Vector spaces define the type of vectors used by each VSA model. They determine how random vectors are generated and how similarity is measured.

Overview

When Spaces Matter

Most users don't interact with spaces directly—VSA.create() automatically selects the correct space for each model. However, understanding spaces helps when:

• Choosing a model for specific hardware constraints
• Debugging similarity computations
• Implementing custom models

Space Details

Bipolar Space

Values: Each dimension is -1 or +1.

Random Generation: Uniform choice of {-1, +1} per dimension.

Similarity: Cosine similarity: $\(\text{sim}(a, b) = \frac{a \cdot b}{||a|| \cdot ||b||}\)$

Used by: MAP, HRR

from holovec import VSA

model = VSA.create('MAP', dim=1000)
v = model.random()
# v contains only -1 and +1 values
print(set(v.tolist()))  # {-1.0, 1.0}

Complex Space

Values: Unit phasors e^iθ where θ ∈ [0, 2π).

Random Generation: Uniform random phase per dimension.

Similarity: Real part of complex cosine: $\(\text{sim}(a, b) = \text{Re}\left(\frac{a \cdot \bar{b}}{||a|| \cdot ||b||}\right)\)$

Used by: FHRR

model = VSA.create('FHRR', dim=1000)
v = model.random()
# v contains complex numbers on the unit circle
print(abs(v[0]))  # ~1.0 (unit magnitude)

Binary Space

Values: Each dimension is 0 or 1.

Random Generation: Bernoulli(0.5) per dimension.

Similarity: Normalized Hamming distance: $\(\text{sim}(a, b) = 1 - \frac{\text{hamming}(a, b)}{d}\)$

Used by: BSC

model = VSA.create('BSC', dim=1000)
v = model.random()
print(set(v.tolist()))  # {0.0, 1.0}

Sparse Space

Values: Sparse binary vectors with fixed number of 1s.

Random Generation: k random positions set to 1, rest are 0.

Similarity: Overlap coefficient: $\(\text{sim}(a, b) = \frac{|a \cap b|}{k}\)$

Used by: BSDC

Parameters: - dim: Total dimensions - sparsity: Number of active bits (k)

model = VSA.create('BSDC', dim=10000, sparsity=100)
v = model.random()
print(v.sum())  # ~100 active bits

Sparse Segment Space

Values: Segment-structured sparse binary vectors.

Random Generation: Vector divided into segments, exactly one 1 per segment.

Similarity: Segment overlap.

Used by: BSDC-SEG

Parameters: - dim: Total dimensions - segments: Number of segments

model = VSA.create('BSDC_SEG', dim=10000, segments=100)
v = model.random()
# Each segment of 100 dimensions has exactly one 1

Matrix Space

Values: Unitary matrices (U†U = I).

Random Generation: Random orthogonal/unitary matrix.

Similarity: Normalized trace: $\(\text{sim}(A, B) = \frac{\text{Re}(\text{tr}(A^\dagger B))}{d}\)$

Used by: GHRR, VTB

Similarity Properties

All spaces guarantee these properties:

Choosing a Space (via Model)

You don't choose spaces directly—you choose a model, and the space follows:

See Also

• Models — Model comparison
• Architecture Overview — How spaces fit in
• Core Concepts — Mathematical foundations