MF mechanism consistently performs at least as well as the best prior methods.

Motivated by our analysis results, we propose new data augmentation methods aimed at enhancing model generalization.

Tensor decompositions (TDs) serve as a powerful tool for analyzing multiway data.

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.

Across fields such as machine learning, social science, geography, considerable attention has been given to models that factorize a nonnegative matrix into the product of two or three matrices, subject to nonnegative or row-sum-to-1 constraints. Although these models are to a large extend similar or even equivalent, they are presented under different names, and their similarity is not well known. This paper highlights similarities among five popular models, latent budget analysis (LBA), latent class analysis (LCA), end-member analysis (EMA), probabilistic latent semantic analysis (PLSA), and nonnegative matrix factorization (NMF). We focus on an essential issue-identifiability-of these models and prove that the solution of LBA, EMA, LCA, PLSA is unique if and only if the solution of NMF is unique. We also provide a brief review for algorithms of these models. We illustrate the models with a time budget dataset from social science, and end the paper with a discussion of closely related models such as archetypal analysis.

Empirical risk minimization (ERM) may be used to generalize the result from train to test data.