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We investigate gradient descent training of wide neural networks and the corresponding implicit bias in function space. For univariate regression, we show that the solution of training a width-n shallow ReLU network is within n−1/2 of the function which fits the training data and whose difference from the initial function has the smallest 2-norm of the second derivative weighted by a curvature penalty that depends on the probability distribution that is used to initialize the network parameters. We compute the curvature penalty function explicitly for various common initialization procedures. For instance, asymmetric initialization with a uniform distribution yields a constant curvature penalty, and thence the solution function is the natural cubic spline interpolation of the training data. For stochastic gradient descent we obtain the same implicit bias result. We obtain a similar result for different activation functions. For multivariate regression we show an analogous result, whereby the second derivative is replaced by the Radon transform of a fractional Laplacian. For initialization schemes that yield a constant penalty function, the solutions are polyharmonic splines. Moreover, we show that the training trajectories are captured by trajectories of smoothing splines with decreasing regularization strength. 

We investigate gradient descent training of wide neural networks and the corresponding implicit bias in function space. For univariate regression, we show that the solution of training a width-n shallow ReLU network is within n1/2 of the function which fits the training data and whose difference from the initial function has the smallest 2-norm of the second derivative weighted by a curvature penalty that depends on the probability distribution that is used to initialize the network parameters. We compute the curvature penalty function explicitly for various common initialization procedures. For instance, asymmetric initialization with a uniform distribution yields a constant curvature penalty, and thence the solution function is the natural cubic spline interpolation of the training data. For stochastic gradient descent we obtain the same implicit bias result. We obtain a similar result for different activation functions. For multivariate regression we show an analogous result, whereby the second derivative is replaced by the Radon transform of a fractional Laplacian. For initialization schemes that yield a constant penalty function, the solutions are polyharmonic splines. Moreover, we show that the training trajectories are captured by trajectories of smoothing splines with decreasing regularization strength. 

The linker of the nucleoskeleton and cytoskeleton (LINC) complex is composed of two transmembrane proteins: the KASH domain protein localized to the outer nuclear membrane and the SUN domain protein to the inner nuclear membrane. In budding yeast, the sole SUN domain protein, Mps3, is thought to pair with either Csm4 or Mps2, two KASH-like proteins, to form two separate LINC complexes. Here, we show that Mps2 mediates the interaction between Csm4 and Mps3 to form a heterotrimeric telomere-associated LINC (t-LINC) complex in budding yeast meiosis. Mps2 binds to Csm4 and Mps3, and all three are localized to the telomere. Telomeric localization of Csm4 depends on both Mps2 and Mps3; in contrast, Mps2’s localization depends on Mps3 but not Csm4. Mps2-mediated t-LINC complex regulates telomere movement and meiotic recombination. By ectopically expressing CSM4 in vegetative yeast cells, we reconstitute the heterotrimeric t-LINC complex and demonstrate its ability to tether telomeres. Our findings therefore reveal the heterotrimeric composition of the t-LINC complex in budding yeast and have implications for understanding variant LINC complex formation. 

Lode gold deposits, which are currently the world’s major gold supply, have been shown to be generated mostly by phase separation of metamorphic fluids and/or interaction between these fluids and wall rocks. Here we use garnet oxygen isotopes by secondary ion mass spectrometry to document the crucial role of magmatic hydrothermal fluids and their mixing with meteoric water in the formation of the world-class Dongping gold deposit in the North China Craton. Garnet grains from quartz veins of various paragenetic stages and the mineralized alteration envelope at Dongping have dynamic δ 18 O variations of 3.8 to −11.0‰, with large intragrain fluctuations up to 5.3‰. These values correspond to calculated δ 18 O values of 6.1 to −9.1‰ for the hydrothermal fluids from which the garnet formed. The isotope data, notably the cyclic alternation in δ 18 O within individual garnet grains, are best interpreted to reflect multiple pulses of magmatically derived fluids and subsequent mixing of each pulse with variable amounts of meteoric water. The results presented here allow us to quantify the significant interplay between magmatic hydrothermal fluids and meteoric water that spanned the entire mineralization history and triggered gold deposition of a lode gold deposit. This study highlights the potential use of in situ oxygen isotope analysis of garnet in tracing the origin and evolution of hydrothermal fluids in the Earth’s crust that may have formed other giant ore deposits.