Generalized Holographic Reduced Representations

GHRR extends FHRR from scalar phasors to m×m unitary matrices, enabling non-commutative binding for compositional structures like trees, graphs, and nested relationships.

Properties

When to Use

• Order-sensitive relationships (A→B ≠ B→A)
• Hierarchical structures (parent-child, cause-effect)
• Knowledge graphs with directed edges
• When binding order must be recoverable
• State-of-the-art applications (2024)

Theory

Vector Space

GHRR vectors are arrays of m×m unitary matrices:

where each \(U_j^\dagger U_j = I\) (unitary property).

Binding Operation

Binding is element-wise matrix multiplication:

Matrix multiplication is non-commutative: \(A_j B_j \neq B_j A_j\) in general.

Unbinding Operation

Unbinding uses conjugate transpose:

This provides exact recovery: unbind(bind(a, b), b) = a.

Diagonality Control

GHRR's non-commutativity is controlled by diagonality parameter:

When m=1, GHRR reduces to FHRR.

Bundling Operation

Bundling sums matrices and projects back to unitary via polar decomposition (SVD):

Capacity Analysis

GHRR has similar capacity to FHRR but trades scalar dimensions for matrix richness:

*Note: GHRR uses effective dimensions = dim × m² (e.g., dim=100, m=3 → 900 effective). Capacity is measured per scalar dimension.

Code Examples

Basic Usage

from holovec import VSA

# Create GHRR model with 3×3 matrices
model = VSA.create('GHRR', dim=100, matrix_size=3)

# Generate random hypervectors
a = model.random(seed=1)
b = model.random(seed=2)

# Test non-commutativity
ab = model.bind(a, b)
ba = model.bind(b, a)
print(model.similarity(ab, ba))  # ~0.0 (very different)

# Exact recovery still works
c = model.bind(a, b)
a_recovered = model.unbind(c, b)
print(model.similarity(a, a_recovered))  # 1.0

Controlling Diagonality

# Maximally non-commutative
model_nc = VSA.create('GHRR', dim=100, matrix_size=3, diagonality=0.0)

# Commutative (FHRR-like)
model_c = VSA.create('GHRR', dim=100, matrix_size=3, diagonality=1.0)

# Measure commutativity
a, b = model_nc.random(seed=1), model_nc.random(seed=2)
print(model_nc.test_non_commutativity(a, b))  # ~0.0

Directional Relationships

# Encode "A causes B" vs "B causes A"
cause = model.random(seed=1)
effect = model.random(seed=2)

# Order encodes direction
a_causes_b = model.bind(cause, effect)  # A → B
b_causes_a = model.bind(effect, cause)  # B → A

# These are different!
print(model.similarity(a_causes_b, b_causes_a))  # ~0.0

Comparison with Similar Models

Memory and Performance

Typical usage: m=3 balances expressiveness and efficiency.

References

• Yeung, W., Zou, A., & Imani, F. (2024). Generalized Holographic Reduced Representations. arXiv:2405.09689
• Plate, T. A. (2003). Holographic Reduced Representations

See Also

• Models Overview — Compare all models
• Model-FHRR — Scalar version
• Model-VTB — Alternative non-commutative model