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Suppose we have a typical logdet function $\mathcal{L}$ $$ \mathcal{L} = \log\vert \mathbf{I} + \mathbf{A}\mathbf{S} \vert + \mathbf{q}^T(\mathbf{A}^{-1} + \mathbf{S})^{-1} \mathbf{q}, $$ where $\mathbf{S}$ is a Symmetric Positive Semi-Definite Matrix, $\mathbf{q}$ is a column vector, $\mathbf{I}$ is an identity matrix. The local minima of $\mathcal{L}$ w.r.t $\mathbf{A}$ could be found by solving $\mathbf{A}$ ($\frac{\partial \mathcal{L}}{\partial \mathbf{A}} = \mathbf{0}$), we get the closed form solution of $\mathbf{A}$ $$ \mathbf{A} = \mathbf{S}^{-1}(\mathbf{S} - \mathbf{q}\mathbf{q}^T)\mathbf{S}^{-1} $$

The problem is, what if I want to constrain the descent path of the logdet function by imposing extra structures on $\mathbf{A}$ ? For example,

1. $\mathbf{A}$ is p.s.d and have a simple structure, which constrains that at each step, $\mathbf{A}$ should be: $\mathbf{A}=r\mathbf{I}$.

2. or we constrain that $\mathbf{A}$ is p.s.d and have a band-p Symmetric Toeplitz structure, i.e., \begin{equation} \mathbf{A} = r \begin{pmatrix} 1 & p & 0 & 0 \\\ p & 1 & p & 0 \\\ 0 & p & 1 & p \\\ 0 & 0 & p & 1 \end{pmatrix}. \end{equation}

How could I analyze the saddle point of $\mathcal{L}$ with these constrains ? Will these constrains affect achieving the local minima of $\mathcal{L}$ and provide a more robust result that more likely to find the true minimal that has this type of structure ?

Suppose we have a typical logdet function $\mathcal{L}$ $$ \mathcal{L} = \log\vert \mathbf{I} + \mathbf{A}\mathbf{S} \vert - \mathbf{q}^T(\mathbf{A}^{-1} + \mathbf{S})^{-1} \mathbf{q}, $$ where $\mathbf{S}$ is a Symmetric Positive Semi-Definite Matrix, $\mathbf{q}$ is a column vector, $\mathbf{I}$ is an identity matrix. The local minima of $\mathcal{L}$ w.r.t $\mathbf{A}$ could be found by solving $\mathbf{A}$ ($\frac{\partial \mathcal{L}}{\partial \mathbf{A}} = \mathbf{0}$), we get the closed form solution of $\mathbf{A}$ $$ \mathbf{A} = \mathbf{S}^{-1}(\mathbf{S} - \mathbf{q}\mathbf{q}^T)\mathbf{S}^{-1} $$

The problem is, what if I want to constrain the descent path of the logdet function by imposing extra structures on $\mathbf{A}$ ? For example,

1. $\mathbf{A}$ is p.s.d and have a simple structure, which constrains that at each step, $\mathbf{A}$ should be: $\mathbf{A}=r\mathbf{I}$.

2. or we constrain that $\mathbf{A}$ is p.s.d and have a band-p Symmetric Toeplitz structure, i.e., \begin{equation} \mathbf{A} = r \begin{pmatrix} 1 & p & 0 & 0 \\\ p & 1 & p & 0 \\\ 0 & p & 1 & p \\\ 0 & 0 & p & 1 \end{pmatrix}. \end{equation}

How could I analyze the saddle point of $\mathcal{L}$ with these constrains ? Will these constrains affect achieving the local minima of $\mathcal{L}$ and provide a more robust result that more likely to find the true minimal that has this type of structure ?

Suppose we have a typical logdet function $\mathcal{L}$ $$ \mathcal{L} = \log\vert \mathbf{I} + \mathbf{A}\mathbf{S} \vert + \mathbf{q}^T(\mathbf{A}^{-1} + \mathbf{S})^{-1} \mathbf{q}, $$ where $\mathbf{S}$ is a Symmetric Positive Semi-Definite Matrix, $\mathbf{q}$ is a column vector, $\mathbf{I}$ is an identity matrix. The local minima of $\mathcal{L}$ w.r.t $\mathbf{A}$ could be found by solving $\mathbf{A}$ ($\frac{\partial \mathcal{L}}{\partial \mathbf{A}} = \mathbf{0}$), we get the closed form solution of $\mathbf{A}$ $$ \mathbf{A} = \mathbf{S}^{-1}(\mathbf{S} - \mathbf{q}\mathbf{q}^T)\mathbf{S}^{-1} $$

The problem is, what if I want to constrain the descent path of the logdet function by imposing extra structures on $\mathbf{A}$ ? For example,

1. $\mathbf{A}$ is p.s.d and have a simple structure constrain that each step, $\mathbf{A}$ should have the following structure: $\mathbf{A}=r\mathbf{I}$.

2. $\mathbf{A}$ is p.s.d and have a band-p Symmetric Toeplitz structure, i.e.,

\begin{equation} \mathbf{A} = r \begin{pmatrix} 1 & p & 0 & 0\\ p & 1 & p & 0\\ 0 & p & 1 & p\\ 0 & 0 & p & 1 \end{pmatrix} \end{equation}

How could I analyze the saddle point of $\mathcal{L}$ with these constrains ? Will these constrains affect some of the local minima of $\mathcal{L}$ and provide a more robust result that more likely to find the true minimal that has this type of structure ?

Suppose we have a typical logdet function $\mathcal{L}$ $$ \mathcal{L} = \log\vert \mathbf{I} + \mathbf{A}\mathbf{S} \vert + \mathbf{q}^T(\mathbf{A}^{-1} + \mathbf{S})^{-1} \mathbf{q}, $$ where $\mathbf{S}$ is a Symmetric Positive Semi-Definite Matrix, $\mathbf{q}$ is a column vector, $\mathbf{I}$ is an identity matrix. The local minima of $\mathcal{L}$ w.r.t $\mathbf{A}$ could be found by solving $\mathbf{A}$ ($\frac{\partial \mathcal{L}}{\partial \mathbf{A}} = \mathbf{0}$), we get the closed form solution of $\mathbf{A}$ $$ \mathbf{A} = \mathbf{S}^{-1}(\mathbf{S} - \mathbf{q}\mathbf{q}^T)\mathbf{S}^{-1} $$

The problem is, what if I want to constrain the descent path of the logdet function by imposing extra structures on $\mathbf{A}$ ? For example,

1. $\mathbf{A}$ is p.s.d and have a simple structure, which constrains that at each step, $\mathbf{A}$ should be: $\mathbf{A}=r\mathbf{I}$.

2. or we constrain that $\mathbf{A}$ is p.s.d and have a band-p Symmetric Toeplitz structure, i.e., \begin{equation} \mathbf{A} = r \begin{pmatrix} 1 & p & 0 & 0 \\\ p & 1 & p & 0 \\\ 0 & p & 1 & p \\\ 0 & 0 & p & 1 \end{pmatrix}. \end{equation}

How could I analyze the saddle point of $\mathcal{L}$ with these constrains ? Will these constrains affect achieving the local minima of $\mathcal{L}$ and provide a more robust result that more likely to find the true minimal that has this type of structure ?