The dynamical stability of the iterates during training plays a key role in determining the minima obtained by optimization algorithms. For example, stable solutions of gradient descent (GD) correspond to flat minima, which have been associated with favorable features. While prior work often relies on linearization to determine stability, it remains unclear whether linearized dynamics faithfully capture the full nonlinear behavior. Recent work has shown that GD may stably oscillate near a linearly unstable minimum and still converge once the step size decays, indicating that linear analysis can be misleading. In this work, we explicitly study the effect of nonlinear terms. Specifically, we derive an exact criterion for stable oscillations of GD near minima in the multivariate setting. Our condition depends on high-order derivatives, generalizing existing results. Extending the analysis to stochastic gradient descent (SGD), we show that nonlinear dynamics can diverge in expectation even if a single batch is unstable. This implies that stability can be dictated by a single batch that oscillates unstably, rather than an average effect, as linear analysis suggests. Finally, we prove that if all batches are linearly stable, the nonlinear dynamics of SGD are stable in expectation.

Continuous attractors offer a unique class of solutions for storing continuous-valued variables in recurrent system states for indefinitely long time intervals.

Data from naturalistic systems are often modeled asgenerated from an underlying low dimensional manifold embedded inahighdimensional measurement space.

Continuous attractors offer a unique class of solutions for storing continuous-valued variables in recurrent system states for indefinitely long time intervals.Unfortunately, continuous attractors suffer from severe structural instability in general---they are destroyed by most infinitesimal changes of the dynamical law that defines them.This fragility limits their utility especially in biological systems as their recurrent dynamics are subject to constant perturbations.We observe that the bifurcations from continuous attractors in theoretical neuroscience models display various structurally stable forms.Although their asymptotic behaviors to maintain memory are categorically distinct, their finite-time behaviors are similar.We build on the persistent manifold theory to explain the commonalities between bifurcations from and approximations of continuous attractors.Fast-slow decomposition analysis uncovers the existence of a persistent slow manifold that survives the seemingly destructive bifurcation, relating the flow within the manifold to the size of the perturbation. Moreover, this allows the bounding of the memory error of these approximations of continuous attractors.Finally, we train recurrent neural networks on analog memory tasks to support the appearance of these systems as solutions and their generalization capabilities.Therefore, we conclude that continuous attractors are functionally robust and remain useful as a universal analogy for understanding analog memory.

Tubular trees, such as blood vessels and lung airways, are essential for material transport within the human body. Accurately detecting their centerlines with correct tree topology is critical for clinical tasks such as diagnosis, treatment planning, and surgical navigation. In these applications, maintaining high recall is crucial, as missing small branches can result in fatal mistakes caused by incomplete assessments or undetected abnormalities. We present RefTr, a 3D image-to-graph model for centerline generation of vascular trees via recurrent refinement of confluent trajectories. RefTr uses a Producer-Refiner architecture based on a Transformer decoder, where the Producer proposes a set of initial confluent trajectories that are recurrently refined by the Refiner to produce final trajectories, which forms the centerline graph. The confluent trajectory representation enables refinement of complete trajectories while explicitly enforcing a valid tree topology. The recurrent refinement scheme improves precision and reuses the same Refiner block across multiple steps, yielding a 2.4x reduction in decoder parameters compared to previous SOTA. We also introduce an efficient non-maximum suppression algorithm for spatial tree graphs to merge duplicate branches and boost precision. Across multiple public centerline datasets, RefTr achieves superior recall and comparable precision to previous SOTA, while offering faster inference and substantially fewer parameters, demonstrating its potential as a new state-of-the-art framework for vascular tree analysis in 3D medical imaging.

Neural Information Processing Systems http://nips.cc/

A central challenge in neuroscience is understanding how neural system implements computation through its dynamics. We propose a nonlinear time series model aimed at characterizing interpretable dynamics from neural trajectories. Our model assumes low-dimensional continuous dynamics in a finite volume. It incorporates a prior assumption about globally contractional dynamics to avoid overly enthusiastic extrapolation outside of the support of observed trajectories. We show that our model can recover qualitative features of the phase portrait such as attractors, slow points, and bifurcations, while also producing reliable long-term future predictions in a variety of dynamical models and in real neural data.

DS, including RNNs, that refer to topological (qualitative) changes in a system's