Is it possible to define a Hessian Matrix for a Matrix-valued function?
$begingroup$
So I'm doing a project on optimization (non-negative matrix factorization), which I know is not convex, from this question:
Why does the non-negative matrix factorization problem non-convex?
However this was addressed only for the scalar case which is not my project's focus.
My question is: How am I supposed to define a gradient and a Hessian matrix for more general cases. Is it possible? Is it still called a Hessian matrix or is it some sort of tensor (of which I don't really know much).
My function:
$$ f = min_{W, H} leftlVert X ; - ; WH rightrVert_{F}^{2} $$
Which is equivalent to:
$$ min_{W, H} f = trlbrack (X ; - ; WH)(X ; - ; WH)^{T} rbrack = trlbrack (X ; - ; WH)^{T}(X ; - ; WH) rbrack $$
I know how to calculate the partial derivatives of this function, and I acutally have:
$$ frac{partial f}{partial W} = frac{partial trlbrack (X ; - ; WH)(X ; - ; WH)^{T} rbrack}{partial W} = -2XH^{T} + 2WHH^{T}$$
Which is the same result as equation (24) found in this document:
http://cal.cs.illinois.edu/~johannes/research/matrix%20calculus.pdf
Applying the same idea, i calculated the other partial derivative:
$$ frac{partial f}{partial H} = frac{partial trlbrack (X ; - ; WH)(X ; - ; WH)^{T} rbrack}{partial H} = -2W^{T}X + 2W^{T}WH$$
If both of these are correct, is it possible to define a vector with entries that are matrices such that:
$$ nabla f(W, H) = left(frac{partial f}{partial W} ;; frac{partial f}{partial H} right)$$
And what would be the way to compute the Hessian? (If it makes sense).
Please do not mark as duplicate. The question
Defining the Hessian of a function that takes general matrices as an input has no appropriate answer on how to compute these but rather an example to disprove the OP's method and an insult.
matrix-equations matrix-calculus nonlinear-optimization
asked Dec 24 '18 at 10:08
Maganna DevMaganna Dev
33
$endgroup$
|
show 2 more comments
$begingroup$
So I'm doing a project on optimization (non-negative matrix factorization), which I know is not convex, from this question:
Why does the non-negative matrix factorization problem non-convex?
However this was addressed only for the scalar case which is not my project's focus.
My question is: How am I supposed to define a gradient and a Hessian matrix for more general cases. Is it possible? Is it still called a Hessian matrix or is it some sort of tensor (of which I don't really know much).
My function:
$$ f = min_{W, H} leftlVert X ; - ; WH rightrVert_{F}^{2} $$
Which is equivalent to:
$$ min_{W, H} f = trlbrack (X ; - ; WH)(X ; - ; WH)^{T} rbrack = trlbrack (X ; - ; WH)^{T}(X ; - ; WH) rbrack $$
I know how to calculate the partial derivatives of this function, and I acutally have:
$$ frac{partial f}{partial W} = frac{partial trlbrack (X ; - ; WH)(X ; - ; WH)^{T} rbrack}{partial W} = -2XH^{T} + 2WHH^{T}$$
Which is the same result as equation (24) found in this document:
http://cal.cs.illinois.edu/~johannes/research/matrix%20calculus.pdf
Applying the same idea, i calculated the other partial derivative:
$$ frac{partial f}{partial H} = frac{partial trlbrack (X ; - ; WH)(X ; - ; WH)^{T} rbrack}{partial H} = -2W^{T}X + 2W^{T}WH$$
If both of these are correct, is it possible to define a vector with entries that are matrices such that:
$$ nabla f(W, H) = left(frac{partial f}{partial W} ;; frac{partial f}{partial H} right)$$
And what would be the way to compute the Hessian? (If it makes sense).
Please do not mark as duplicate. The question
Defining the Hessian of a function that takes general matrices as an input has no appropriate answer on how to compute these but rather an example to disprove the OP's method and an insult.
matrix-equations matrix-calculus nonlinear-optimization
asked Dec 24 '18 at 10:08
Maganna DevMaganna Dev
33
$endgroup$
$begingroup$
What's your function $f$?
$endgroup$
– user550103
Dec 24 '18 at 11:50
$begingroup$
@user550103 I've added the function.
$endgroup$
– Maganna Dev
Dec 24 '18 at 12:07
$begingroup$
Your gradients look correct.
$endgroup$
– user550103
Dec 24 '18 at 12:19
$begingroup$
@user550103 What troubles me are the dimensions of each partial derivative. if $$ frac{partial f}{partial W} in R^{m times r}$$ and $$ frac{partial f}{partial H} in R^{r times n}$$, does the gradient i want to define exist? Does the Hessian exist?
