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Creating Intelligence: A Computational Foundation for AGI

T0 review · 3 major / 0 minor · reviewed 2026-07-01 · grok-4.3

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Pith's one-line read Subset pattern matching in sparse binary networks unifies associative learning and serves as the core mechanism of cognition in both the cerebellum and neocortex.

desk verdict The paper claims a new set-theoretic foundation for AGI via subset pattern matching and topological plasticity but supplies no mechanisms, derivations, or evidence to back any of it up. read the letter →

arxiv 2606.31819 v1 pith:QH42XOFA submitted 2026-06-30 cs.AI

Peter Overmann This is my paper · ORCID
classification cs.AI
keywords associativememorysubsetpatternmatchingsparsebinaryrepresentationshyperdimensionalcomputingtopologicalplasticityneuroanatomymappingconstant-timeretrievalAGIarchitecture
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper advances a computational theory of mind that replaces continuous weights and matrix operations with discrete sets and sparse binary representations drawn from set theory and hyperdimensional computing. It argues that associative memory arises automatically from network topologies that include a combinatorially expanded hidden layer, with learning occurring through changes in network topology rather than adjustments to scalar weights. Under this view, a single algorithm based on subset pattern matching and exact nearest-neighbor search handles both auto-associative and hetero-associative retrieval in constant time, directly linking perceptual sparse distributed representations to symbolic sparse holographic representations. The framework maps these operations onto neuroanatomy by proposing that the cerebellum and neocortex each run variants of the same subset-matching procedure, positioning it as the fundamental engine of cognition. Because the operations are purely discrete, the architecture is claimed to map directly onto in-memory hardware without continuous bottlenecks, opening a route to synthetic intelligence at human-level energy efficiency.

What carries the argument

Subset pattern matching and exact nearest-neighbor search performed on sparse binary representations within combinatorially expanded hidden layers.

What would settle it

An explicit construction of a combinatorially expanded sparse binary network that fails to retrieve stored patterns via subset matching, or a direct anatomical or physiological measurement showing that cerebellar or neocortical circuits do not perform the described pattern-matching operations.

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Extended reading notes

Core claim

The paper claims that associative memory and cognition reduce to information retrieval via subset pattern matching and exact nearest-neighbor search performed on sparse binary sets; that this single mechanism unifies auto-associative and hetero-associative learning; that both the cerebellum and the neocortex implement variants of it; and that the required network topologies produce the behavior through combinatorial expansion of a hidden layer combined with topological plasticity rather than weight tuning.

Load-bearing premise

Associative memory and learning emerge naturally once a network contains a combinatorially expanded hidden layer and plasticity acts on topology rather than on scalar weights.

Editorial extensions

If this is right

  • Auto-associative and hetero-associative learning are performed by one algorithm rather than separate mechanisms.
  • The system retrieves information in constant time without iterative matrix operations.
  • Perceptual data and symbols are bridged directly through sparse distributed and holographic representations.
  • The architecture maps onto in-memory hardware implementations that avoid continuous arithmetic.
  • Cognition in both the cerebellum and neocortex reduces to variants of the same subset-matching procedure.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Hardware realizations could be tested by measuring energy per inference against conventional neural-network accelerators on the same associative-retrieval tasks.
  • If the topological-plasticity rule is made explicit, one could simulate small networks to check whether combinatorial expansion alone suffices for stable memory without additional regularization.
  • The proposed unification suggests that disorders affecting pattern completion might be modeled as disruptions in subset-matching capacity rather than in weight matrices.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 0 minor

Summary. The paper introduces a computational theory of mind based on set theory and hyperdimensional computing using sparse binary set representations instead of continuous weights. It claims that associative memory emerges naturally from network topologies with a combinatorially expanded hidden layer via topological plasticity (rather than scalar weight updates), that this unifies auto- and hetero-associative learning under a single algorithm of subset pattern matching plus exact nearest-neighbor search with constant-time complexity, that the framework bridges perceptual sparse distributed representations and symbolic sparse holographic representations, and that both the cerebellum and neocortex implement variants of this algorithm, enabling direct translation to efficient in-memory hardware for AGI.

