AGISystem2 Strategies vs. HRR

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This document analyzes the relationship between AGISystem2's HDC strategies and Holographic Reduced Representations (HRR) as defined by Tony Plate (1995). Two of our strategies—Sparse Polynomial and Metric-Affine—are original contributions developed for this system. Metric-Affine Elastic (EMA) is an extension of Metric-Affine focused on large superpositions. EXACT is a lossless “bitset polynomial” exploration used as an upper bound for retrievability and decoding behavior.

Note on EMA: Metric-Affine Elastic keeps the same XOR bind and L1 similarity as Metric-Affine, but changes bundling (chunked bundles; bounded depth). Geometry (D bytes) is configurable in both strategies; EMA does not auto-grow D during a session in the current runtime. See the EMA theory page for details.
Note on EXACT: EXACT is intentionally not “HRR-like”. It is a lossless, session-local representation with a quotient-like UNBIND (UNBIND is not required to equal BIND). See the EXACT theory page and DS25.

What is HRR?

Holographic Reduced Representations (HRR), introduced by Tony Plate in his 1995 thesis and 2003 book, is a specific implementation of holographic computing with these defining characteristics:

Property Classic HRR (Plate)
Representation Real-valued vectors (continuous)
Binding Operation Circular convolution: c[k] = Σᵢ a[i] × b[(k-i) mod n]
Unbinding Circular correlation: a ≈ c ⊛ b⁻¹
Bundling Vector addition (superposition)
Similarity Dot product / Cosine similarity
Key Property Distributed, holographic encoding

HRR vs. VSA Family

HRR is one member of the broader Vector Symbolic Architectures (VSA) family, which includes:

All VSA members share the holographic principle (distributed representation, compositional binding, content-addressable retrieval) but differ in implementation details.

Strategy Analysis

1. Dense-Binary Standard VSA

Assessment: VSA Member, Not Classic HRR

Dense-Binary is a standard implementation of Binary Spatter Codes (BSC), following Pentti Kanerva's Hyperdimensional Computing paradigm. It shares the holographic principle with HRR but uses different mathematics.

Property HRR Dense-Binary Match?
Values Real (continuous) Binary {0, 1} ❌ No
Binding Circular convolution XOR (component-wise) ❌ No
Bundling Vector addition Majority vote ⚠️ Analog
Similarity Dot product Hamming distance ⚠️ Analog
XOR cancellation in binding Approximate Exact (XOR) ✅ Better
Holographic property Yes Yes ✅ Yes
Dense-Binary is NOT HRR, but IS a valid VSA with holographic properties.

2. Sparse Polynomial (SPHDC) ORIGINAL

Assessment: Novel Paradigm, NOT HRR

Sparse Polynomial HDC is a fundamentally different paradigm that we developed specifically for AGISystem2. It does NOT follow the HRR model and introduces several novel concepts not found in existing VSA literature.

Property HRR Sparse Polynomial Match?
Representation Fixed-length real vector Set of k integers ❌ Fundamentally different
Binding Circular convolution O(n log n) Cartesian XOR O(k²) ❌ Novel operation
Bundling Vector addition Set union + Min-Hash ❌ Set-theoretic
Similarity Dot product Jaccard index ❌ Different metric
Sparsification Not applicable Min-Hash sampling ❌ Novel
Memory model Dense vector Sparse set (k elements) ❌ Sparse by design
Holographic property Full distribution Partial (set overlap) ⚠️ Limited

What Makes SPHDC Novel?

  1. Set-Based Representation: Instead of fixed-length vectors, concepts are sets of k integers from a virtually infinite space (2⁶⁴). This is fundamentally different from all existing VSA approaches.
  2. Cartesian XOR Binding: The binding operation computes the Cartesian product of two sets with XOR: A BIND B = {a xor b | a ∈ A, b ∈ B}. This creates |A| × |B| elements, then sparsifies back to k.
  3. Min-Hash Sparsification: We apply locality-sensitive hashing (Min-Hash) to maintain constant-size representations while preserving similarity relationships. This combines HDC with streaming algorithms.
  4. Jaccard Similarity: Using set intersection/union for similarity is natural for set-based representations but novel in the context of symbolic reasoning.
"SPHDC represents a departure from the continuous-vector paradigm of HRR. It asks: what if we represent concepts as fingerprints (sets of hashes) rather than points in a continuous space? The result is a system optimized for symbolic manipulation with bounded memory."
SPHDC is NOT HRR. It is an original contribution combining HDC principles with set theory and streaming algorithms.

3. Metric-Affine ORIGINAL

Assessment: HRR-Inspired Hybrid

Metric-Affine is a novel hybrid that combines HRR's continuous-value philosophy with Binary HDC's XOR binding. It's closer to HRR than Dense-Binary but still distinct.

Property HRR Metric-Affine Match?
Values Real (continuous) Byte [0-255] (quasi-continuous) ✅ Similar spirit
Binding Circular convolution Byte-wise XOR ❌ Different
Bundling Vector addition Arithmetic mean ✅ Affine operation
Similarity Dot product L₁ (Manhattan) ⚠️ Both metric
Holographic property Full distribution Full distribution ✅ Yes
XOR cancellation in binding Approximate Exact (XOR) ✅ Better
Interpolation Natural (real) Natural (byte mean) ✅ Yes

What Makes Metric-Affine Novel?

  1. Fuzzy-Boolean Hybrid: It bridges the gap between discrete (binary) and continuous (HRR) representations by using byte values that allow 256 gradations per channel.
  2. XOR on Continuous Values: Unlike HRR's circular convolution, Metric-Affine uses XOR—but on multi-bit bytes, giving exact XOR cancellation while retaining continuous bundling.
  3. Affine Bundling: The arithmetic mean creates smooth interpolations between concepts (the "affine" in the name). This is conceptually similar to HRR's vector addition but normalized.
  4. Compact Holography: At only 32 bytes per vector (vs 4KB for Dense-Binary or variable for HRR), it's the most memory-efficient holographic representation we know of.
  5. Shifted Baseline: The ~0.67 random similarity baseline (vs 0.5 for binary) requires different threshold tuning but provides finer discrimination in the "related" range.
"Metric-Affine asks: can we have the algebraic simplicity of XOR binding (exact cancellation) while retaining the smooth bundling and interpolation of continuous representations? The answer is yes, by operating on bytes rather than bits."
Metric-Affine is a novel HRR-inspired hybrid combining continuous values with discrete XOR binding.

Summary: Are They HRR?

Strategy Is it HRR? Is it VSA? Is it Novel? Classification
Dense-Binary ❌ No ✅ Yes (BSC) ❌ No Standard HDC
Sparse Polynomial ❌ No ⚠️ Loosely ✅ Yes Original paradigm
Metric-Affine ⚠️ Inspired ✅ Yes ✅ Yes Novel hybrid

Theoretical Contributions

AGISystem2's Original Contributions to HDC/VSA:

  1. Set-Based HDC (SPHDC): First system to use finite integer sets with Cartesian XOR binding and Min-Hash sparsification as a complete HDC strategy.
  2. Fuzzy-Boolean HDC (Metric-Affine): First system to combine exact XOR binding with continuous-value bundling in a compact 32-byte representation.
  3. Multi-Strategy Architecture: A unified interface allowing runtime selection between fundamentally different HDC paradigms while maintaining identical reasoning semantics.

References

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