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We propose a landmark-constrained algorithm, LA-VDM (Landmark Accelerated Vector Diffusion Maps), to accelerate the Vector Diffusion Maps (VDM) framework built upon the Graph Connection Laplacian (GCL), which captures pairwise connection relationships within complex datasets. LA-VDM introduces a novel two-stage normalization that effectively address nonuniform sampling densities in both the data and the landmark sets. Under a manifold model with the frame bundle structure, we show that we can accurately recover the parallel transport with landmark-constrained diffusion from a point cloud, and hence asymptotically LA-VDM converges to the connection Laplacian. The performance and accuracy of LA-VDM are demonstrated through experiments on simulated datasets and an application to nonlocal image denoising.

The theory enables a systematicclassification of all existing G-CNNs in terms of their symmetry group, base space,andfieldtype.

In addition, we present an alternative design that permits a near-optimal sublinear decoding timeofO(klog2k+klogn).

Apple could unveil Gemini-powered Siri in Feb. Apple Creator Studio is now available: What's included, how much it costs and what it means for creators The new subscription bundle is now available for $13 per month or $129 per year. Apple has been leaning harder on services for several years now. That part of the business brought in tens of billions of dollars in revenue last year alone, and the company says 2025 was a record year . With Apple Creator Studio, Apple is extending that strategy further into professional creative software. Apple Creator Studio is a new subscription bundle that packages several of the company's pro apps under a single monthly or yearly fee.