Significance. If the core claims were substantiated with explicit topology definitions, plasticity rules, and derivations, the work would offer a discrete, biologically mapped alternative to matrix-based neural networks with potential advantages in energy efficiency and unification of associative memory types. The absence of any such mechanisms, proofs, or examples in the manuscript prevents assessment of whether these properties actually hold or reduce to standard operations.

major comments (3)
  1. Abstract: The central claim that 'associative memory emerges naturally from network topologies featuring a combinatorially expanded hidden layer' and that 'learning is driven by topological plasticity' is asserted without any definition of the topology, the plasticity rule, a derivation showing emergence, or an example demonstrating the property; this makes the unification claim unevaluable.
  2. Abstract: The assertion that the architecture operates with 'constant-time complexity' via 'subset pattern matching and exact nearest-neighbor search' and 'translates directly into in-memory hardware' is presented without any formal definition of the matching procedure, complexity analysis, or hardware mapping; no section supplies the required algorithm or proof.
  3. Abstract: The neuroanatomical mapping stating that 'both the cerebellum and the neocortex implement variants of this algorithm' is offered as a direct consequence but without any supporting correspondence, circuit-level description, or reference to specific neuroanatomical data that would ground the claim.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thoughtful review and for highlighting areas where the manuscript's high-level presentation requires additional formalization to allow proper evaluation. The work is a conceptual proposal for a set-based framework, and we agree that the abstract and main text would benefit from explicit definitions, algorithms, and examples. We address each major comment below and will revise accordingly.

read point-by-point responses

  1. Referee: Abstract: The central claim that 'associative memory emerges naturally from network topologies featuring a combinatorially expanded hidden layer' and that 'learning is driven by topological plasticity' is asserted without any definition of the topology, the plasticity rule, a derivation showing emergence, or an example demonstrating the property; this makes the unification claim unevaluable.

    Authors: We acknowledge that the abstract is highly condensed and does not supply the requested definitions or derivations. The manuscript frames the topology as a bipartite graph between input and a combinatorially expanded hidden layer using sparse binary set representations, with topological plasticity implemented as dynamic edge addition/removal driven by subset co-occurrence. The unification of auto- and hetero-associative memory follows from the same subset-matching operation. To make these claims evaluable, the revised manuscript will add a dedicated section containing (1) a formal definition of the topology and plasticity rule, (2) a short derivation showing emergence of associative recall, and (3) a concrete numerical example. revision: yes

  2. Referee: Abstract: The assertion that the architecture operates with 'constant-time complexity' via 'subset pattern matching and exact nearest-neighbor search' and 'translates directly into in-memory hardware' is presented without any formal definition of the matching procedure, complexity analysis, or hardware mapping; no section supplies the required algorithm or proof.

    Authors: The constant-time claim rests on representing items as sparse sets and performing exact subset matching via hash-table lookup, which is O(1) average-case with standard data structures; the hardware mapping targets content-addressable or in-memory compute fabrics that natively support set intersection. The current manuscript states these properties at a high level without pseudocode or analysis. In revision we will insert an algorithms subsection with (a) pseudocode for the matching procedure, (b) a complexity argument, and (c) a brief mapping to existing in-memory hardware primitives. revision: yes

  3. Referee: Abstract: The neuroanatomical mapping stating that 'both the cerebellum and the neocortex implement variants of this algorithm' is offered as a direct consequence but without any supporting correspondence, circuit-level description, or reference to specific neuroanatomical data that would ground the claim.

