Saddlepoint Approximation for Linear Function of Uniformly-Distributed Data

We follow the method of Section 2.3.2. Let $ {\bf x}$ be a set of $ N$ independent uniform-distributed RV in $ [0,1]$ Let $ {\bf A}$ be an $ N$-by-$ D$ full-rank matrix and let $ {\bf z}$ be the $ D\times 1$ feature vector

$\displaystyle {\bf z}={\bf A}^\prime {\bf x}.$

When $ {\bf x}\in{\cal R}^N$ is uniformly distributed, the CGF is given by

$\displaystyle c_z($$ \lambda$$\displaystyle )=\sum_{n=1}^N \log
\left( \frac{\exp\left( \sum_{i=1}^D \lambda_i A_{n,i}\right)-1}{\sum_{i=1}^D \lambda_i A_{n,i} }\right).$

For conciseness, define

$\displaystyle w_n=\sum_{i=1}^D \lambda_i A_{n,i}.$

Then,

$\displaystyle c_z($$ \lambda$$\displaystyle )=\sum_{n=1}^N \log
\left( \frac{e^{w_n}-1}{w_n}\right),$

$\displaystyle \frac{\partial}{\partial \lambda_i} c_z($$ \lambda$$\displaystyle )=\sum_{n=1}^N
\left( \frac{e^{w_n}}{e^{w_n}-1} - \frac{1}{w_n}\right) A_{n,i},$

and

$\displaystyle \frac{\partial^2}{\partial \lambda_i \partial \lambda_j} c_z($$ \lambda$$\displaystyle )=\sum_{n=1}^N
\left( \frac{e^{w_n}}{e^{w_n}-1} - \frac{(e^{w_n})^2}{(e^{w_n}-1)^2} + \frac{1}{w_n^2}\right) A_{n,i} A_{n,j}.$

From these expressions, we may find the saddle-point (2.12).



Baggenstoss 2017-05-19