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| -rw-r--r-- | ictp-saifr_colloquium.tex | 29 |
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diff --git a/ictp-saifr_colloquium.tex b/ictp-saifr_colloquium.tex index 1d823a7..e1c78bb 100644 --- a/ictp-saifr_colloquium.tex +++ b/ictp-saifr_colloquium.tex @@ -863,6 +863,35 @@ \end{frame} \begin{frame} + \frametitle{A simple model of nonlinear least squares} + \begin{columns} + \begin{column}{0.9\textwidth} + Data: $x_i\mapsto J^i$, $N\times\cdots\times N$ $p$-tensors of Gaussian random numbers\\ + \hphantom{Data: }$y_i\mapsto V_0$, a constant \\ + + \medskip + + Parameters: $\pmb a$ is an $N$-dimensional vector constrained to live on the sphere \\ + + \medskip + + Model: $\hat f(J\mid\pmb a)$ is the homogeneous degree-$p$ polynomial + \[ + \hat f(J\mid \pmb a) + =\sum_{i_1=1}^N\cdots\sum_{i_p=1}^NJ_{i_1,\ldots,i_p}a_{i_1}\cdots a_{i_p} + \] + Cost function: the usual + \[ + \chi^2(\pmb a\mid J^1,\ldots,J^M) + =\sum_{i=1}^M\left[ + V_0-\hat f(J^i\mid\pmb a) + \right]^2 + \] + \end{column} + \end{columns} +\end{frame} + +\begin{frame} \frametitle{The Euler characteristic \boldmath{$\chi$}} \begin{columns} \begin{column}{0.5\textwidth} |