    Authors: The mapping is motivated by the known use of sparse distributed representations in both regions and by circuit motifs (e.g., parallel-fiber to Purkinje-cell connectivity) that can realize subset matching. The manuscript currently presents this as a high-level analogy without detailed circuit correspondences or citations. The revision will expand the neuroanatomy discussion with (1) explicit references to sparse-coding literature, (2) a circuit-level sketch for each structure, and (3) a table comparing algorithmic operations to known anatomical features. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No derivation chain or equations supplied; claims asserted without reductions to check.

full rationale

The provided abstract and context contain no equations, derivations, self-citations, or explicit mechanisms. The paper asserts that associative memory emerges from combinatorial topologies and unifies learning under subset pattern matching, but supplies neither the topology definition, plasticity rule, nor any formal steps that could reduce to inputs by construction. Because no load-bearing derivation exists in the text, none of the enumerated circularity patterns can be exhibited via quote and reduction. This is the normal case of an absent chain rather than a circular one.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

No specific free parameters, axioms, or invented entities can be identified from the abstract alone; the proposal references set theory and hyperdimensional computing but provides no details on what is assumed versus derived.

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Cite this review

Pith. "Pith review of Creating Intelligence: A Computational Foundation for AGI." pith.science (2026). https://pith.science/paper/QH42XOFA

@misc{pith2026260631819,
  author       = {Pith},
  title        = {Pith review of: Creating Intelligence: A Computational Foundation for AGI},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QH42XOFA}},
  note         = {Machine review of arXiv:2606.31819}
}
read the original abstract

This work introduces a new computational theory of mind grounded in set theory and hyperdimensional computing. Whereas traditional neural networks rely on continuous weights and matrix multiplication, this framework works with sparse binary data. It represents information as discrete sets, directly modeling biological neural population codes. I demonstrate that associative memory emerges naturally from network topologies featuring a combinatorially expanded hidden layer. Learning is driven by topological plasticity rather than scalar weight adjustments. This architecture unifies auto-associative and hetero-associative learning under a single core algorithm: information retrieval via subset pattern matching and exact nearest-neighbor search. Operating with constant-time complexity, these mechanisms bridge perceptual data (sparse distributed representations) and symbols (sparse holographic representations) without continuous bottlenecks. Mapping this framework to neuroanatomy, I propose that both the cerebellum and the neocortex implement variants of this algorithm, making subset pattern matching the fundamental engine of cognition. Because it relies on discrete logic rather than matrix arithmetic, this algorithm translates directly into in-memory hardware. This opens a new route toward synthetic intelligence with human-level energy efficiency.

Figures

Figures reproduced from arXiv: 2606.31819 by the authors.

