All Questions
Tagged with matrix-decomposition optimization
77
questions
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30
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Optimization of eigenvalue of matrix with discrete variables
Suppose we have a matrix $H$ like this, where $H_{ii}$ is a discrete variable (for example out of $[1,2,3]$) and $H_{ij}$ is a value that depends on the two adjacent values ($H_{ii}$ and $H_{jj}$). $H$...
- 31
1
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0
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27
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A low-rank approximation problem with rank constraints
I am seeking a solution or some ideas to address the following problem:
$$
\begin{aligned}
&\text { minimize }_{\widehat{A}, \widehat{B}} \quad\|A-\widehat{A}\|_2 + \|B-\widehat{B}\|_2 \\ &\...
- 11
0
votes
0
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25
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Best method for sequential small size Hermitian smallest eigenpair problem
I am working on a perhaps rather strange problem. To find the smallest eigenvalue and its eigenvector, for a large number (a few billions) of small (20 * 20 to 200 * 200) Hermitian matrices. These ...
- 31
2
votes
1
answer
71
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Minimize $||A-AWW^TA^T||_F$ w.r.t. $W$
Given $n \in \mathbb{N}$ and $A \in \{0,1\}^{n \times n}$, we aim to find
$$\arg \min_{W \in \mathbb{R}^{n \times n}} f(W) = ||A-AWW^TA^T||_F,$$
where $||\cdot||_F$ represents the Frobenius norm with
$...
- 2,150
1
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0
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39
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Eigenvalue-like problem for two matrices with coupled blocks
For given two real-symmetric matrices $A$ and $B$, both are of size $M \times M$, I'm trying to find two matrices $C_1$ (size $M\times N_1$) and $C_2$ (size $M\times N_2$) such that
$$
AC_1 = C_1\...
- 11
3
votes
0
answers
48
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When do symmetrical matrices commute?
For context I was taking an optimization course and at one point we used the restriction by line to prove the concavity of $\log det(X)$. Let $g(t) = \log det(X + tV)$ with $V \in S_{n}$ and $t \in \...
- 31
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6
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Which divergence or measure is more suitable for graph clustering application using symmetric NMF
Symmetric NMF is a well know tool used for graph clustering applications.
Given a similarity matrix $X \in \mathcal{R}^{n\times n}$, symmetric NMF seek to factorize it into o production of the form $...
- 562
1
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0
answers
48
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Minimizing a log-determinant divergence based function
For a problem of the form $X\approx S$, where $X$ is an $n\times n$ symmetric but not always PD matrix, and $S$ its approximate, I am trying to minimize the log-determinant divergence between $X$ and $...
- 562
1
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0
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100
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Low rank approximation with MSE approach
Suppose that we have a matrix $\mathbf{A}\in\mathbb{R}^{n\times p}$. If we want to find an approximation of $\mathbf{A}$ with maximum rank $k$, which is denoted as $\tilde{\mathbf{A}}$, is it possible ...
- 11
0
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1
answer
65
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Maximize trace(UV) subject to $U$ and $V$ being Orthogonal
I am facing an optimization problem where I want to maximize $\operatorname{trace} (U V)$ with respect to $U$ and $V$ where $U$ and $V$ are orthogonal. How can I find the optimal $U$ and $V$?
This is ...
- 43
1
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0
answers
60
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Best inverse / minimization solution of ill conditioned matrix and underdetermined system
I have an ill-conditioned matrix G representing Green's function estimates of a strain model for a rock mass that has been instrumented with optical fiber measuring strain. My problem is seemingly the ...
- 11
5
votes
3
answers
140
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Intuition of $D\leftarrow XC^{T}\text{diag}(C1_n)^{-1}$ update rule in matrix factorization
I am reading this paper where they use Matrix Factorization over Attention mechanism in their Hamburger model. In section 2.2.2 they say,
Vector Quantization (VQ) (Gray & Neuhoff, 1998), a ...
- 875
0
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0
answers
21
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How to solve the system of equation $y=\operatorname{Diag}(Bz) x $ with respect $B$ and z
Assume I have the following system of equations:
$$y = \operatorname{Diag}(h) x$$
where $x,y, h \in {\bf R}^N$ and suppose that I know them.
I also know that $h$ can be factorized as: $$h=Bz$$
where $...
- 888
3
votes
1
answer
105
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Nearest semi-orthogonal matrix with fixed row
I want to find the nearest (in the Frobenius sense) $n\times d$ (where $n > d$) matrix $X$ to a given matrix $A$ of the same size such that the $d$ columns of $X$ are orthonormal .
Since $X$ would ...
- 65
0
votes
1
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76
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Maximization on trace of quadratic and linear terms under orthonormal constraints
I have the following optimization problem
$$ \max_{R: RR^{T}=I} \mbox{Tr} \left( M \left( R A R^{T} - K R^{T} \right) \right) $$
where:
$A$ is a rank-one square matrix (assume the first row that are ...
- 31