$endgroup$
– Maganna Dev
Dec 24 '18 at 12:22
$begingroup$
If I am understanding your problem, then the gradients are matrices. Yes, they are matrices, which is as expected...
$endgroup$
– user550103
Dec 24 '18 at 12:25
|
show 2 more comments
$begingroup$
So I'm doing a project on optimization (non-negative matrix factorization), which I know is not convex, from this question:
Why does the non-negative matrix factorization problem non-convex?
However this was addressed only for the scalar case which is not my project's focus.
My question is: How am I supposed to define a gradient and a Hessian matrix for more general cases. Is it possible? Is it still called a Hessian matrix or is it some sort of tensor (of which I don't really know much).
My function:
$$ f = min_{W, H} leftlVert X ; - ; WH rightrVert_{F}^{2} $$
Which is equivalent to:
$$ min_{W, H} f = trlbrack (X ; - ; WH)(X ; - ; WH)^{T} rbrack = trlbrack (X ; - ; WH)^{T}(X ; - ; WH) rbrack $$
I know how to calculate the partial derivatives of this function, and I acutally have:
$$ frac{partial f}{partial W} = frac{partial trlbrack (X ; - ; WH)(X ; - ; WH)^{T} rbrack}{partial W} = -2XH^{T} + 2WHH^{T}$$
Which is the same result as equation (24) found in this document:
http://cal.cs.illinois.edu/~johannes/research/matrix%20calculus.pdf
Applying the same idea, i calculated the other partial derivative:
$$ frac{partial f}{partial H} = frac{partial trlbrack (X ; - ; WH)(X ; - ; WH)^{T} rbrack}{partial H} = -2W^{T}X + 2W^{T}WH$$
If both of these are correct, is it possible to define a vector with entries that are matrices such that:
$$ nabla f(W, H) = left(frac{partial f}{partial W} ;; frac{partial f}{partial H} right)$$
And what would be the way to compute the Hessian? (If it makes sense).
Please do not mark as duplicate. The question
Defining the Hessian of a function that takes general matrices as an input has no appropriate answer on how to compute these but rather an example to disprove the OP's method and an insult.
matrix-equations matrix-calculus nonlinear-optimization
asked Dec 24 '18 at 10:08
Maganna DevMaganna Dev
33
$endgroup$
So I'm doing a project on optimization (non-negative matrix factorization), which I know is not convex, from this question:
Why does the non-negative matrix factorization problem non-convex?
However this was addressed only for the scalar case which is not my project's focus.
My question is: How am I supposed to define a gradient and a Hessian matrix for more general cases. Is it possible? Is it still called a Hessian matrix or is it some sort of tensor (of which I don't really know much).
My function:
$$ f = min_{W, H} leftlVert X ; - ; WH rightrVert_{F}^{2} $$
Which is equivalent to:
$$ min_{W, H} f = trlbrack (X ; - ; WH)(X ; - ; WH)^{T} rbrack = trlbrack (X ; - ; WH)^{T}(X ; - ; WH) rbrack $$
I know how to calculate the partial derivatives of this function, and I acutally have:
$$ frac{partial f}{partial W} = frac{partial trlbrack (X ; - ; WH)(X ; - ; WH)^{T} rbrack}{partial W} = -2XH^{T} + 2WHH^{T}$$
Which is the same result as equation (24) found in this document:
http://cal.cs.illinois.edu/~johannes/research/matrix%20calculus.pdf
Applying the same idea, i calculated the other partial derivative:
$$ frac{partial f}{partial H} = frac{partial trlbrack (X ; - ; WH)(X ; - ; WH)^{T} rbrack}{partial H} = -2W^{T}X + 2W^{T}WH$$
If both of these are correct, is it possible to define a vector with entries that are matrices such that:
$$ nabla f(W, H) = left(frac{partial f}{partial W} ;; frac{partial f}{partial H} right)$$
And what would be the way to compute the Hessian? (If it makes sense).
Please do not mark as duplicate. The question
Defining the Hessian of a function that takes general matrices as an input has no appropriate answer on how to compute these but rather an example to disprove the OP's method and an insult.
matrix-equations matrix-calculus nonlinear-optimization
matrix-equations matrix-calculus nonlinear-optimization
asked Dec 24 '18 at 10:08
Maganna DevMaganna Dev
33
asked Dec 24 '18 at 10:08
Maganna DevMaganna Dev
33
edited Dec 24 '18 at 12:02
Maganna Dev
asked Dec 24 '18 at 10:08
Maganna DevMaganna Dev
33
asked Dec 24 '18 at 10:08
Maganna DevMaganna Dev
33
asked Dec 24 '18 at 10:08
Maganna DevMaganna Dev
33
33
$begingroup$
What's your function $f$?
