All Questions
Tagged with upper-lower-bounds normed-spaces
18
questions with no upvoted or accepted answers
5
votes
0
answers
84
views
Gettings bounds for seminorms from bound of absolute value
On a compact domain $\Omega \subseteq \mathbb{R}^d$, we have a function $u(x) \in C^{\infty}(\Omega)$ with an approximation $u_h(x)$ with the following properties:
$$
|u(x) - u_h (x)| \leq C h^{m+1} |...
- 115
4
votes
0
answers
379
views
Need to improve upper bound for $\| (uv^T + B)^{-1} \|$ (Sherman-Morrison formula)
I have a matrix $A \in \mathbb{C}^{n \times n}$ in the form $A = uv^T + B$, where $u,v \in \mathbb{C}^n$ and $B \in \mathbb{C}^{n \times n}$. I know $A$ is invertible and want to find an upper bound ...
- 6,574
3
votes
0
answers
244
views
Upper bound the maximum column sum of a particular stochastic matrix
Let $(x_i)_{i=1}^N$ be a set of vectors in $\mathbb{R}^D$. Define the matrix $W \in \mathbb{R}^{N \times N}$ as:
$W_{ij} = \frac{\exp(-||x_i-x_j||^2)}{\sum_k \exp(-||x_i-x_k||^2)}$
i.e. row $i$ of $...
- 51
2
votes
0
answers
83
views
Upper bound in bayesian regression setting
Let $y_i = x_i^\top \beta + \epsilon_i$, $i=1,\ldots,n$; where $\epsilon_i$ are i.i.d. following a distribution with mean zero and unit variance, i.e., $\epsilon_i \sim P_{\epsilon_i}(0,1)$, $i=1,\...
- 81
1
vote
0
answers
54
views
Greatest lower bound and smallest upper bound of Frobenius norm of product of matrices
Let $A,B \in M_p(R)$ be symmetric matrices, $A$ is given and non-singular, $\|A^{-1}\|.\|B\| < 1$ where $\|.\|$ is Frobenius norm.
I would like to find the lower bound and upper bound of $\|A^{-1}....
- 11
1
vote
0
answers
345
views
Bound on L2-Norm of Probability Distributions?
I am working on a problem where I have continuous probability distributions $p$ over a bounded domain $D$, i.e., $\forall x$ $p(x) \geq 0$ and $\|p\|_1 = \int_D p(x) dx = 1$. However, I also want $p$ ...
- 127
1
vote
0
answers
485
views
Upper bound on the norm of an inverse of a positive definite matrix
Given two real-valued symmetric positive-definite matrices $A$ and $B$. Assume that $A\succeq I$. Let $\Delta = B-A$. We are interested in a bound on $\Vert B^{-1}-A^{-1}\Vert$ in terms of $\Vert \...
1
vote
0
answers
95
views
If the norm of the difference between two unit vector is lower bounded by a positive constant, does it mean that the inner product is upper bounded?
Let $x,y$ be two vectors with $\lVert x \rVert = \lVert y \rVert =1$ and $\lVert x-y \rVert \geq \delta$, where $\delta \gt 0$. Is it possible to show that, $1-(x^Ty)^2 \geq \delta^2$?
My Approach:
$$\...
1
vote
0
answers
76
views
Probability of ell-1 norms of vertices of the rotated Hamming cube
Let $O$ be a $d$-dimensional rotation matrix (i.e., it has real entries and $OO^T = O^TO = I$). Let $\mathbf{x}$ be a uniformly random bitstring of length $d$, i.e., $\mathbf{x} \sim U(\{0,1\}^d)$. In ...
- 658
1
vote
0
answers
78
views
Trace and 2-norm of linear combination of outer products
Suppose that $c_i \in \mathbb{R}-\{0\}, B_i \in \mathbb{R}^{k \times m}, \alpha \in \mathbb{R}^k$ with $\|\alpha\|_2 = 1$. Consider the following linear combination of outer products:
$$M = \sum_{i=1}...
- 1,180
1
vote
0
answers
60
views
2-norm of product of multiple matrices
Suppose that $A \in \mathbb{R}^{m,n}$ and $B \in \mathbb{R}^{n,m}$ and $A$ has a bounded 2-norm. We know that $AB$ is a positive semi-definite matrix with $\|AB\|_2 \leq 1$. Further, assume that $C \...
- 1,180
1
vote
0
answers
99
views
Norm equivalence constants
Take a polynomial $g\in\mathbb{R}[\mathbf{x}]$, in $n$ variables and having some degree $d$, with $g(\mathbf{x})\geq 0$. We define the $p$-norms of $g$ as
$$
\vert \vert g \vert \vert _{p} = \left( \...
- 177
0
votes
0
answers
23
views
The lower bound of Frobenius norm of matrices product.
Let $A,B \in M_p(R)$ be symmetric matrices, $A$ is given and non-singular, where $\|.\|$ is Frobenius norm.
I would like to find the lower bound of $\|A^{-1}.B.(A+B)^{-1}\|$:
I have the result:
$$\|A^{...
- 11
0
votes
0
answers
23
views
How to find an upper bound for a norm with power less than 1?
I need to find an upper bound for $$\|x-y\|_2^\nu$$ in which $\nu \in [0,1]$. I was wondering if $\|x-y\|_2^2$ can be considered as an upper bound. or $\|x\|+\|y\|$?
0
votes
0
answers
381
views
upper bound for quadratic form in terms of vector norm and eigenvalues
I have a quadratic form. if Q, P and M are positive and symmetric matrices.
$$(-x^T Q x - 2 x^T Q e - e^T Q e) + (y^T M y + 2 x^T P y + 2 e^T P y )$$
how can I get an upper bound for this quadratic ...
- 1