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| -rw-r--r-- | paper/sections/results.tex | 8 |
1 files changed, 4 insertions, 4 deletions
diff --git a/paper/sections/results.tex b/paper/sections/results.tex index 25f540a..86f4c32 100644 --- a/paper/sections/results.tex +++ b/paper/sections/results.tex @@ -296,16 +296,16 @@ In our case we have: \bigg[x_i^{t+1}\frac{f''f-f'^2}{f^2}(\inprod{\theta^*}{x^t})\\ -(1-x_i^{t+1})\frac{f''(1-f) + f'^2}{(1-f)^2}(\inprod{\theta^*}{x^t})\bigg] \end{multline*} -It is interesting to observe that the Hessian of $\mathcal{L}$ can be seen as +Observe that the Hessian of $\mathcal{L}$ can be seen as a re-weighted Gram matrix of the observations. In other words, the restricted eigenvalue condition expresses that the observed set of active nodes are not too collinear with each other. -In the specific case of ``logistic cascades'' (where $f$ is the logistic +In the specific case of ``logistic cascades'' (when $f$ is the logistic function), the Hessian simplifies to $\nabla^2\mathcal{L}(\theta^*) = \frac{1}{|\mathcal{T}|}XX^T$ where $X$ is the design matrix $[x^1 \ldots -x^\mathcal{|T|}]$. The restricted eigenvalue condition is equivalent in this -case to the assumption made in the Lasso analysis of \cite{bickel:2009}. +x^\mathcal{|T|}]$. The (RE) condition is then equivalent +to the assumption made in the Lasso analysis of \cite{bickel:2009}. \paragraph{(RE) with high probability} |