Tagged Questions
0
votes
1answer
20 views
Why does the non-negative matrix factorization problem non-convex?
Supposing $\mathbf{X}\in\mathbb{R}_+^{m\times n}$, $\mathbf{Y}\in\mathbb{R}_+^{m\times r}$, $\mathbf{W}\in\mathbb{R}_+^{r\times n}$, the non-negative matrix factorization problem is defined as:
...
0
votes
1answer
29 views
Positive semidefinite Matrix examples query
This might be really dumb question but I've just started dealing with such matrices. I would like to know why $$\begin{bmatrix} 0 & 1 \\ 1 & x \end{bmatrix}$$ cannot be a positive semidefinite ...
0
votes
1answer
47 views
Rewriting a quadratic Matrix equation as a quadratic vector equation
Consider the set of $N \times N$ matrices $\{W_i\}_{i=1}^{i=L}$, set of $N \times 1$vectors $\{g_i\}_{i=1}^{i=L}$ and $\{h_i\}_{i=1}^{i=L}$. Now consider the following sum
\begin{align}
...
0
votes
1answer
59 views
relation between size of matrix and condition number
I have a matrix A of size NxM. Is there any relationship between size of a matrix A with the condition number ? I am computing the pseudo inverse (pinv in matlab ) ...
2
votes
1answer
63 views
Proof for certain matrix results?
There are certain results of matrices that Stephen Boyd uses often in his book on Convex optimization. Can someone provide me proof for the results I have enumerated below:
If $B \in S^n$ and $A \in ...
0
votes
2answers
66 views
Are any matrices positive semidefinite, non-negative, and not diagonally dominated?
If so, I'd appreciate any examples. Thanks.
3
votes
1answer
64 views
decomposing PSD block matrix into two PSD block matrices
Given $Q = \left( \begin{array}{ccc}
A + B & C \\
C^T & D\end{array} \right) $, where we know that $Q, A, B, D$ are all positive semi-definite, square, but not necessarily equally sized ...
2
votes
0answers
144 views
Optimization problem about large matrices
I'd like to solve the following optimization problem:
Find non-negative scalar $a$, $b$, $c$ to minimize
$\| (D-(aA+bB+cC+D^{-1})^{-1})y\|^2+2\operatorname{trace}((aA+bB+cC+D^{-1})^{-1})$
where ...
1
vote
2answers
124 views
Linear optimization problem: Minimizing a linear function over an affine set.
The problem is as follows:
Give an explicit solution of the linear optimization problem below.
$$
\text{minimize}\ c^Tx \\
\text{subject to}\ Ax\ =\ b
$$
No other information is given.
My ...
3
votes
2answers
53 views
About the convexity of Ky Fan's norm
As we know, the Ky Fan norm is convex, and so is the Ky Fan k-norm. My question is, does this imply that the difference between them is a non-convex function, since it results from "difference between ...
1
vote
1answer
260 views
Armijo's rule line search
I have read a paper (http://www.seas.upenn.edu/~taskar/pubs/aistats09.pdf) which describes a way to solve an optimization problem involving Armijo's rule, cf. p363 eq 13.
The variable is $\beta$ ...
0
votes
2answers
62 views
Any idea which matrix theorem this is?
I came across a theorem that boyd uses to convert the simplex to the form of a polyhedra. I don't know anything about this theorem. Theorem states:
If $B$ has rank $k$, then we can find two matrices ...
2
votes
2answers
60 views
Geometric difference between $x^TAx$ and $x^TAx + b^Tx + c$
What is the difference between $x^TAx$ and $x^TAx + b^Tx + c$ geometrically? Some analogous examples from quadriatic equations would be great.
1
vote
1answer
49 views
On the convexity of the element-wise norm 1 of a pseudoinverse
Let us define $\|A\|_1$ the element wise norm 1 of a matrix $A \in \mathbb{R}^{n \times m}$ as
$$
\|A\|_1= \sum_{i,j} |A_{i,j}|.
$$
Obviously, this function is convex over $\mathbb{R}^{n \times m}$. ...
0
votes
1answer
43 views
matrix completion by rank minimization
In matrix completion, the starting point is often stated as: the optimization problem for matrix completion:
min(X): (1/2) ||X-M||^2
s.t. rank(X)<= r
Where X is the reconstructed matrix and M ...
0
votes
1answer
62 views
Matrix computations problem: rank, pseudo inverse,…
Suppose we are given two arbitrary $m \times n$ matrices, $A$, $B$, where we know $B$ has full column rank. Let $m>>n$.
Can we always find a square $m \times m $ matrix $X$, such that $A=XB$? I ...
1
vote
1answer
87 views
Convex Combination of Hermitian Matrices
Assume all the matrices I discuss about are $N \times N$. Consider any two hermitian matrices $A_1$ and $A_2$ which are indefinite. The question is, In general, for any $A_1$ and $A_2$, does there ...
2
votes
1answer
86 views
Positive values for a set of quadratic forms of Hermitian Matrices. (To find a set of vectors in which a hermitian matrix is positive definite)
Assume all matrices I discuss about are $N \times N$ and the vectors conform with dimensions. Consider the following set of Quadratic inequalities where all the matrices $A_i$ are hermitian.
...
0
votes
0answers
20 views
Does post-processing of NMF output preserve the objective function value inequality?
Assume $F,B$ are fixed and my function is $O(C) = \frac{1}{2} ||F-BC^T||^2_F$.
I know that $O(C_1)<O(C_2)$. Now, if i do same post processing on columns $C_1$ and $C_2$, for example, entries in the ...
1
vote
2answers
285 views
Proof of Convexity?
Given a positive semidefinite matrix $A$, is $\operatorname{Tr}X^TAX$ a convex function in $X$? Am looking for a proof of convexity or non-convexity, whichever is true.
0
votes
1answer
105 views
Correctly adding constraints to Ax=b
I have a function of the form
$$
E(\mathbf{x})=E_1(\mathbf{x})+E_2(\mathbf{x})
=\sum_i\|\ldots\|^2+\sum_j\|\ldots\|^2
$$
and want to solve the optimization problem
$$
...
2
votes
1answer
401 views
Maximum likelihood covariance estimation of Gaussian
I was reading these notes on matrix calculus
http://research.microsoft.com/en-us/um/people/minka/papers/matrix/minka-matrix.pdf
and I could not figure out how to go from equation (30) to (31).
Any ...
