All Questions
Tagged with upper-lower-bounds normed-spaces
60
questions
0
votes
1
answer
90
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Randomness in the norm of sum of vectors
Let $x_1,x_2,\ldots,x_n \in \mathbb{R}^d$ be vectors and $a_1, a_2, \ldots, a_n \in \mathbb{R}$ be random iid scalars distributed by $N~(0,\sigma^2).$
Then, is it possible to lower bound the ...
-1
votes
1
answer
55
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$\ell_2$ vs $\ell_{\infty}$ induced norm of a square matrix
I'm wondering if the result of this post $\| A \|_{L^2} \le \| A \|_{\infty}$ for symmetric matrices $A$ applies to square matrices that are not symmetric. Of course, for an asymmetrical square ...
1
vote
0
answers
76
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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 ...
5
votes
0
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84
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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:
$$
|u(x) - u_h (x)| \leq C h^{m+1} |...
1
vote
1
answer
36
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Wrong defined/boundless $g \rightarrow\sum_{n = 1}^\infty\frac{g(\frac1n)}{2^n}$
Let's consider:
$$f\colon (C[0,1], \Vert\cdot \Vert_1) \ni g\rightarrow \sum_{n=1}^\infty\frac{g(\frac1n)}{2^n}$$
I'm trying to check if this object is well defined.
where $\Vert f \Vert_1 = \int_0^1|...
3
votes
3
answers
363
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Why $\|(A + cI)^{-1}x\|\leq \frac{\|x\|}{\lambda_{\min}(A)}$
Given $A\succ 0$ (positive-definite) and $c>0$, I am trying to show
$$\|(A + cI)^{-1}x\|\leq \frac{\|x\|}{\lambda_{\min}(A)} \tag{1}$$
using information like this but without success so far. Could ...
1
vote
1
answer
51
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Inconclusive inequality result: show $(id - f)^{-1}$ is bounded
I have the following statement to prove:
Let $V$ be a normed vector space (so not necessarily complete or finite dimensional) over $\mathbb{R}$. Take any norm $\Vert \cdot \Vert_V$ on $V$ and let $\...
1
vote
2
answers
3k
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Bound on l1 norm given bound on l2 norm
While doing self-study exercices, I found the following bound without explanation and was not able to see why it is always the case. I found some examples, it seems legitimate but I am unable to ...
1
vote
1
answer
1k
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Is the 2-norm of a matrix bounded by the maximum of its 1-norm and Infinity-norm?
I am implementing the algorithm in "Approximating the Logarithm of a Matrix to Specified Accuracy" by Sheung Hun Cheng, Nicholas J. Higham, Charles S. Kenny, Alan J. Laub, 2001.
In this ...
1
vote
1
answer
130
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Bounding $\|A^n-B^n\|_F$ by $\|A-B\|_F$
Given
$$ \epsilon = \|A-B\|_F, $$
it is clear that
$$ \epsilon^n > \|(A-B)^n\|_F, $$
which follows from submultiplicativity.
I wonder if something related can be said about $\|A^n-B^n\|_F$?
0
votes
1
answer
53
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Bound for $\Vert\sum_iA_i^\dagger A_i\Vert_\infty$ given that $\Vert\sum_i A_i\Vert_\infty$ is small
Let $A_i$ be matrices such that
$$\left\Vert \sum_i A_i \right\Vert_\infty \leq \varepsilon,$$
where $\Vert\cdot\Vert_\infty$ is the operator norm and is equal to the largest singular value of its ...
3
votes
2
answers
400
views
How to find upper and lower bound
Let $\Sigma \in S_{++}^n$ be a symmteric positive definte matrix with all diagonal entries one. Let $U \in R^{n \times k_1}$, $W \in R^{n \times k_2}$, $\Lambda \in R^{k_1 \times k_1}$ and $T \in R^{...
2
votes
1
answer
51
views
alternative asymptotic bounds
I have an $n$ by 1 vector of weights $w$, and an $n$ by $k$ matrix, $\Gamma$. I have that $w'w$ is $\mathcal{O}(1)$, $\frac{\Gamma'\Gamma}{n}=\mathcal{O}(1)$ and $\frac{\Gamma\Gamma'}{n}=\mathcal{O}(1)...
0
votes
1
answer
28
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Deducing properties of the $\ell_3$ norm from the $\ell_1$ and $\ell_2$ norms
Suppose we have a function $f: [0,1] \rightarrow \mathbb{R}$, $f(x) \geq 0$ normalised so that $\|f\|_1 = 1$, where
$$
\| f\|_p = \left( \int_0^1 f(x)^p d x \right)^{1/p}.
$$
Moreover, we know that $\|...
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,\...