Ask what should change when the equation is manipulated, then let the visualization test that expectation.
Foundation Lab
Representation Learning & Embedding Geometry
Word & token embeddings in LMs, vision embeddings in CLIP-like models, multimodal embeddings in Gemini and GPT-4V
#10EmbeddingsRepresentations
key equation
\mathcal L = - \mathbb E \left[ \log \frac{\exp(\mathrm{sim}(f(x),g(y))/\tau)}{\sum_{y'} \exp(\mathrm{sim}(f(x), g(y'))/\tau)} \right]Selected Foundation Object
Keep the equation fixed; move through the evidence.
Concept 10 of 100EmbeddingsRepresentations / Phase 5: Representation & interpretability
Current question
Geometric explanation of anisotropy (representations bunch along a few directions) and how normalization/whitening alter behavior
\mathcal L = - \mathbb E \left[ \log \frac{\exp(\mathrm{sim}(f(x),g(y))/\tau)}{\sum_{y'} \exp(\mathrm{sim}(f(x), g(y'))/\tau)} \right]Use the runnable panel, the key equation, and canonical papers as separate forms of evidence for the same object.
The useful learning product is the reusable mechanism you can carry into another model, paper, or engineering tradeoff.
This atlas page keeps the working demo; the domain notebook carries the fuller Intuition -> Math -> Code -> Demo sequence.
Why It Matters for Modern Models
- Word & token embeddings in LMs, vision embeddings in CLIP-like models, multimodal embeddings in Gemini and GPT-4V
- Latent spaces of Stable Diffusion designed so distances correspond to semantic similarity
What Tutorials Skip
What is still poorly explained in textbooks and papers:
- Geometric explanation of anisotropy (representations bunch along a few directions) and how normalization/whitening alter behavior
- Visuals showing how representations evolve across layers (local to global features)
Interactive Visualization
Core Math (Optional Deep Dive)
If you want intuition first, start with the key equation and the visualization. Come back here for the full walkthrough.
Key Equation
Learn a mapping such that inner products or distances reflect meaningful relations.
Contrastive objective (InfoNCE-style):
This pushes "positive" pairs together, "negatives" apart; at optimum, it maximizes a lower bound on mutual information between views.