Sphere
2 long-form posts on Sphere: machine-learning research by Taha Bouhsine, each built around live, in-browser interactive visualizations.
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What a Weight Can Be, in JAX/Flax NNX
A runnable companion: compute the price list of a kernel in JAX. The eigenvalues are the kernel's spectral density, found with an FFT; the RKHS norm of a weight is a sum over them. A corner is affordable only under a Sobolev kernel, and the same numbers place the Yat kernel: universal and smooth, roomier than a Gaussian but not a Sobolev space.
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What Can a Weight Be?
Once a kernel gives a weight a home, a second question follows: what is the weight allowed to be? Not all reproducing kernel Hilbert spaces are the same. A Sobolev space lets the weight have a sharp corner; a Gaussian's space forbids it; on normalized data the home is a sphere graded by spherical harmonics. A kernel is secretly a price list for roughness, and that list decides everything. Four interactive panels.