a perfectly standard defenition of the 2-norm of a matrix that isn't the Frobenius norm. Wny not write $| delta |_F$, since that's what you mean?
– Brian Borchers
Jul 24 at 15:23
There's a perfectly standard defenition of the 2-norm of a matrix that isn't the Frobenius norm. Wny not write $| delta |_F$, since that's what you mean?
– Brian Borchers
Jul 24 at 15:23
1
1
@MichaelGrant Thanks, I am familiar with both definitions. I was just trying to get OP to clarify, since I have seen some people use $|cdot|_2$ to refer to Frobenius norm, while [most] others use it to refer to operator norm.
– angryavian
Jul 27 at 17:05
@MichaelGrant Thanks, I am familiar with both definitions. I was just trying to get OP to clarify, since I have seen some people use $|cdot|_2$ to refer to Frobenius norm, while [most] others use it to refer to operator norm.
– angryavian
Jul 27 at 17:05
1
1
Fair enough. I consider the Frobenius interpretation incorrect. Let’s fight the good fight!
– Michael Grant
Jul 27 at 17:06
Fair enough. I consider the Frobenius interpretation incorrect. Let’s fight the good fight!
– Michael Grant
Jul 27 at 17:06
 |Â
show 4 more comments
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What does $|delta|_2$ mean when $delta$ is a matrix?
– angryavian
Jul 24 at 4:04
4
proceedings.mlr.press/v28/yang13e.pdf
– LinAlg
Jul 24 at 13:59
1
There's a perfectly standard defenition of the 2-norm of a matrix that isn't the Frobenius norm. Wny not write $| delta |_F$, since that's what you mean?
– Brian Borchers
Jul 24 at 15:23
1
@MichaelGrant Thanks, I am familiar with both definitions. I was just trying to get OP to clarify, since I have seen some people use $|cdot|_2$ to refer to Frobenius norm, while [most] others use it to refer to operator norm.
– angryavian
Jul 27 at 17:05
1
Fair enough. I consider the Frobenius interpretation incorrect. Let’s fight the good fight!
– Michael Grant
Jul 27 at 17:06