I think we can get additional information from the topological representation. We can look at the relationship between the different level sets under different cumulative probabilities. Although this requires evaluating the model over the whole dataset.
Let’s say we’ve trained a continuous normalizing flow model (which are equivalent to ordinary differential equations). These kinds of model require that the input and output dimensionality are the same, but we can narrow the model as the depth increases by directing many of those dimensions to isotropic gaussian noise. I haven’t trained any of these models before, so I don’t know if this works in practice.
Here is an example of the topology of an input space. The data may be knotted or tangled, and includes noise. The contours show level sets Si={x∣p(x)>pi}.
The model projects the data into a high dimensionality, then projects it back down into an arbitrary basis, but in the process untangling knots. (We can regularize the model to use the minimum number of dimensions by using an L1 activation loss
Lastly, we can view this topology as the Cartesian product of noise distributions and a hierarchical model. (I have some ideas for GAN losses that might be able to discover these directly)
We can use topological structures like these as anchors. If a model is strong enough, they will correspond to real relationships between natural classes. This means that very similar structures will be present in different models. If these structures are large enough or heterogeneous enough, they may be unique, in which case we can use them to find transformations between (subspaces of) the latent spaces of two different models trained on similar data.
I think we can get additional information from the topological representation. We can look at the relationship between the different level sets under different cumulative probabilities. Although this requires evaluating the model over the whole dataset.
Let’s say we’ve trained a continuous normalizing flow model (which are equivalent to ordinary differential equations). These kinds of model require that the input and output dimensionality are the same, but we can narrow the model as the depth increases by directing many of those dimensions to isotropic gaussian noise. I haven’t trained any of these models before, so I don’t know if this works in practice.
Here is an example of the topology of an input space. The data may be knotted or tangled, and includes noise. The contours show level sets Si={x∣p(x)>pi}.
The model projects the data into a high dimensionality, then projects it back down into an arbitrary basis, but in the process untangling knots. (We can regularize the model to use the minimum number of dimensions by using an L1 activation loss
Lastly, we can view this topology as the Cartesian product of noise distributions and a hierarchical model. (I have some ideas for GAN losses that might be able to discover these directly)
We can use topological structures like these as anchors. If a model is strong enough, they will correspond to real relationships between natural classes. This means that very similar structures will be present in different models. If these structures are large enough or heterogeneous enough, they may be unique, in which case we can use them to find transformations between (subspaces of) the latent spaces of two different models trained on similar data.