Week 15
$$\gdef \sam #1 {\mathrm{softargmax}(#1)}$$
$$\gdef \vect #1 {\boldsymbol{#1}} $$
$$\gdef \matr #1 {\boldsymbol{#1}} $$
$$\gdef \E {\mathbb{E}} $$
$$\gdef \V {\mathbb{V}} $$
$$\gdef \R {\mathbb{R}} $$
$$\gdef \N {\mathbb{N}} $$
$$\gdef \relu #1 {\texttt{ReLU}(#1)} $$
$$\gdef \D {\,\mathrm{d}} $$
$$\gdef \deriv #1 #2 {\frac{\D #1}{\D #2}}$$
$$\gdef \pd #1 #2 {\frac{\partial #1}{\partial #2}}$$
$$\gdef \set #1 {\left\lbrace #1 \right\rbrace} $$
% My colours
$$\gdef \aqua #1 {\textcolor{8dd3c7}{#1}} $$
$$\gdef \yellow #1 {\textcolor{ffffb3}{#1}} $$
$$\gdef \lavender #1 {\textcolor{bebada}{#1}} $$
$$\gdef \red #1 {\textcolor{fb8072}{#1}} $$
$$\gdef \blue #1 {\textcolor{80b1d3}{#1}} $$
$$\gdef \orange #1 {\textcolor{fdb462}{#1}} $$
$$\gdef \green #1 {\textcolor{b3de69}{#1}} $$
$$\gdef \pink #1 {\textcolor{fccde5}{#1}} $$
$$\gdef \vgrey #1 {\textcolor{d9d9d9}{#1}} $$
$$\gdef \violet #1 {\textcolor{bc80bd}{#1}} $$
$$\gdef \unka #1 {\textcolor{ccebc5}{#1}} $$
$$\gdef \unkb #1 {\textcolor{ffed6f}{#1}} $$
% Vectors
$$\gdef \vx {\pink{\vect{x }}} $$
$$\gdef \vy {\blue{\vect{y }}} $$
$$\gdef \vb {\vect{b}} $$
$$\gdef \vz {\orange{\vect{z }}} $$
$$\gdef \vtheta {\vect{\theta }} $$
$$\gdef \vh {\green{\vect{h }}} $$
$$\gdef \vq {\aqua{\vect{q }}} $$
$$\gdef \vk {\yellow{\vect{k }}} $$
$$\gdef \vv {\green{\vect{v }}} $$
$$\gdef \vytilde {\violet{\tilde{\vect{y}}}} $$
$$\gdef \vyhat {\red{\hat{\vect{y}}}} $$
$$\gdef \vycheck {\blue{\check{\vect{y}}}} $$
$$\gdef \vzcheck {\blue{\check{\vect{z}}}} $$
$$\gdef \vztilde {\green{\tilde{\vect{z}}}} $$
$$\gdef \vmu {\green{\vect{\mu}}} $$
$$\gdef \vu {\orange{\vect{u}}} $$
% Matrices
$$\gdef \mW {\matr{W}} $$
$$\gdef \mA {\matr{A}} $$
$$\gdef \mX {\pink{\matr{X}}} $$
$$\gdef \mY {\blue{\matr{Y}}} $$
$$\gdef \mQ {\aqua{\matr{Q }}} $$
$$\gdef \mK {\yellow{\matr{K }}} $$
$$\gdef \mV {\lavender{\matr{V }}} $$
$$\gdef \mH {\green{\matr{H }}} $$
% Coloured math
$$\gdef \cx {\pink{x}} $$
$$\gdef \ctheta {\orange{\theta}} $$
$$\gdef \cz {\orange{z}} $$
$$\gdef \Enc {\lavender{\text{Enc}}} $$
$$\gdef \Dec {\aqua{\text{Dec}}}$$
Practicum part A
When encountering the data with multiple outputs for a single input, feed-forward networks cannot capture such implicit dependencies. Instead, latent-variable energy-based models (EBMs) come to the rescue. We developed a toy ellipse example with a fixed input and the optimal model formulation. Then, we applied latent-variable EBMs to inference the best latent variables that can learn the implicit relationships.
Practicum part B
This section starts from introducing a relaxed version of free energy by modifying the “temperature” to smooth the energy function. Then we demonstrate how to train EBMs by minimizing loss functionals with several examples. Finally we give a concrete example of self-supervised learning, where we train a EBM to learn a horn-like data manifold.