Learning between the peaks: sharp asymptotics for kernel ridge regression under power-law anisotropy

We study kernel ridge regression under anisotropic Gaussian data, where the input covariance decays as a power law with exponent $α\geq 0$ for polynomial inner-product kernels. We derive asymptotically sharp expressions for the kernel spectrum and the generalization error in the polynomial high-dimensional regime $n=Θ(d^κ)$, revealing how anisotropy reshapes the learning curves. For weak anisotropy ($0<α<1$), the problem remains effectively high-dimensional and retains some features of the isotropic case, while departing from it in others: the variance still peaks at integer sample complexities $κ\in\mathbb{N}$, but these peaks are progressively damped as $α$ grows; meanwhile, for targets strongly aligned with the data's principal directions, the bias drops at fractional sample complexities, decoupling the bias transitions from the interpolation peaks. For strong anisotropy ($α> 1$), the effective dimension of the problem is constant, and the variance stops depending on sample size altogether, plateauing under ridgeless interpolation or vanishing at an explicit rate under fixed ridge penalty. The bias undergoes a sharp transition governed by the target's decay rate: below a threshold, learning is abrupt rather than gradual; above it, the bias decays as a power law that recovers the classical source and capacity rates. We finally specialize these results to single-index targets, showing how the alignment of the index with the data's principal directions determines the effect of anisotropy on learning. Together, our results clarify how the input geometry shapes the kernel features and fundamentally impacts its generalization properties.

Beyond Isotropic Assumptions: Continuity-Constrained Segmentation and GPU Morphometry for Nanoscale GBM Analysis

Confocal microscopy of optically cleared and swelled tissue resolves complex biological structures in 3D, but such acquisitions are highly anisotropic: along the under-sampled axial direction the structure can appear discontinuous, hampering reconstruction and automated quantitative analysis. The usual remedy upsamples the axial dimension to an isotropic volume before training a segmentation model, which requires dense annotations in the upsampled space, a prohibitive labeling burden. We present an end-to-end, GPU-accelerated framework that overcomes this without additional annotations. The model is trained on the native acquisition volume; random rotation of training patches leverages the well-resolved lateral plane to supply the missing axial information, and a z-axis continuity loss keeps neighboring slices consistent. We adapt both a convolutional (3D U-Net) and a transformer (SwinUNETR) backbone, aggregate overlapping patches by Gaussian consensus, and compute point-spread-function-corrected membrane thickness by ray-surface intersection on the GPU. We apply the method to the glomerular basement membrane (GBM), a thin, highly convoluted part of the kidney's filtration barrier that grows more irregular in disease. Segmentation accuracy matches inter-expert agreement. Continuity-aware training improves reconstruction smoothness and suppresses a periodic terracing artifact at minimal accuracy cost. We quantify GBM thickness across the reconstructed 3D surface and capture disease-related thickening, enabling fully automated anisotropic 3D morphometry of biological structures without dense volumetric labels or image restoration.