A significant task in statistics and machine learning currently revolves around generating samples from a target distribution that is only accessible via a dataset.

Unlike distance metric learning where the subsequent tasks utilizing the estimated distance metric is the usual focus, the proposal focuses on the estimated metric characterizing the geometry structure. Despite the illustrated taxi and MNIST examples, it is still open to finding more compelling applications that target the data space geometry. Interpreting mathematical concepts such as Riemannian metric and geodesic in the context of potential application (e.g., cognition and perception research where similarity measures are common) could be inspiring. Our proposal requires sufficiently dense data, which could be demanding, especially for high-dimensional data due to the curse of dimensionality. Dimensional reduction (e.g., manifold embedding as in the MNIST example) can substantially alleviate the curse of dimensionality, and the dense data requirement will more likely hold true.

The key quantity of interest here is the Riemannian metric, which characterizes the Riemannian geometry and defines straight lines and derivatives on the manifold.

Monte Carlo stopping rules in a manner that is both sample efficient and robust to estimation error.

The remarkable capability of over-parameterised neural networks to generalise effectively has been explained by invoking a "simplicity bias": neural networks prevent overfitting by initially learning simple classifiers before progressing to

By analyzing the intrinsic dimension (ID) and neighbor composition, we find that the representations evolve similarly in transformers trained on protein language tasks and image reconstruction tasks.

A central problem in machine learning theory is to characterize how learning dynamics select particular solutions among the many compatible with the training objective, a phenomenon, called implicit bias, which remains only partially characterized. In the present work, we identify a general mechanism, in terms of an explicit geometric correction of the learning dynamics, for the emergence of implicit biases, arising from the interaction between continuous symmetries in the model's parametrization and stochasticity in the optimization process. Our viewpoint is constructive in two complementary directions: given model symmetries, one can derive the implicit bias they induce; conversely, one can inverse-design a wide class of different implicit biases by computing specific redundant parameterizations. More precisely, we show that, when the dynamics is expressed in the quotient space obtained by factoring out the symmetry group of the parameterization, the resulting stochastic differential equation gains a closed form geometric correction in the stationary distribution of the optimizer dynamics favoring orbits with small local volume. We compute the resulting symmetry induced bias for a range of architectures, showing how several well known results fit into a single unified framework. The approach also provides a practical methodology for deriving implicit biases in new settings, and it yields concrete, testable predictions that we confirm by numerical simulations on toy models trained on synthetic data, leaving more complex scenarios for future work. Finally, we test the implicit bias inverse-design procedure in notable cases, including biases toward sparsity in linear features or in spectral properties of the model parameters.