Figure 1
Figure 1. A sparse set with 10 out of 2,000 possible elements. This image should convey the scale of the representational space defined by these small subsets. There are approximately 2.76×1026 unique subsets containing exactly 10 elements — exceeding any volume of information that we may want to encode. The tendency towards orthogonality of hypervectors aligns with subset combinatorics, particularly regarding the probability… view at source ↗
Figure 2
Figure 2. Hetero-associative topology with 6 inputs (bottom) and 7 outputs (top) which are fully connected to the hidden layer. This type of feed-forward topology, with its physically distinct input and output layers, is inherently hetero-associative. It requires a secondary incoming pathway for training the output layer. The input and output layers generally have different dimensions, representing information from distinct d… view at source ↗
Figure 3
Figure 3. Auto-associative topology with 6 input/output dimensions (blue). The 15-node latent space is shown in red. Both hetero-associative and auto-associative topologies are technically single-layer 20 [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figures from the paper (54 more)
Figure 4
Figure 4. Figure 4: Left: Recurrent network topology, storing autoassociations A → A. Right: Feed-forward network with two inputs for heteroassociations A → B. In a biological network, the hidden layer can manifest either as a massive population of neurons (instantly recognizable in the c…
Figure 5
Figure 5. Figure 5: SHR storage capacity as a function of memory density ρ for hyperparame￾ters N = 1, 000 and P = 10. Two complementary metrics quantify retrieval accuracy: the average overlap between stored and retrieved sets (number of correctly matched elements) and their average Hamm…
Figure 6
Figure 6. Figure 6: Retrieval accuracy as a function of memory density ρ. The memory cliff phenomenon occurs well beyond the nominal capacity mark ρ ∗ = 0.5. Auto-associative memory has a higher capacity than hetero-associative memory with 35 [PITH_FULL_IMAGE:figures/full_fig_p035_6.png]
Figure 7
Figure 7. Figure 7: Memory capacity to store associations with SHR population P = 10, as a function of dimension N. 36 [PITH_FULL_IMAGE:figures/full_fig_p036_7.png]
Figure 8
Figure 8. Figure 8: Retrieval accuracy with 3 random elements dropped, as a function of memory density ρ. Beyond handling missing data, the algorithm uses subset pattern matching to extract known information from inputs polluted with additive noise. It filters out elements that are not pa…
Figure 9
Figure 9. Figure 9: Retrieval accuracy with 15 random elements added, as a function of memory density ρ. The average overlap decreases as the memory density increases. Examining the output of individual queries, we observe that most outcomes are either perfect or empty. Zero results occur…
Figure 10
Figure 10. Figure 10: Probability of false-positive responses to random queries, as a function of memory density ρ. This simulation uses heteroassociations with N = 1, 000 and P = 10. 7 Computing With Sets In our computational framework, sets act as the foundational data structure — specif…
Figure 11
Figure 11. Figure 11: Expected size of a superposition as a function of the number of bundled SHRs, assuming N = 1, 000 and P = 10. As the bundle grows, its subsets increasingly resemble unrelated patterns. This phenomenon (crosstalk) can be quantified as follows: Consider an item memory f…
Figure 12
Figure 12. Figure 12: Usable bundle size at which false-positive matches are unlikely, as a function of dimensionality N. This simulation uses a fixed SHR population P = 10. The usable bundle size is around 10 for small memory configurations. This aligns with the structural limitations of …
Figure 13
Figure 13. Figure 13: A bidirectional associative memory, realized as an auto-associative memory operating on a bipartite representation. The two blocks may differ in dimensions, representing information from distinct domains. This asymmetric bidirectional setup emulates a pair of hetero-a…
Figure 14
Figure 14. Figure 14: Memory space parametrization of a bipartite auto-associative memory. When storing a collection of random autoassociations into this memory, the cross￾correlating division reaches its nominal capacity at memory density ρ ∗ = 0.5 first. The two self-correlating division…
Figure 15
Figure 15. Figure 15: Undirected hypergraph with 6 nodes and 5 hyperedges. Vertices are encoded, as usual, via random SHRs with uniform population. Apart from spurious overlaps, these representations are mostly dissimilar. Training the memory involves one write operation per edge: > a b c …
Figure 16
Figure 16. Figure 16: Average number of iterations needed to extract one token from a bundle, as a function of the number of bundled tokens. This is simulated with hyperparameters N = 2, 000 and P = 10. The results are consistent with Miller’s magical number seven, plus or minus two. 67 […
Figure 17
Figure 17. Figure 17: Auto-associative memory storing a temporal sequence. Recurrent permu￾tations create a pointer chain. This auto-associative system automatically learns the associations between the items and the pointer chain when receiving a sequence of items. To trigger inference, we…
