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UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction

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abstract

UMAP (Uniform Manifold Approximation and Projection) is a novel manifold learning technique for dimension reduction. UMAP is constructed from a theoretical framework based in Riemannian geometry and algebraic topology. The result is a practical scalable algorithm that applies to real world data. The UMAP algorithm is competitive with t-SNE for visualization quality, and arguably preserves more of the global structure with superior run time performance. Furthermore, UMAP has no computational restrictions on embedding dimension, making it viable as a general purpose dimension reduction technique for machine learning.

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  • abstract UMAP (Uniform Manifold Approximation and Projection) is a novel manifold learning technique for dimension reduction. UMAP is constructed from a theoretical framework based in Riemannian geometry and algebraic topology. The result is a practical scalable algorithm that applies to real world data. The UMAP algorithm is competitive with t-SNE for visualization quality, and arguably preserves more of the global structure with superior run time performance. Furthermore, UMAP has no computational restrictions on embedding dimension, making it viable as a general purpose dimension reduction technique

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On the continuum limit of t-SNE for data visualization

stat.ML · 2026-04-13 · unverdicted · novelty 8.0

t-SNE converges in the large-data limit to a non-convex variational energy with attraction and repulsion terms that admits a unique smooth minimizer but infinitely many discontinuous ones in one dimension.

Trajectory Geometry of Transformer Representations Across Layers

cs.LG · 2026-06-08 · unverdicted · novelty 7.0

Transformer representations form trajectories showing semantic convergence in middle-to-late layers, higher curvature on reasoning tasks, bifurcation on ambiguous tokens, and a consistent three-phase cosine similarity pattern across GPT-2, TinyLlama, and Qwen2.5.

Robust and Efficient Guardrails with Latent Reasoning

cs.AI · 2026-05-27 · unverdicted · novelty 7.0

COLAGUARD matches explicit-reasoning guardrail performance on safety benchmarks while delivering 12.9X speedup and 22.4X token reduction by propagating hidden states instead of generating text.

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