$endgroup$
– user550103
Dec 24 '18 at 11:50
$begingroup$
@user550103 I've added the function.
$endgroup$
– Maganna Dev
Dec 24 '18 at 12:07
$begingroup$
Your gradients look correct.
$endgroup$
– user550103
Dec 24 '18 at 12:19
$begingroup$
@user550103 What troubles me are the dimensions of each partial derivative. if $$ frac{partial f}{partial W} in R^{m times r}$$ and $$ frac{partial f}{partial H} in R^{r times n}$$, does the gradient i want to define exist? Does the Hessian exist?
$endgroup$
– Maganna Dev
Dec 24 '18 at 12:22
$begingroup$
If I am understanding your problem, then the gradients are matrices. Yes, they are matrices, which is as expected...
$endgroup$
– user550103
Dec 24 '18 at 12:25
|
show 2 more comments
$begingroup$
What's your function $f$?
$endgroup$
– user550103
Dec 24 '18 at 11:50
$begingroup$
@user550103 I've added the function.
$endgroup$
– Maganna Dev
Dec 24 '18 at 12:07
$begingroup$
Your gradients look correct.
$endgroup$
– user550103
Dec 24 '18 at 12:19
$begingroup$
@user550103 What troubles me are the dimensions of each partial derivative. if $$ frac{partial f}{partial W} in R^{m times r}$$ and $$ frac{partial f}{partial H} in R^{r times n}$$, does the gradient i want to define exist? Does the Hessian exist?
$endgroup$
– Maganna Dev
Dec 24 '18 at 12:22
$begingroup$
If I am understanding your problem, then the gradients are matrices. Yes, they are matrices, which is as expected...
$endgroup$
– user550103
Dec 24 '18 at 12:25
$begingroup$
What's your function $f$?
$endgroup$
– user550103
Dec 24 '18 at 11:50
$begingroup$
What's your function $f$?
$endgroup$
– user550103
Dec 24 '18 at 11:50
$begingroup$
@user550103 I've added the function.
$endgroup$
– Maganna Dev
Dec 24 '18 at 12:07
$begingroup$
@user550103 I've added the function.
$endgroup$
– Maganna Dev
Dec 24 '18 at 12:07
$begingroup$
Your gradients look correct.
$endgroup$
– user550103
Dec 24 '18 at 12:19
$begingroup$
Your gradients look correct.
$endgroup$
– user550103
Dec 24 '18 at 12:19
$begingroup$
@user550103 What troubles me are the dimensions of each partial derivative. if $$ frac{partial f}{partial W} in R^{m times r}$$ and $$ frac{partial f}{partial H} in R^{r times n}$$, does the gradient i want to define exist? Does the Hessian exist?
$endgroup$
– Maganna Dev
Dec 24 '18 at 12:22
$begingroup$
@user550103 What troubles me are the dimensions of each partial derivative. if $$ frac{partial f}{partial W} in R^{m times r}$$ and $$ frac{partial f}{partial H} in R^{r times n}$$, does the gradient i want to define exist? Does the Hessian exist?
$endgroup$
– Maganna Dev
Dec 24 '18 at 12:22
$begingroup$
If I am understanding your problem, then the gradients are matrices. Yes, they are matrices, which is as expected...
$endgroup$
– user550103
Dec 24 '18 at 12:25
$begingroup$
If I am understanding your problem, then the gradients are matrices. Yes, they are matrices, which is as expected...
$endgroup$
– user550103
Dec 24 '18 at 12:25
|
show 2 more comments
1 Answer
1
active
oldest
votes
$begingroup$
Define a new matrix $$Y=WH-X$$
Write the function in terms of this new variable
$$f = |Y|^2_F = Y:Y$$
where a colon denotes the trace/Frobenius product, i.e. $,,A:B={rm tr}(A^TB)$
Find its differential and gradients.
$$eqalign{
df &= 2Y:dY cr
&= 2Y:(dW,H+W,dH) cr
&= 2YH^T:dW + 2W^TY:dH cr
&= 2(WH-X)H^T:dW + 2W^T(WH-X):dH cr
frac{partial f}{partial W} &= 2(WH-X)H^T,quad
frac{partial f}{partial H} = 2W^T(WH-X) cr
}$$
Since the gradients are themselves matrices, the hessians will be 4th order tensors which cannot be represented in matrix notation.
One way to approach the hessian is to use vectorization which flattens matrices into vectors.
