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Holographic Reduced Representation
archive 2025-07-28 Description, source and code snippet are the archive's method entry.
Holographic Reduced Representations are a simple mechanism to represent an associative array of key-value pairs in a fixed-size vector. Each individual key-value pair is the same size as the entire associative array; the array is represented by the sum of the pairs. Concretely, consider a complex vector key r = (aᵣ[1]e^(iφᵣ[1]), aᵣ[2]e^(iφᵣ[2]), …), which is the same size as the complex vector value x. The pair is "bound" together by element-wise complex multiplication, which multiplies the moduli and adds the phases of the elements:
y = r ⊗x
y = (aᵣ[1]aₓ[1]e^(i(φᵣ[1]+φₓ[1])), aᵣ[2]aₓ[2]e^(i(φᵣ[2]+φₓ[2])), …)
Given keys r₁, r₂, r₃ and input vectors x₁, x₂, x₃, the associative array is:
c = r₁ ⊗x₁ + r₂ ⊗x₂ + r₃ ⊗x₃
where we call c a memory trace. Define the key inverse:
r⁻¹ = (aᵣ[1]⁻¹e^(-iφᵣ[1]), aᵣ[2]⁻¹e^(-iφᵣ[2]), …)
To retrieve the item associated with key rₖ, we multiply the memory trace element-wise by the vector r⁻¹ₖ. For example:
r₂⁻¹ ⊗c = r₂⁻¹ ⊗(r₁ ⊗x₁ + r₂ ⊗x₂ + r₃ ⊗x₃)
r₂⁻¹ ⊗c = x₂ + r⁻¹₂ ⊗(r₁ ⊗x₁ + r₃ ⊗x3)
r₂⁻¹ ⊗c = x₂ + noise
The product is exactly x₂ together with a noise term. If the phases of the elements of the key vector are randomly distributed, the noise term has zero mean.
Source: Associative LSTMs
Papers archive 2025-07-28
4 shown of 4, newest first. Repository counts are the archive's code-links table. A Syntology line states what Syntology ran from that paper's harvested code; it is per sample and not a correctness claim.
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Improved Cleanup and Decoding of Fractional Power Encodings 30 Nov 2024 · 0 repositories · arXiv:2412.00488
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Audio Fingerprinting with Holographic Reduced Representations 19 Jun 2024 · 0 repositories · arXiv:2406.13139
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Residual and Attentional Architectures for Vector-Symbols 18 Jul 2022 · 0 repositories · arXiv:2207.08953
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Associative Long Short-Term Memory 9 Feb 2016 · 3 repositories · arXiv:1602.03032
Tasks archive 2025-07-28
5 tasks the archive attaches to papers tagged with this method, by distinct papers. A task without a page in the catalog is plain text.
| Task | Papers |
|---|---|
| Image Classification | 1 |
| Memorization | 1 |
| Position | 1 |
| Retrieval | 1 |
| image-classification | 1 |
Usage over time archive 2025-07-28
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Categories archive 2025-07-28
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