Geometry
The geometry of representations: simplices, frames, cones, and the metrics learned objectives actually impose.
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Where Does a Weight Live?
A standard neuron's weight and its input never actually meet: one is a point you can see, the other an arrow off in its own space, joined only by a shadow. This is what a reproducing kernel Hilbert space fixes: it gives input and weight one shared address, where the optimal weight is built from the data itself and sits right next to it. Four interactive panels.
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What a Finite Kernel Buys an MLP
Replace the activation function with a finite, explicit, positive-definite kernel, the Yat kernel, and an MLP stops being a stack of linear maps glued by a nonlinearity. It becomes a kernel machine, with locality, attribution, geometry, capacity control, and a feature map you can write down.
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Opposite Is Not Different: The Cosine-Similarity Bug in CLIP and Contrastive Learning
Maximum difference between two unit vectors is orthogonality (cos = 0), not opposition (cos = −1). CLIP, InfoNCE, and SimCLR have been optimizing for the wrong target for years.
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Activations Are Bad for Geometry
ReLU, GELU, and friends factor into a layer's Jacobian as a diagonal modulation that wrecks the geometry of the data manifold. Why pointwise activations are a representational bug.