TensorView

Rank-3 Model

Formula / Preset:

Palette:

\(f(t, y, x) =\) Vars: t, y, x, T, H, W, PI, sin, cos, exp, sqrt

Hover or click any 3D voxel or dataset cell to inspect

Time Dimension (\(t\)): t = 0 / 3 (0.00s)

Explode Slices: 0.45

Ghost Inactive Slices

Isolate Current Slice

Configure Tensor Shape \(\mathcal{T} \in \mathbb{R}^{T \times H \times W}\)

Customize the rank-3 dimensions. Mode-1 is the Time/Depth axis (\(T\)), Mode-2 is Rows/Height (\(H\)), and Mode-3 is Columns/Width (\(W\)).

Time Axis (\(T\))

Time frames

Height Axis (\(H\))

Matrix rows

Width Axis (\(W\))

Matrix cols

Total Elements: 64 floating-point values.

📖 Understanding Rank-3 Tensors in Mathematics & ML

1. What is a Rank-3 Tensor?

In multilinear algebra, a rank-3 (or 3rd-order) tensor \(\mathcal{T} \in \mathbb{R}^{T \times H \times W}\) is a 3-dimensional array of numbers addressed by three indices:

\(\mathcal{T}_{t, i, j} \quad \text{where } t \in [0, T-1], \; i \in [0, H-1], \; j \in [0, W-1]\)

A scalar is rank-0, a vector is rank-1, a matrix is rank-2, and a block/cube of numbers is rank-3.

2. The Three Canonical Slices

Slicing a rank-3 tensor by holding one index constant produces a 2D matrix:

Frontal Slices: \(\mathcal{T}_{t, :, :}\) (holding time/depth fixed) — video frames or matrix snapshots over time.
Horizontal Slices: \(\mathcal{T}_{:, i, :}\) (holding row fixed) — cross-sections across all time.
Lateral Slices: \(\mathcal{T}_{:, :, j}\) (holding col fixed) — vertical planes through the tensor.

3. Real-World Applications

Computer Vision & Video: \((T \text{ seconds/frames}, H \text{ height}, W \text{ width})\).
Deep Learning Batches: \((B \text{ batch size}, S \text{ sequence length}, D \text{ embedding dimension})\).
Spatiotemporal Physics: Wave equations, fluid dynamics, and temperature diffusion over 2D surfaces in time.
Neuroimaging & Medicine: 3D MRI / CT volumetric voxel scans.

4. Tensor Decompositions (CP & Tucker)

Just as matrices have Singular Value Decomposition (SVD), rank-3 tensors can be decomposed into sums of rank-1 outer products:

\(\mathcal{T} \approx \sum_{r=1}^R \mathbf{a}_r \circ \mathbf{b}_r \circ \mathbf{c}_r\)

Use the Explode Slices slider in TensorView to separate the layers and inspect how rank-1 components combine!