Other concurrent works address, e.g., learning from soft interventions with polynomial mixing [

For a small training set, CNNs behave similarly.

In this work we consider the problem of discretization of neural operators in a general setting. Using category theory, we give a no-go theorem that shows that diffeomorphisms between Hilbert spaces may not admit any continuous approximations by diffeomorphisms on finite spaces, even if the discretization is non-linear. This shows how infinite-dimensional Hilbert spaces and finite-dimensional vector spaces fundamentally differ. A key take-away is that to obtain discretization invariance, considerable effort is needed to ensure that finite-dimensional approximations of neural operator converge not only as sequences of functions, but that their representations converge in a suitable sense as well. With this perspective, we give several positive results. We first show that strongly monotone diffeomorphism operators always admit finite-dimensional strongly monotone diffeomorphisms. Next we show that bilipschitz neural operators may always be written via the repeated alternating composition of strongly monotone neural operators and invertible linear maps. We also show that such operators may be inverted locally via iteration provided that such inverse exists. Finally, we conclude by showing how our framework may be used `out of the box' to prove quantitative approximation results for discretization of neural operators.

Linear mixed-effect models provide a natural baseline for estimating disease progression using longitudinal data. They provide interpretable models at the cost of modeling assumptions on the progression profiles and their variability across subjects. A significant improvement is to embed the data in a Riemannian manifold and learn patient-specific trajectories distributed around a central geodesic. A few interpretable parameters characterize subject trajectories at the cost of a prior choice of the metric, which determines the shape of the trajectories. We extend this approach by learning the metric from the data allowing more flexibility while keeping the interpretability. Specifically, we learn the metric as the push-forward of the Euclidean metric by a diffeomorphism. This diffeomorphism is estimated iteratively as the composition of radial basis functions belonging to a reproducible kernel Hilbert space. The metric update allows us to improve the forecasting of imaging and clinical biomarkers in the Alzheimer's Disease Neuroimaging Initiative (ADNI) cohort. Our results compare favorably to the 56 methods benchmarked in the TADPOLE challenge.

Understanding why deep nets can classify data in large dimensions remains a challenge. It has been proposed that they do so by becoming stable to diffeomorphisms, yet existing empirical measurements support that it is often not the case. We revisit this question by defining a maximum-entropy distribution on diffeomorphisms, that allows to study typical diffeomorphisms of a given norm. We confirm that stability toward diffeomorphisms does not strongly correlate to performance on benchmark data sets of images. By contrast, we find that the stability toward diffeomorphisms relative to that of generic transformations $R_f$ correlates remarkably with the test error $\epsilon_t$. It is of order unity at initialization but decreases by several decades during training for state-of-the-art architectures. For CIFAR10 and 15 known architectures, we find $\epsilon_t\approx 0.2\sqrt{R_f}$, suggesting that obtaining a small $R_f$ is important to achieve good performance. We study how $R_f$ depends on the size of the training set and compare it to a simple model of invariant learning.

Recent theoretical work established the unsupervised identifiability of quantized factors under any diffeomorphism. The theory assumes that quantization thresholds correspond to axis-aligned discontinuities in the probability density of the latent factors. By constraining a learned map to have a density with axis-aligned discontinuities, we can recover the quantization of the factors. However, translating this high-level principle into an effective practical criterion remains challenging, especially under nonlinear maps. Here, we develop a criterion for unsupervised disentanglement by encouraging axis-aligned discontinuities. Discontinuities manifest as sharp changes in the estimated density of factors and form what we call cliffs. Following the definition of independent discontinuities from the theory, we encourage the location of the cliffs along a factor to be independent of the values of the other factors. We show that our method, Cliff, outperforms the baselines on all disentanglement benchmarks, demonstrating its effectiveness in unsupervised disentanglement.

This is achieved by generalizing the multilayer kernel introduced in the context of convolutional kernel networks and by studying the geometry of the corresponding reproducing kernel Hilbert space.

We consider the optimization of cost functionals on infinite dimensional manifolds and derive a variational approach to accelerated methods on manifolds.