Figure 18
Figure 18. Figure 18: Generic schematic symbol of a topological associative memory component. Signal flow components — any node that does not encapsulate a memory — are shown as circles [PITH_FULL_IMAGE:figures/full_fig_p080_18.png]
Figure 19
Figure 19. Figure 19: Generic schematic symbol of a signal flow component. Pathways, drawn as one-directional arrows, connect the circuit components. At the tick phase of each two-phase execution cycle, data propagates down the pathways to the receiving components where it is processed (th…
Figure 20
Figure 20. Figure 20: A pathway on which a permutation is applied to the signal. In this computational framework, pathways normally transport pure sets (population codes), but can be extended to carry multisets. This is by no means a departure from our fundamental premise of representing c…
Figure 21
Figure 21. Figure 21: Multiset pathway are displayed as thicker, colored lines. Pathways do not alter the dimensionality of the transported data. In the dataflow specification, this condition is enforced by assigning hyperparameters (dimension N together with the reference population P) no…
Figure 22
Figure 22. Figure 22: A memory component that processes a triple pathway via block coding. 81 [PITH_FULL_IMAGE:figures/full_fig_p081_22.png]
Figure 23
Figure 23. Figure 23: Schematic symbol of an input component. Output components form the circuit’s outbound gateway. Each output line corre￾sponds to one block of data. Postprocessing and decoding functions can be plugged into individual output components [PITH_FULL_IMAGE:figures/full_fig…
Figure 24
Figure 24. Figure 24: Schematic symbol of an output component. Unlike any other component types, inputs and outputs do not delay the flow of data. A circuit defines a stateful function f of m input blocks that returns n output blocks. f : (A1, . . . , Am) → (B1, . . . , Bn) Each function c…
Figure 25
Figure 25. Figure 25: A circuit, encapsulated as a modular circuit component. We can set up an embedded circuit to operate at a different speed from its enclosing circuit. For instance, a high-frequency perception processor may run at a higher clock speed than a symbolic module at the top …
Figure 26
Figure 26. Figure 26: Schematic symbol of a delay component. In this framework, the simplest recurrent network consists of a delay component with an external feedback line [PITH_FULL_IMAGE:figures/full_fig_p083_26.png]
Figure 27
Figure 27. Figure 27: A basic recurrent architecture. Here the incoming signal is bundled with the delayed feedback. This system requires stochastic subsampling at every step to prevent runaway signal populations. If this feedback system stops receiving inputs, it continues to emit the las…
Figure 28
Figure 28. Figure 28: Schematic symbol of a latch component which holds and repeats the last above-threshold signal. A dedicated k-winners-take-all (kWTA) component integrates incoming signals, bounding time (a sliding window of execution cycles), space (a population capacity), or a combin…
Figure 29
Figure 29. Figure 29: A kWTA component calculates the top-k signal over a temporally aggre￾gated multiset. Complementing stochastic subsampling as a model for “leaky” pathways or the lossy memory components, we model additive noise through a circuit component that emits a random pattern at…
Figure 30
Figure 30. Figure 30: Random noise generator. Auto-associative circuit components incorporate a topological memory instance with a shared input and output layer and an internal feedback loop. Inputs and outputs have the same overall dimensions and can be divided into non-overlapping partit…
Figure 31
Figure 31. Figure 31: Auto-associative circuit component based on the replacement update rule. Typically used as item memory. Completes the query pattern while removing noise [PITH_FULL_IMAGE:figures/full_fig_p085_31.png]
Figure 33
Figure 33. Figure 33: Auto-associative memory using the augmentation update rule forms the union of the query and the retrieved pattern [PITH_FULL_IMAGE:figures/full_fig_p085_33.png]
Figure 36
Figure 36. Figure 36: The difference update rule retains the symmetric difference between the query and the retrieved pattern. This gives rise to generative functionality similar to the complement update rule. Hetero-associative circuit components can be categorized according to their lear…
Figure 37
Figure 37. Figure 37: An associator circuit component that learns and retrieves heteroassocia￾tions of a context signal with a teaching signal [PITH_FULL_IMAGE:figures/full_fig_p086_37.png]
Figure 39
Figure 39. Figure 39: A heteroencoder circuit components works as a classifier, categorizing input patterns and generating a unique token for each detected cluster. 11 Recurrence and Temporal Memory 11.1 Recurrence As a defining property of cognitive architectures, recurrence is characteri…
Figure 40
Figure 40. Figure 40: A basic recurrent circuit. Rate limiting is applied by the circuit component. When processing a data stream, the state of this system is dominated by the most recent inputs, while older signals gradually fade. We can instead limit the feedback signal prior to merging …
