Multiply-Add-Permute

MAP uses element-wise multiplication for binding, which is self-inverse. It's one of the simplest VSA models, making it ideal for hardware deployment and neuromorphic computing.

Properties

When to Use

• Hardware deployment (FPGA, ASIC)
• Neuromorphic computing
• Real-time systems with strict latency requirements
• When self-inverse property simplifies logic
• Resource-constrained environments

Theory

Vector Space

MAP vectors use bipolar values:

Each dimension is independently sampled with equal probability.

Binding Operation

Binding is element-wise multiplication (Hadamard product):

For bipolar vectors, this is equivalent to XOR: - (+1)(+1) = +1 - (+1)(-1) = -1 - (-1)(+1) = -1 - (-1)(-1) = +1

Self-Inverse Property

Binding is its own inverse:

Since \(b_i \cdot b_i = 1\) for all bipolar values.

Bundling Operation

Bundling sums vectors and applies majority vote:

Ties are broken by the first vector's value.

Capacity Analysis

MAP has moderate capacity due to its ~0.5 noise floor (random bipolar similarity):

Code Examples

Basic Usage

from holovec import VSA

# Create MAP model
model = VSA.create('MAP', dim=10000)

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

# Bind (same as unbind for MAP)
c = model.bind(a, b)

# Unbind using bind (self-inverse)
a_recovered = model.bind(c, b)  # or model.unbind(c, b)

print(model.similarity(a, a_recovered))  # 1.0 (exact)

Self-Inverse Demonstration

# Binding is self-inverse
c = model.bind(a, b)

# Both methods recover a:
a_via_bind = model.bind(c, b)
a_via_unbind = model.unbind(c, b)

# They're identical
print(model.similarity(a_via_bind, a_via_unbind))  # 1.0

Hardware-Friendly Operations

import numpy as np

# Convert bipolar to binary for hardware
a_bipolar = model.random()
a_binary = (a_bipolar + 1) / 2  # {-1,+1} → {0,1}

# XOR binding in binary
def xor_bind(a_bin, b_bin):
    return np.logical_xor(a_bin, b_bin).astype(float)

Comparison with Similar Models

Hardware Implementation

MAP is ideal for hardware because:

1. Binary operations: Bipolar can be stored as single bits
2. XOR binding: Single gate per dimension
3. Majority bundling: Popcount and threshold
4. No multiply-accumulate: Pure logic operations

FPGA Resource Estimate (10,000 dim)

References

• Kanerva, P. (2009). Hyperdimensional Computing: An Introduction
• Schlegel, K., et al. (2022). A comparison of vector symbolic architectures

See Also

• Models Overview — Compare all models
• Model-BSC — Binary version
• Performance — Benchmarks