The kernel trick explained: how lifting one feature makes rings linearly separable
techNmak · x · 2026-10-31
A clear explainer of the kernel trick using a classic example: a linear classifier can't separate a cluster from a surrounding ring, but adding the feature z=x₁²+x₂² (squared distance from origin) makes the classes separable by a plane in 3D.
Key points:
- SVM training depends only on inner products, and kernels K(a,b)=φ(a)ᵀφ(b) compute them in the transformed space without ever building the transformed vectors.
- The Gaussian RBF kernel corresponds to an infinite-dimensional feature space, yet is computed from a simple exponential of squared distances.
- Trade-offs: pairwise kernel evaluations get expensive on large datasets, and choosing a kernel doesn't guarantee an easy problem.
More from Research
- DeepMind's Gemini agent pipeline cracks rules for 11 unresolved BB(6) Turing machines — rvp · 2026-10-31
- Christophe Giraud's Introduction to High-Dimensional Statistics, 2nd edition — FrnkNlsn · 2026-10-31
- Beating the compiler at signed 32-bit division: multiply-shift tricks cut latency — lemire · 2026-10-31
- NetHack port challenge: AI transpilers hit 99.5% correctness but tank readability — davidbau · 2026-10-31
- AIGENIE: Open-Source R Package Uses LLMs to Auto-Generate and Validate Psychometric Scales — GolinoHudson · 2026-10-31
- LLM judges are strongly biased toward position, researcher warns to randomize everything — antoine_chaffin · 2026-10-31