All Questions
7
questions
0
votes
1
answer
75
views
Lower bounding $v^TA^{\dagger}v$ where A is a sum of rank-1 matrices
This is related to a previous question I asked (Upper bounding $v^TAv$ where $A$ is the inverse of a sum of rank-$1$ matrices and $v$ is a vector).
Let $(x_i)_{1 \leq i \leq n}$ be vectors of $\...
- 192
3
votes
3
answers
363
views
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 ...
- 865
1
vote
1
answer
51
views
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 $\...
- 155
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
2
votes
1
answer
227
views
Show that $\|A\|_{\infty} \leq \sqrt n \|A\|_2$
Question:
Let $A \in \Bbb R^{m\times n}$. Show that $$\|A\|_{\infty} \leq \sqrt n \|A\|_2$$
Attempt:
First, I tried invoking the SVD (Singular Value Decomposition) of $A$:
$$\|A\|_\infty = \|UDV\|...
- 4,945
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
0
votes
1
answer
103
views
Sharp Lower Bound for Entrywise 1-norm of a Real Semi-Orthogonal Matrix
Let $A$ be a real-valued $m\times n$ ($m>n>1$) matrix such that $A^TA=I$, what is the sharp lower bound for $\|A\|_1=\|\operatorname{vec}(A)\|_1=\sum|A_{i,j}|$?
Since one can show the ...
- 823