Projection
A projection is a mapping or operation that produces a selected or derived representation of an input, often by retaining specified components or mapping it to a subspace.
Math and Architectures of Deep Learning
A transformation that maps points from a higher-dimensional space to a lower-dimensional one by discarding coordinates or components.
- How it is used
- Reduces data representations by retaining variation near a lower-dimensional subspace.
- How deep it goes
- Derives least-squares projection through residual orthogonality and develops PCA, scalar components, and topic-space projections.
- Examples
- Uses column-space perpendiculars, nearly planar 3D data, narrow 2D point clouds, and document-term matrices.
- What you can do after
- Evaluate/understand how projections support least-squares fitting, dimensionality reduction, compression, and topic representations.
Read how Math and Architectures of Deep Learning explains projection →
Natural Language Processing in Action, Second Edition
A projection is the mapping of a vector onto a specified line or axis to obtain its component along that direction.
- How it is used
- Uses vector projections to reduce feature vectors to classification scores and to construct attention representations.
- How deep it goes
- Defines projection and demonstrates its computation, while deferring the underlying LDA mathematics and focusing attention on operational linear algebra.
- Examples
- Demonstrates centroid-axis toxicity scores for comments and query, key, and value transformations in self-attention.
- What you can do after
- Apply it via NumPy dot products to score vectors, or understand how attention creates query, key, and value representations.
Read how Natural Language Processing in Action, Second Edition explains projection →
Geometry for Programmers
A mapping that sends a point or vector to a corresponding point or vector on a specified geometric object or subspace.
- How it is used
- Uses projection to solve linear systems iteratively and compute geometric distances for lines, segments, planes, triangles, and rectangles.
- How deep it goes
- Defines perpendicular closest-point projection, connects it to dot products, and gives branching distance procedures without pursuing numerical-method internals.
- Examples
- Uses intersecting lines, vector projections, triangle meshes, segments, and a two-dimensional rectangle signed-distance function.
- What you can do after
- Apply projection to classify segment locations, compute point-to-object distances, and implement simple signed-distance geometry.
The AI Pocket Book
A linear transformation that maps vectors into another space, used here as part of transformer attention computations.
- How it is used
- Uses projections to reshape embeddings, produce token probabilities, and connect representations across modalities.
- How deep it goes
- Defines matrix-based dimensionality changes and gives applications, but stops short of deriving the mathematics or interpreting learned dimensions.
- Examples
- Uses animal-topic dimensionality reduction, a PyTorch linear layer, token probabilities, and image-to-text embedding alignment.
- What you can do after
- Apply it via PyTorch by placing a linear projection after an embedding operation.
Advanced Algorithms and Data Structures
A projection maps points onto a selected axis or lower-dimensional subspace by retaining their coordinates in that direction.
- How it is used
- Uses coordinate reduction to screen geometric intersections, split spatial-search nodes, and visualize multidimensional optimization.
- How deep it goes
- Develops projection from coordinate intuition to sorting, variance-based partitioning, and one-dimensional intersection tests, without formal projection matrices.
- Examples
- Shows projected segments, eight-point tree splits along the y axis, and two-dimensional views of higher-dimensional optimization functions.
- What you can do after
- Apply it via coordinate comparisons for intersection screening and via sorted projected values for variance-minimizing spatial splits.
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Fabulous Adventures in Data Structures and Algorithms
A projection is a mapping or function that selects or derives specified components or values from a structured input.
- How it is used
- Transforms distribution outcomes, joint-roll pairs, and Cartesian-product tuples into values useful to the surrounding computation.
- How deep it goes
- Explains mapping, support collapse, grouped weight calculation, greatest-common-divisor reduction, and composition with joint distributions.
- Examples
- Uses colored die faces, roll-with-advantage maxima, Cartesian-product results, and posterior calculations from filtered joint distributions.
- What you can do after
- Build projected discrete distributions and combine projections with joint distributions to model categories, maxima, minima, and sums.
Read how Fabulous Adventures in Data Structures and Algorithms explains projection →
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