For example,
$$eqalign{
G &= frac{partial f}{partial W} = 2WHH^T - 2XH^T cr
dG &= 2,dW,HH^T cr
{rm vec}(dG) &= 2,{rm vec}(dW,HH^T) cr
dg &= 2,(HH^Totimes I),dw cr
nabla_{ww}f &= 2,(HH^Totimes I) cr
}$$
Working through the other hessians
$$eqalign{
nabla_{hh}f &= 2,(Iotimes W^TW) cr
nabla_{wh}f &= 2(Hotimes W) + 2(Iotimes Y)K cr
nabla_{hw}f &= 2(H^Totimes W^T) + 2(Y^Totimes I)K cr
}$$
where $K$ is the Commutation Matrix which can be used to vectorize the transpose of a matrix
$${rm vec}(X^T) = K,{rm vec}(X)$$
answered Dec 24 '18 at 13:40
greggreg
8,1651822
$endgroup$
add a comment |
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1 Answer
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Define a new matrix $$Y=WH-X$$
Write the function in terms of this new variable
$$f = |Y|^2_F = Y:Y$$
where a colon denotes the trace/Frobenius product, i.e. $,,A:B={rm tr}(A^TB)$
Find its differential and gradients.
$$eqalign{
df &= 2Y:dY cr
&= 2Y:(dW,H+W,dH) cr
&= 2YH^T:dW + 2W^TY:dH cr
&= 2(WH-X)H^T:dW + 2W^T(WH-X):dH cr
frac{partial f}{partial W} &= 2(WH-X)H^T,quad
frac{partial f}{partial H} = 2W^T(WH-X) cr
}$$
Since the gradients are themselves matrices, the hessians will be 4th order tensors which cannot be represented in matrix notation.
One way to approach the hessian is to use vectorization which flattens matrices into vectors.
For example,
$$eqalign{
G &= frac{partial f}{partial W} = 2WHH^T - 2XH^T cr
dG &= 2,dW,HH^T cr
{rm vec}(dG) &= 2,{rm vec}(dW,HH^T) cr
dg &= 2,(HH^Totimes I),dw cr
nabla_{ww}f &= 2,(HH^Totimes I) cr
}$$
Working through the other hessians
$$eqalign{
nabla_{hh}f &= 2,(Iotimes W^TW) cr
nabla_{wh}f &= 2(Hotimes W) + 2(Iotimes Y)K cr
nabla_{hw}f &= 2(H^Totimes W^T) + 2(Y^Totimes I)K cr
}$$
where $K$ is the Commutation Matrix which can be used to vectorize the transpose of a matrix
$${rm vec}(X^T) = K,{rm vec}(X)$$
answered Dec 24 '18 at 13:40
greggreg
8,1651822
$endgroup$
add a comment |
$begingroup$
Define a new matrix $$Y=WH-X$$
Write the function in terms of this new variable
$$f = |Y|^2_F = Y:Y$$
where a colon denotes the trace/Frobenius product, i.e. $,,A:B={rm tr}(A^TB)$
Find its differential and gradients.
$$eqalign{
df &= 2Y:dY cr
&= 2Y:(dW,H+W,dH) cr
&= 2YH^T:dW + 2W^TY:dH cr
&= 2(WH-X)H^T:dW + 2W^T(WH-X):dH cr
frac{partial f}{partial W} &= 2(WH-X)H^T,quad
frac{partial f}{partial H} = 2W^T(WH-X) cr
}$$
Since the gradients are themselves matrices, the hessians will be 4th order tensors which cannot be represented in matrix notation.
One way to approach the hessian is to use vectorization which flattens matrices into vectors.
For example,
$$eqalign{
G &= frac{partial f}{partial W} = 2WHH^T - 2XH^T cr
dG &= 2,dW,HH^T cr
{rm vec}(dG) &= 2,{rm vec}(dW,HH^T) cr
dg &= 2,(HH^Totimes I),dw cr
nabla_{ww}f &= 2,(HH^Totimes I) cr
}$$
Working through the other hessians
$$eqalign{
nabla_{hh}f &= 2,(Iotimes W^TW) cr
nabla_{wh}f &= 2(Hotimes W) + 2(Iotimes Y)K cr
nabla_{hw}f &= 2(H^Totimes W^T) + 2(Y^Totimes I)K cr
}$$
where $K$ is the Commutation Matrix which can be used to vectorize the transpose of a matrix
$${rm vec}(X^T) = K,{rm vec}(X)$$
answered Dec 24 '18 at 13:40
greggreg
8,1651822
$endgroup$
add a comment |
$begingroup$
Define a new matrix $$Y=WH-X$$
Write the function in terms of this new variable
$$f = |Y|^2_F = Y:Y$$
where a colon denotes the trace/Frobenius product, i.e. $,,A:B={rm tr}(A^TB)$
Find its differential and gradients.
