Reading and Writing with Projections

2025-07-10 · Source: Chris McCormick · Field: Technology & Digital — Artificial Intelligence & Machine Learning · Depth: Intermediate, medium

Summary

Transformers store, retrieve, and modify data using "feature directions" and "projections" within their model space, enabling them to pack more features than available dimensions. This concept is illustrated by encoding and decoding speaker settings (bass, volume, treble) into two dimensions. Initially, axis-aligned directions are used, but neural networks typically learn arbitrary, near-orthogonal directions, leading to slight interference and recovery errors. Transformers update values by encoding adjustments as residual vectors. While packing three features into two dimensions results in significant interference, higher dimensionality, such as a 4,096-embedding, allows for greater feature density with minimal interference due to the "blessing of dimensionality." Models achieve this by learning functional groups of directions and leveraging sparsity, where only a subset of directions carries data at any given time.

Key takeaway

For AI Engineers designing or debugging Transformer architectures, understanding how models use feature directions and projections is crucial. This mechanism allows models to store more features than their embedding dimensions, but imperfect orthogonality can introduce interference. You should consider the implications of dimensionality and sparsity in feature representation to optimize model capacity and minimize data corruption during updates.

Key insights

Transformers use projections and feature directions to store and modify data, enabling more features than dimensions.

Principles

• Orthogonal directions minimize interference in data storage.
• High dimensionality reduces interference when packing features.
• Sparsity is key to avoiding catastrophic interference.

Method

Data is encoded by multiplying input values with a projection matrix, and decoded by multiplying the encoded vector with the same matrix. Adjustments are encoded as residual vectors and added to the encoded data.

In practice

• Use projection matrices to encode and decode feature values.
• Design systems with high dimensionality for dense feature packing.
• Employ sparse activation to manage interference in high-dimensional spaces.

Topics

• Transformers
• Feature Directions
• High-Dimensional Spaces
• Projection Matrices
• LLM Architecture

Best for: AI Engineer, Machine Learning Engineer, AI Student

Related on AIssential

• Transformers Can Overcome the Curse of Dimensionality: A Theoretical Study from an Approximation Perspective — Transformers can overcome the curse of dimensionality, demonstrating strong expressive capabilities for function approximation.
• The Geometry of Representational Failures in Vision Language Models — VLM multi-object failures stem from geometric interference between concept vectors in a shared latent space.
• Why Transformer Representations Tend to Live on Hyperspheres — Transformer representations cluster on hyperspheres due to normalization and high-dimensional geometry, making direction paramount.
• Different Layers, Different Manifolds: Module-Wise Weight-Space Geometry in Transformer Optimization — Transformer optimization benefits from module-specific manifold constraints, with Stiefel for attention and DGram for MLP layers.
• Relational Rank Geometry in Transformers: Detecting and Steering Hidden-State Relation Frames — Transformers encode multi-argument relations as detectable and steerable rank-indexed hidden-state geometries.
• Learning to Recall with Transformers Beyond Orthogonal Embeddings — Transformer storage capacity scales multiplicatively with sample size, embedding dimension, and sequence length under random embeddings.

Open in AIssential →

Editorial summary, takeaway, and curation by AIssential. Original article published by Chris McCormick.