Neural Collapse
Neural collapse and the terminal phase of training: why class means form a simplex equiangular tight frame, and what the Welch bound says about it.
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The Three States of Information, in JAX
A runnable companion to The Three States of Information: train tiny models in JAX and measure the three states directly: the feature-covariance spectrum collapsing from high-rank (random) to a C−1-mode frame (structured), the distributional simplicity bias that fits low-order structure first (organized), the neural-collapse simplex where class-mean cosines lock onto −1/(C−1), and the alignment/uniformity split of contrastive learning running on two separate clocks. Four live JAX visualizations, every number an eigenvalue or a loss.
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The Three States of Information
Representations learned by a network pass through three states, like matter: random (high-entropy, no structure), organized (clusters, local order), and structured (a maximally-separated simplex, global order). The transitions between them are exactly the loss plateaus you see when training: the flat stretch is where the representation reorganizes before that reorganization shows up in the loss. Built from live in-browser training runs.
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Latent on the Spectrum, in JAX
A runnable companion to Latent on the Spectrum: build a codebook as the spectral embedding of a label kernel in JAX (classical MDS with square-root eigenvalue scaling), watch a flat spectrum become the simplex and a graded one become the horseshoe, measure kernel-target alignment, split a representation into its between-class prototype frame and within-class information spectrum, and watch neural collapse grind the information to zero.
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Latent on the Spectrum: Why Cats Sit Closer to Dogs Than to Cars
The regular simplex is the perfect codebook only when classes are strangers, and real labels are not strangers. A latent space is a lossy, finite-dimensional encoding of a label-similarity kernel: the codebook is the top eigenmodes of that kernel, the information rides in the modes below them, and the Welch bound sets the geometry of that channel. A follow-up to the Welch-bound post with live in-browser experiments: steer a codebook from simplex to taxonomy, spend a dimension budget, watch neural collapse grind the information spectrum to zero, read dark knowledge off a wandering feature, and see a structured codebook make better mistakes.
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Auditing Latent Space Geometry in JAX
A runnable companion to the Welch-bound latent-space post: generate GIFs and implement the JAX metrics that tell you whether embeddings are collapsing, wasting rank, forming a simplex, or pressing against the Welch floor.
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What Makes a Good Latent Space? The Welch Bound and the Simplex
The hidden codebook inside representation learning: why collapse happens, why opposition is a trap, why class means form a simplex, and why the Welch bound sets the best geometry when too many concepts share too few dimensions.