Brian MacWhinney
Last edited: August 8, 2025Professor Brian MacWhinney is a professor of psychology, modern languages, and language technology at CMU.
Brown v. Board of Education
Last edited: August 8, 2025Brown v. Board of Education is a landmark case in the US. This lead for schools to be integrated, and many children were taken out of school out of protest due to the subsequent integration movement between schools.
Brownian Motion
Last edited: August 8, 2025Brownian Motion is the pattern for measuring the convergence of random walk through continuous timing.
discrete random walk
discrete random walk is a tool used to construct Brownian Motion. It is a random walk which only takes on two discrete values at any given time: \(\Delta\) and its additive inverse \(-\Delta\). These two cases take place at probabilities \(\pi\) and \(1-\pi\).
Therefore, the expected return over each time \(k\) is:
\begin{equation} \epsilon_{k} = \begin{cases} \Delta, p(\pi) \\ -\Delta, p(1-\pi) \end{cases} \end{equation}
Broyden-Fletcher-Goldfarb-SHanno (BFGS)
Last edited: August 8, 2025Broyden's Method
Last edited: August 8, 2025Broyden’s method is an approximate method to estimate the Jacobian. We give the root-finding variant here (i.e. the search direction is for finding \(F\qty(x) = 0\) instead of \(\min F\)).
For function \(F\), let: \(J^{(0)} = I\). For every step
- compute \(\Delta c^{(q)}\) from \(J^{(q)}\Delta c^{(q)} = -F\qty(c^{(q)})\)
- compute \(\arg\min_{\alpha} F\qty(c^{(q)} + \alpha \Delta c^{(q)})^{T}F\qty(c^{(q)} + \alpha \Delta c^{(q)})\) for root finding
- compute \(c^{(q+1)} = c^{(q)} + \alpha \Delta c^{(q)}\)
- set \(\Delta c^{(q)} = c^{(q+1)} - c^{(q)}\)
- finally, we can update \(J\) such that…
\begin{equation} J^{(q+1)} = J^{(q)} + \frac{1}{\qty(\Delta c^{(q)})^{T} \Delta c^{(q)}} \qty(F\qty(c^{(q+1)}) - F\qty(c^{(q)}) - J^{(q)}\Delta c^{(q)}) \qty(\Delta c^{(q)})^{T} \end{equation}