Biological neural networks are noisy: their dynamics involve difficult-to-predict fluctuations which are likely due not just to chaos, but also to a variety of low-level biophysical processes. Despite the omnipresence of intrinsic noise, its theoretical consequences are incompletely understood, especially in the context of learning. For example, does noise meaningfully shape representation learning, or is it better viewed as a nuisance factor whose presence or absence changes little about what solution is found and when? Here, we argue that intrinsic noise can significantly modulate representation learning in recurrent neural networks (RNNs), since it can strongly break solution-space degeneracies and in some cases control whether a non-degenerate solution exists at all. We show that this is true for an analytically tractable model of noisy RNN learning with a linear activation function and white noise inputs. We find that, in the so-called rich regime (where weights are initialized with small values), noise strongly modulates the directions in which the read-in, readout, and recurrent weights align. Moreover, we find that noise produces a kind of phase transition: for sufficiently high noise, a non-degenerate solution is not even optimal if parameter scales are limited (e.g., via regularization). These results extend recent analytic work on RNN representation learning and show that noise can act as a fundamental determinant of representations.
Data distributions often possess one or more symmetries: for example, all orientations of any specific dog are included in the distribution of all possible dog pictures. Despite the omnipresence of such symmetries, modern generative modeling approaches may fail to incorporate or learn them, which can lead to lower sample efficiency and poorer generalization. To what extent do diffusion models fail to learn data distribution symmetries, and what sources of symmetry breaking are most responsible? Here, we answer this question in the theoretically tractable setting of linear diffusion models. We show that of three potential sources of symmetry breaking---a non-symmetric initial weight matrix, a noisy learning target, and finite-sample noise---target noise produces the most robust symmetry breaking, since it persists even in the sample-rich regime. Using local laws from random matrix theory, we derive formulas for average symmetry error in fairly general cases involving symmetry within a subspace and show that smaller symmetries are harder to learn.