Is it possible to define a Hessian Matrix for a Matrix-valued function?












0












$begingroup$


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







share|cite|improve this question











$endgroup$












  • $begingroup$
    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
















0












$begingroup$


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







share|cite|improve this question











$endgroup$












  • $begingroup$
    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














0












0








0





$begingroup$


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







share|cite|improve this question











$endgroup$




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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 24 '18 at 12:02







Maganna Dev

















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


















  • $begingroup$
    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










1 Answer
1






active

oldest

votes


















1












$begingroup$

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
$${rm vec}(X^T) = K,{rm vec}(X)$$






share|cite|improve this answer











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    active

    oldest

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    1












    $begingroup$

    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
    $${rm vec}(X^T) = K,{rm vec}(X)$$






    share|cite|improve this answer











    $endgroup$


















      1












      $begingroup$

      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
      $${rm vec}(X^T) = K,{rm vec}(X)$$






      share|cite|improve this answer











      $endgroup$
















        1












        1








        1





        $begingroup$

        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
        $${rm vec}(X^T) = K,{rm vec}(X)$$






        share|cite|improve this answer











        $endgroup$



        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
        $${rm vec}(X^T) = K,{rm vec}(X)$$







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 24 '18 at 14:19

























        answered Dec 24 '18 at 13:40









        greggreg

        8,1651822




        8,1651822






























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            Cabo Verde

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              Nota: Cabo-verdiano redireciona para este artigo. Para a língua falada em Cabo Verde, veja crioulo cabo-verdiano. Para outros significados, veja Cabo Verde (desambiguação). Coordenadas: 14° 55' N 23° 31' O Cabo Verde República de Cabo Verde Bandeira Brasão de Armas Lema: Unidade, Trabalho, Progresso Hino nacional: Cântico da Liberdade Gentílico: cabo-verdiano [ 1 ] Capital Praia 14° 55' N 23° 31' O Cidade mais populosa Praia Língua oficial Português Governo República unitária semipresidencialista [ 2 ]  - Presidente Jorge Carlos Fonseca  - Primeiro-ministro Ulisses Correia e Silva História    - Descoberta 1460   - Independência 00 • de Portugal 5 de Julho de 1975   - Multipartidarismo 13 de Janeiro de 1990  Área    - Total 4 033 km² (146.º)  - Água (%) 0,08 População    - Estimativa para 2017 560 899 [ 3 ] hab. (173.º) ...
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            Gyllenstierna

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