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### Computing the expectation of a quadratic matrix form involving Bernoulli and Gaussian distributed matrices

$\newcommand{\ka}{\kappa}\newcommand{\si}{\sigma}$We have $\ka^2=ZHZ^TZHZ^T$ and hence, for $i$ and $j$ in $[n]:=\{1,\dots,n\}$, \begin{equation} (E\ka^2)_{ij}=\sum_{k,l,m,s,r}EZ_{ik}H_{kl}Z_{ml}...
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### Tree decomposition of graphs with low height

One way to address this question is to think of a tree decomposition as an unrooted tree. Then, it is easy to see that its diameter is related to the depth by a factor of 2. Searching for diameter, ...

### Computing the expectation of a quadratic matrix form involving Bernoulli and Gaussian distributed matrices

If $H$ has iid $N(0,1)$ entries, write $\kappa=\langle Z^TZ, H\rangle$ using the usual Frobenius inner product $\langle A,B\rangle = trace[A^TB]$. Conditionally on $Z$, $\kappa\sim N(0,\|Z^TZ\|_F^2)$ ...
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### On algebraic topology of coset complexes without geometry

Regarding your question (ii): So far as I know, proving the Nerve Lemma requires either a homotopy type argument, or a spectral sequence argument. You can find the spectral sequence argument in Ken ...
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