$$eqalign{
df &= 2Y:dY cr
&= 2Y:(dW,H+W,dH) cr
&= 2YH^T:dW + 2W^TY:dH cr
&= 2(WH-X)H^T:dW + 2W^T(WH-X):dH cr
frac{partial f}{partial W} &= 2(WH-X)H^T,quad
frac{partial f}{partial H} = 2W^T(WH-X) cr
}$$
Since the gradients are themselves matrices, the hessians will be 4th order tensors which cannot be represented in matrix notation.
One way to approach the hessian is to use vectorization which flattens matrices into vectors.
For example,
$$eqalign{
G &= frac{partial f}{partial W} = 2WHH^T - 2XH^T cr
dG &= 2,dW,HH^T cr
{rm vec}(dG) &= 2,{rm vec}(dW,HH^T) cr
dg &= 2,(HH^Totimes I),dw cr
nabla_{ww}f &= 2,(HH^Totimes I) cr
}$$
Working through the other hessians
$$eqalign{
nabla_{hh}f &= 2,(Iotimes W^TW) cr
nabla_{wh}f &= 2(Hotimes W) + 2(Iotimes Y)K cr
nabla_{hw}f &= 2(H^Totimes W^T) + 2(Y^Totimes I)K cr
}$$
where $K$ is the Commutation Matrix which can be used to vectorize the transpose of a matrix
$${rm vec}(X^T) = K,{rm vec}(X)$$
answered Dec 24 '18 at 13:40
greggreg
8,1651822
$endgroup$
Define a new matrix $$Y=WH-X$$
Write the function in terms of this new variable
$$f = |Y|^2_F = Y:Y$$
where a colon denotes the trace/Frobenius product, i.e. $,,A:B={rm tr}(A^TB)$
Find its differential and gradients.
$$eqalign{
df &= 2Y:dY cr
&= 2Y:(dW,H+W,dH) cr
&= 2YH^T:dW + 2W^TY:dH cr
&= 2(WH-X)H^T:dW + 2W^T(WH-X):dH cr
frac{partial f}{partial W} &= 2(WH-X)H^T,quad
frac{partial f}{partial H} = 2W^T(WH-X) cr
}$$
Since the gradients are themselves matrices, the hessians will be 4th order tensors which cannot be represented in matrix notation.
One way to approach the hessian is to use vectorization which flattens matrices into vectors.
For example,
$$eqalign{
G &= frac{partial f}{partial W} = 2WHH^T - 2XH^T cr
dG &= 2,dW,HH^T cr
{rm vec}(dG) &= 2,{rm vec}(dW,HH^T) cr
dg &= 2,(HH^Totimes I),dw cr
nabla_{ww}f &= 2,(HH^Totimes I) cr
}$$
Working through the other hessians
$$eqalign{
nabla_{hh}f &= 2,(Iotimes W^TW) cr
nabla_{wh}f &= 2(Hotimes W) + 2(Iotimes Y)K cr
nabla_{hw}f &= 2(H^Totimes W^T) + 2(Y^Totimes I)K cr
}$$
where $K$ is the Commutation Matrix which can be used to vectorize the transpose of a matrix
$${rm vec}(X^T) = K,{rm vec}(X)$$
answered Dec 24 '18 at 13:40
greggreg
8,1651822
edited Dec 24 '18 at 14:19
answered Dec 24 '18 at 13:40
greggreg
8,1651822
answered Dec 24 '18 at 13:40
greggreg
8,1651822
answered Dec 24 '18 at 13:40
greggreg
8,1651822
8,1651822
add a comment |
add a comment |
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$begingroup$
What's your function $f$?
$endgroup$
– user550103
Dec 24 '18 at 11:50
$begingroup$
@user550103 I've added the function.
$endgroup$
– Maganna Dev
Dec 24 '18 at 12:07
$begingroup$
Your gradients look correct.
$endgroup$
– user550103
Dec 24 '18 at 12:19
$begingroup$
@user550103 What troubles me are the dimensions of each partial derivative. if $$ frac{partial f}{partial W} in R^{m times r}$$ and $$ frac{partial f}{partial H} in R^{r times n}$$, does the gradient i want to define exist? Does the Hessian exist?
$endgroup$
– Maganna Dev
Dec 24 '18 at 12:22
$begingroup$
If I am understanding your problem, then the gradients are matrices. Yes, they are matrices, which is as expected...
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– user550103
Dec 24 '18 at 12:25