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We then generated data using 30 latent variables with 100 % of active weights. Data was simulated for 800 samples and 1600 features and the latent variables explained 75% of the

E.1 Identifiability of the latent structure for sparse and dense factors 175

total variability (due to the higher density of the weights).

Again, we compared three models: a first model with no spike-and-slab prior; a second model with spike-and-slab priors on the weights and a third model with both spike-and-slab priors on the weights and on the factors. Like in Section E.1.1, the three models were fitted three times with random initialisations of the latent variables in order to assess the robustness of the inference process. Correlation of the inferred weights with the simulated weights showed that none of the models could identify the true latent weights (Fig. E.4). Across our three random trials, inference was only reproducible for the model with spike-and-slab priors on both weights and factors (Fig. E.4), although the inferred weights were in any case rotated compared to the true simulated weights.

Fig. E.3 Correlation of the simulated weights with weights inferred with a model without any spike-and-slab prior (left), with spike-and-slab priors on the weights (middle) and with both spike-and-slab priors on the weights and on the factors (right). The generative latent factors were all dense (100% of active weights).

Taken together, these results confirm the usefulness of sparsity-inducing priors for the identification of sparse latent factors, while showing that dense latent factors pose in any case an identifiability problem.

Fig. E.4 Robustness of weights inference for a model without any spike-and-slab prior (left), with spike-and-slab priors on the weights (middle) and with both spike-and-slab priors on the weights and on the factors (right). Each heat map shows the correlation matrix between all inferred weights for all three separate runs, so that off-diagonal blocks correspond to correlation plots between runs. The generative latent factors were all dense (100% of active weights).

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