Tensor and Contraction
A tensor maps each named index in its shape to a value.
A shape is an unordered set of named axes, so { N = 4, C = 3 } and { C = 3, N = 4 } identify the same tensor.
An ordered representation of that shape behaves like a familiar multidimensional array.
A tensor index supplies one value for every axis, so { N = 4, C = 3 } includes indices such as { N: 0, C: 0 } and { N: 0, C: 1 }.
| Tensor | Dimension | Example | Named shape |
|---|---|---|---|
| Scalar | 0D | 5.2 | {} |
| Vector | 1D | [1, 2, 3] | { I = 3 } |
| Matrix | 2D | a 2 × 4 grid | { I = 2, J = 4 } |
| Image batch | 4D | four RGB images | { N = 4, C = 3, H = 256, W = 512 } |
Tensor contraction multiplies two inputs elementwise and reduces every shared axis that is absent from the output. Every contraction consists of broadcast, multiply, and reduce steps.
| Operation | Einsum | Broadcast | Multiply | Reduce |
|---|---|---|---|---|
| Dot product | (I, I \rightarrow 1) | None. | (x_i y_i) | (\sum_i x_i y_i) |
| GEMV | (IJ, J \rightarrow I) | (x) across (I). | (A_{ij} x_j) | (y_i = \sum_j A_{ij} x_j) |
| GEMM | (IK, KJ \rightarrow IJ) | (A) across (J) and (B) across (I). | (A_{ik} B_{kj}) | (C_{ij} = \sum_k A_{ik} B_{kj}) |
This page owns the introductory math only. Mapping, movement, and engine contracts remain in their respective reference chapters.