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    The chain rule applies in some of the cases, but unfortunately does not apply in matrix-by-scalar derivatives or scalar-by-matrix derivatives (in the latter case, mostly involving the trace operator applied to matrices).

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zaq123321

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What's the matrix norm in J

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2Â¥2015-11-04 00:09:04
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pluiefox

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ÒýÓûØÌû:
2Â¥: Originally posted by zaq123321 at 2015-11-04 00:09:04
What's the matrix norm in J

J ÊÇ [p - [c1 ; c2]] µÄ¶þ·¶ÊýµÄƽ·½ÔÙ³ËÒÔ¶þ·ÖÖ®Ò»£¬ÎªÁ˼ò»¯ÎÊÌâ¿ÉÒÔ°Ñ J ÖØÐ¶¨ÒåΪ£º
J = ||p||£¬¼´ J Ϊ p µÄÒ»·¶Êý£¬ÏµÊýȫΪ1µÄpµÄ·ÖÁ¿µÄÏßÐÔ×éºÏ¡£
¶ÔÓڸĹýµÄʽ×Ó£¬ÎÒ°´ÔªËØÍƵ¼£¬Ïȵõ½ÁË{∂J / ∂Wij}£¬Õâ¸ö½á¹û¿ÉÒÔÈÏΪÊÇÕýÈ·µÄ¡£½Ó×ÅÎÒ°´¾ØÕóÏòÁ¿ÍƵ¼£¨²»¿¼ÂǾØÕóÇóµ¼¹æÔò£©£¬½á¹ûÊÇ£º
{∂J / ∂Wij} = ((J/p) * (p/q)£© * (q/W)) = [1, 1, ... , 1] * V * [c1;c2]
ÆäÖÐ J/p ±íʾ J ¶Ô p Ç󯫵¼£¬ÆäÓàÒ»Ñù¡£
²¢ÇÒÆ«µ¼Êý J/p ºÍ p/q ÓõķÖ×Ó²¼¾Ö£¬ËùÒÔ J/p ÊÇ 2n άÐÐÏòÁ¿£¬p/q ÊÇ2n x n ¾ØÕ󣬶ø q/W ÔÝʱ²»¿¼ÂÇÕýÈ·ÐԺͲ¼¾ÖÐÎʽֱ½ÓÓà [c1;c2] ±íʾ¡£
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((J/p) * (p/q))' * c'
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3Â¥2015-11-04 09:45:10
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zaq123321

ר¼Ò¹ËÎÊ (ÖøÃûдÊÖ)

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ÒýÓûØÌû:
3Â¥: Originally posted by pluiefox at 2015-11-04 09:45:10
J ÊÇ  µÄ¶þ·¶ÊýµÄƽ·½ÔÙ³ËÒÔ¶þ·ÖÖ®Ò»£¬ÎªÁ˼ò»¯ÎÊÌâ¿ÉÒÔ°Ñ J ÖØÐ¶¨ÒåΪ£º
J = ||p||£¬¼´ J Ϊ p µÄÒ»·¶Êý£¬ÏµÊýȫΪ1µÄpµÄ·ÖÁ¿µÄÏßÐÔ×éºÏ¡£
¶ÔÓڸĹýµÄʽ×Ó£¬ÎÒ°´ÔªËØÍƵ¼£¬Ïȵõ½ÁË{∂J / ∂Wij}£¬Õâ ...

You can write out the squared vector norm as inner product   f=<a, a>=a^Ta.
Here a=(VW-I)(c_1, c_2).
Then df/dW = 2*<a, da/dW>  
You can find how to calculate da/dW from matrix calculus website and get answer.
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4Â¥2015-11-04 10:32:54
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