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Diffstat (limited to 'ictp-saifr_colloquium.tex')
| -rw-r--r-- | ictp-saifr_colloquium.tex | 21 |
1 files changed, 13 insertions, 8 deletions
diff --git a/ictp-saifr_colloquium.tex b/ictp-saifr_colloquium.tex index 8611f15..2d621f2 100644 --- a/ictp-saifr_colloquium.tex +++ b/ictp-saifr_colloquium.tex @@ -85,19 +85,20 @@ \begin{columns} \begin{column}{0.4\textwidth} - Pick a basis of $N$ functions $b_1(x), \ldots, b_N(x)$ + Pick a basis of $N$ functions + \[b_1(x), \ldots, b_N(x)\] - \bigskip + \smallskip Approximate the ground truth \[ \hat f(x\mid a_1,\ldots, a_N)=\sum_{j=1}^Na_jb_j(x) \] - Find $a_1, \ldots, a_N$ minimizing + Find $\pmb a=[a_1, \ldots, a_N]$ minimizing \[ - \chi^2 - =\sum_{i=1}^M\left(y_i-\sum_{j=1}^Na_jb_j(x_i)\right)^2 + \chi^2(\pmb a\mid\pmb x,\pmb y) + =\sum_{i=1}^M\left(y_i-\hat f(x_i\mid\pmb a)\right)^2 \] \end{column} \begin{column}{0.6\textwidth} @@ -193,7 +194,7 @@ \begin{column}{0.5\textwidth} Knowing the ground truth, fit error is \[ - \text{MSE}=\int dx\left(f(x)-\sum_{j=1}^Na_jb_j(x)\right)^2 + \text{MSE}=\int dx\left(f(x)-\hat f(x\mid\pmb a)\right)^2 \] \smallskip @@ -237,12 +238,16 @@ Fit function is a neural network: \[ - \hat f(\mathbf x\mid B_1,\ldots B_L)=\sigma\left(B_L \sigma\left( B_{L-1}\cdots\sigma\left(B_2\sigma (B_1\mathbf x)\right)\cdots\right)\right) + \hat f(\pmb x\mid B_1,\ldots B_L)=\sigma\left(B_L \sigma\left( B_{L-1}\cdots\sigma\left(B_2\sigma (B_1\pmb x)\right)\cdots\right)\right) \] \medskip - $\chi^2$ is called \emph{training error} + $\chi^2(\pmb a\mid\text{data})$ is called \emph{cost} or \emph{objective function} + + \medskip + + $\chi^2(\pmb a^*\mid\text{data})$ is call the \emph{training error} \medskip |