Derivative of a trace with Kronecker product
Say I have the following expression:
$f=mathbf{Tr} (S^T L S)$,where $S=(Bbigotimes A)H$ Then what's the derivative of $f$ with respect to $A$ and $B$ ?
derivatives chain-rule kronecker-product
asked Aug 31 '16 at 6:39
FuliXiong
12
add a comment |
Say I have the following expression:
$f=mathbf{Tr} (S^T L S)$,where $S=(Bbigotimes A)H$ Then what's the derivative of $f$ with respect to $A$ and $B$ ?
derivatives chain-rule kronecker-product
asked Aug 31 '16 at 6:39
FuliXiong
12
add a comment |
Say I have the following expression:
$f=mathbf{Tr} (S^T L S)$,where $S=(Bbigotimes A)H$ Then what's the derivative of $f$ with respect to $A$ and $B$ ?
derivatives chain-rule kronecker-product
asked Aug 31 '16 at 6:39
FuliXiong
12
Say I have the following expression:
$f=mathbf{Tr} (S^T L S)$,where $S=(Bbigotimes A)H$ Then what's the derivative of $f$ with respect to $A$ and $B$ ?
derivatives chain-rule kronecker-product
derivatives chain-rule kronecker-product
asked Aug 31 '16 at 6:39
FuliXiong
12
asked Aug 31 '16 at 6:39
FuliXiong
12
asked Aug 31 '16 at 6:39
FuliXiong
12
asked Aug 31 '16 at 6:39
FuliXiong
12
asked Aug 31 '16 at 6:39
FuliXiong
12
12
add a comment |
add a comment |
1 Answer
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First find the SVD of
$$H=sum_k(sigma_ku_k)v_k^T = sum_kw_kv_k^T$$
Then use the trace/Frobenius product $big(A!:!B={rm Tr}(A^TB)big)$ to rewrite the function before finding its differential and the gradients.
$$eqalign{
f &= L:SS^T crcr
df &= L:(dS,S^T+S,dS^T) cr
&= (L+L^T):dS,S^T cr
&= (L+L^T),S:dS cr
&= (L+L^T),S:(dBotimes A+Botimes dA),H cr
&= sum_k,,(L+L^T),S:(dBotimes A+Botimes dA),w_kv_k^T cr
&= sum_k,,(L+L^T),Sv_k:(dBotimes A+Botimes dA),w_k cr
&= sum_k,,q_k:{rm vec}(AW_k,dB^T+dA,W_kB^T) cr
&= sum_k,,Q_k:(AW_k,dB^T+dA,W_kB^T) cr
&= sum_k,,Q_kBW_k^T:dA + Q_k^TAW_k:dB crcr
frac{partial f}{partial A} &= sum_k,Q_kBW_k^T,,,,,,,,
frac{partial f}{partial B} = sum_k,Q_k^TAW_k cr
}$$
where
$$eqalign{
{rm vec}(Q_k) &= q_k &= (L+L^T),Sv_k cr
{rm vec}(W_k) &= w_k &= sigma_k v_k cr
}$$
answered Oct 15 '16 at 5:28
greg
7,5251821
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
First find the SVD of
$$H=sum_k(sigma_ku_k)v_k^T = sum_kw_kv_k^T$$
Then use the trace/Frobenius product $big(A!:!B={rm Tr}(A^TB)big)$ to rewrite the function before finding its differential and the gradients.
$$eqalign{
f &= L:SS^T crcr
df &= L:(dS,S^T+S,dS^T) cr
&= (L+L^T):dS,S^T cr
&= (L+L^T),S:dS cr
&= (L+L^T),S:(dBotimes A+Botimes dA),H cr
&= sum_k,,(L+L^T),S:(dBotimes A+Botimes dA),w_kv_k^T cr
&= sum_k,,(L+L^T),Sv_k:(dBotimes A+Botimes dA),w_k cr
&= sum_k,,q_k:{rm vec}(AW_k,dB^T+dA,W_kB^T) cr
&= sum_k,,Q_k:(AW_k,dB^T+dA,W_kB^T) cr
&= sum_k,,Q_kBW_k^T:dA + Q_k^TAW_k:dB crcr
frac{partial f}{partial A} &= sum_k,Q_kBW_k^T,,,,,,,,
frac{partial f}{partial B} = sum_k,Q_k^TAW_k cr
}$$
where
$$eqalign{
{rm vec}(Q_k) &= q_k &= (L+L^T),Sv_k cr
{rm vec}(W_k) &= w_k &= sigma_k v_k cr
}$$
answered Oct 15 '16 at 5:28
greg
7,5251821
add a comment |
First find the SVD of
$$H=sum_k(sigma_ku_k)v_k^T = sum_kw_kv_k^T$$
Then use the trace/Frobenius product $big(A!:!B={rm Tr}(A^TB)big)$ to rewrite the function before finding its differential and the gradients.
