### 3.2 Variational objective

We train the encoder \( {q}_{\phi }\left( {z \mid  x}\right) \) and the decoder \( {p}_{\theta }\left( {x \mid  z}\right) \) jointly by maximising the evidence lower bound on the marginal log-likelihood of each observation \( x \) :

\[
\mathcal{L}\left( {\theta ,\phi ;x}\right)  = {\mathbb{E}}_{{q}_{\phi }\left( {z \mid  x}\right) }\left\lbrack  {\log {p}_{\theta }\left( {x \mid  z}\right) }\right\rbrack   - {D}_{\mathrm{KL}}\left( {{q}_{\phi }\left( {z \mid  x}\right) \parallel p\left( z\right) }\right) .\tag{4}
\]

The first term rewards faithful reconstruction; the second keeps the approximate posterior close to the prior \( p\left( z\right)  = \mathcal{N}\left( {0, I}\right) \) . Table 2 reports the error for four latent sizes.

Table 2: Mean squared reconstruction error on the held-out split, in thousandths.

<table><tr><td>Latent size</td><td>Baseline</td><td>Ours</td><td>Change</td></tr><tr><td>16</td><td>9.84</td><td>8.71</td><td>-1.13</td></tr><tr><td>32</td><td>7.62</td><td>6.05</td><td>-1.57</td></tr><tr><td>64</td><td>6.48</td><td>4.93</td><td>-1.55</td></tr><tr><td>128</td><td>6.31</td><td>4.88</td><td>-1.43</td></tr></table>

Increasing \( d \) beyond 64 yields diminishing returns, which is consistent with the capacity analysis in Section 5. We therefore fix \( d = {64} \) in all remaining experiments.