Solution to the Fokker-Planck equation for slowly driven Brownian motion: Emergent geometry and a formula for the corresponding thermodynamic metric.

Considerable progress has recently been made with geometrical approaches to understanding and controlling small out-of-equilibrium systems, but a mathematically rigorous foundation for these methods has been lacking. Towards this end, we develop a perturbative solution to the Fokker-Planck equation for one-dimensional driven Brownian motion in the overdamped limit enabled by the spectral properties of the corresponding single-particle Schrödinger operator. The perturbation theory is in powers of the inverse characteristic timescale of variation of the fastest varying control parameter, measured in units of the system timescale, which is set by the smallest eigenvalue of the corresponding Schrödinger operator.

Critical Point-Finding Methods Reveal Gradient-Flat Regions of Deep Network Losses.

Despite the fact that the loss functions of deep neural networks are highly nonconvex, gradient-based optimization algorithms converge to approximately the same performance from many random initial points. One thread of work has focused on explaining this phenomenon by numerically characterizing the local curvature near critical points of the loss function, where the gradients are near zero. Such studies have reported that neural network losses enjoy a no-bad-local-minima property, in disagreement with more recent theoretical results.

Pure Cerebellar Ataxia with Homozygous Mutations in the PNPLA6 Gene.

Autosomal-recessive cerebellar ataxias (ARCA) are clinically and genetically heterogeneous conditions primarily affecting the cerebellum. Mutations in the PNPLA6 gene have been identified as the cause of hereditary spastic paraplegia and complex forms of ataxia associated with retinal and endocrine manifestations in a field where the genotype-phenotype correlations are rapidly expanding. We identified two cousins from a consanguineous family belonging to a large Zoroastrian (Parsi) family residing in Mumbai, India, who presented with pure cerebellar ataxia without chorioretinal dystrophy or hypogonadotropic hypogonadism.