$$eqalign{
f &= L:SS^T crcr
df &= L:(dS,S^T+S,dS^T) cr
&= (L+L^T):dS,S^T cr
&= (L+L^T),S:dS cr
&= (L+L^T),S:(dBotimes A+Botimes dA),H cr
&= sum_k,,(L+L^T),S:(dBotimes A+Botimes dA),w_kv_k^T cr
&= sum_k,,(L+L^T),Sv_k:(dBotimes A+Botimes dA),w_k cr
&= sum_k,,q_k:{rm vec}(AW_k,dB^T+dA,W_kB^T) cr
&= sum_k,,Q_k:(AW_k,dB^T+dA,W_kB^T) cr
&= sum_k,,Q_kBW_k^T:dA + Q_k^TAW_k:dB crcr
frac{partial f}{partial A} &= sum_k,Q_kBW_k^T,,,,,,,,
frac{partial f}{partial B} = sum_k,Q_k^TAW_k cr
}$$
where
$$eqalign{
{rm vec}(Q_k) &= q_k &= (L+L^T),Sv_k cr
{rm vec}(W_k) &= w_k &= sigma_k v_k cr
}$$
answered Oct 15 '16 at 5:28
greg
7,5251821
add a comment |
First find the SVD of
$$H=sum_k(sigma_ku_k)v_k^T = sum_kw_kv_k^T$$
Then use the trace/Frobenius product $big(A!:!B={rm Tr}(A^TB)big)$ to rewrite the function before finding its differential and the gradients.
$$eqalign{
f &= L:SS^T crcr
df &= L:(dS,S^T+S,dS^T) cr
&= (L+L^T):dS,S^T cr
&= (L+L^T),S:dS cr
&= (L+L^T),S:(dBotimes A+Botimes dA),H cr
&= sum_k,,(L+L^T),S:(dBotimes A+Botimes dA),w_kv_k^T cr
&= sum_k,,(L+L^T),Sv_k:(dBotimes A+Botimes dA),w_k cr
&= sum_k,,q_k:{rm vec}(AW_k,dB^T+dA,W_kB^T) cr
&= sum_k,,Q_k:(AW_k,dB^T+dA,W_kB^T) cr
&= sum_k,,Q_kBW_k^T:dA + Q_k^TAW_k:dB crcr
frac{partial f}{partial A} &= sum_k,Q_kBW_k^T,,,,,,,,
frac{partial f}{partial B} = sum_k,Q_k^TAW_k cr
}$$
where
$$eqalign{
{rm vec}(Q_k) &= q_k &= (L+L^T),Sv_k cr
{rm vec}(W_k) &= w_k &= sigma_k v_k cr
}$$
answered Oct 15 '16 at 5:28
greg
7,5251821
First find the SVD of
$$H=sum_k(sigma_ku_k)v_k^T = sum_kw_kv_k^T$$
Then use the trace/Frobenius product $big(A!:!B={rm Tr}(A^TB)big)$ to rewrite the function before finding its differential and the gradients.
$$eqalign{
f &= L:SS^T crcr
df &= L:(dS,S^T+S,dS^T) cr
&= (L+L^T):dS,S^T cr
&= (L+L^T),S:dS cr
&= (L+L^T),S:(dBotimes A+Botimes dA),H cr
&= sum_k,,(L+L^T),S:(dBotimes A+Botimes dA),w_kv_k^T cr
&= sum_k,,(L+L^T),Sv_k:(dBotimes A+Botimes dA),w_k cr
&= sum_k,,q_k:{rm vec}(AW_k,dB^T+dA,W_kB^T) cr
&= sum_k,,Q_k:(AW_k,dB^T+dA,W_kB^T) cr
&= sum_k,,Q_kBW_k^T:dA + Q_k^TAW_k:dB crcr
frac{partial f}{partial A} &= sum_k,Q_kBW_k^T,,,,,,,,
frac{partial f}{partial B} = sum_k,Q_k^TAW_k cr
}$$
where
$$eqalign{
{rm vec}(Q_k) &= q_k &= (L+L^T),Sv_k cr
{rm vec}(W_k) &= w_k &= sigma_k v_k cr
}$$
answered Oct 15 '16 at 5:28
greg
7,5251821
edited Dec 2 '18 at 21:11
answered Oct 15 '16 at 5:28
greg
7,5251821
answered Oct 15 '16 at 5:28
greg
7,5251821
answered Oct 15 '16 at 5:28
greg
7,5251821
7,5251821
add a comment |
add a comment |
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