Figure 41
Figure 41. Figure 41: A basic recurrent circuit with rate-limited feedback, indicated by a pathway labeled R. Applying a permutation to the feedback encodes each signal’s temporal position within the input sequence [PITH_FULL_IMAGE:figures/full_fig_p087_41.png]
Figure 42
Figure 42. Figure 42: Recurrent architecture with a permuted feedback line, indicated by a pathway labeled P. Here is a recurrent circuit formed by two delay components. The feedback signal is permuted and delayed by one execution cycle [PITH_FULL_IMAGE:figures/full_fig_p087_42.png]
Figure 43
Figure 43. Figure 43: Recurrent architecture with delayed feedback. 87 [PITH_FULL_IMAGE:figures/full_fig_p087_43.png]
Figure 44
Figure 44. Figure 44: Elementary reverberatory circuit. The forward signal (tagged with an exclamation mark) takes priority over the feedback. This circuit samples the input and holds it without loss of information until new input replaces the system’s state. The following diagram shows a …
Figure 45
Figure 45. Figure 45: A circuit with delayed reverberation, bouncing two stable states between delay nodes. A reverberatory circuit feeding into an item memory decomposes bundles into their constituents. An inhibitory feedback line removes the extracted items, one after another, from the r…
Figure 46
Figure 46. Figure 46: A reverberatory circuit combined with an item memory extracts items from a bundle. 11.3 Reservoir architectures Temporal memory emerges from the combination of recurrent architectures and topological associative memory. A system that learns sequences on the fly while …
Figure 47
Figure 47. Figure 47: Reservoir architecture, representing a temporal sequence memory. Taking this basic design as a starting point, we now build a more advanced reservoir architecture, shown in the following circuit diagram. 89 [PITH_FULL_IMAGE:figures/full_fig_p089_47.png]
Figure 48
Figure 48. Figure 48: A reservoir architecture incorporating bigram encoding and a dual-delay reservoir. Here, the reservoir at the center of the circuit is formed by a delayed feedback loop. This increases the reservoir’s temporal depth, allowing it to hold information over a longer time …
Figure 49
Figure 49. Figure 49: Sample digits from the MNIST Handwritten Digit Database The dataset (LeCun 1998) consists of 70,000 handwritten digits, rasterized as 28 ×28 pixel grayscale images. We map the flattened image data, after applying a 1-pixel-radius Gaussian blur, to SDRs via a random pr…
Figure 50
Figure 50. Figure 50: Hetero-associative memory component set up for supervised learning. We train this classifier with 60,000 images in a single pass. Testing with 10,000 images results in an accuracy score of about 96%, out of the box, without any optimization. This demonstrates that top…
Figure 51
Figure 51. Figure 51: Auto-associative memory component storing a visual scene graph. 92 [PITH_FULL_IMAGE:figures/full_fig_p092_51.png]
Figure 52
Figure 52. Figure 52: A scene with three colored features. The features in this image — colors, sizes, relative positions — are encoded as symbolic SHRs. Storing the entire scene in memory involves binding the features to three objects (encoded as latent symbols O1, O2, O3), and binding th…
Figure 53
Figure 53. Figure 53: Hetero-associative memory for symbolic information. Storing 1.9 million associations requires a memory of dimension N = 3, 000 if tokens are encoded with a population P = 10. The memory inherently merges associations that share the same key into a single association. …
Figure 54
Figure 54. Figure 54: A similarity matrix of 9 words, computed as the ARI between their semantic fields. A similar result would be obtained using a topological hetero-associative memory algorithm that returns the raw activated hidden layer as a multiset, skipping the kWTA reduction to spar…
Figure 55
Figure 55. Figure 55: An instance of a frame. Each column represents a slot, binding a top-level node (for instance “capital”) with a filler value (“Paris”) in the context of the specific frame instance (“France”). The first column plays a special role: It binds the class (“country”) to th…
Figure 56
Figure 56. Figure 56: Generative auto-associative memory circuit, using the complement update rule and an external feedback path. Once an initial signal from the memory’s input channel kicks off the process, the walk continues independently. If external input is received during a walk, it …
Figure 57
Figure 57. Figure 57: Random walk on a knowledge graph using the complement update rule. 102 [PITH_FULL_IMAGE:figures/full_fig_p102_57.png]
Figure 58
Figure 58. Figure 58: Random walk on a knowledge graph using the difference update rule. (Animations of these random walk experiments are available on this project’s website.) The setup described above implements an undirected random walk on a directed graph, because querying with a single…
Figure 59
Figure 59. Figure 59: Dendritic on/off switches as the mechanism of topological plasticity. A presynaptic neuron may have multiple connections to the same postsynaptic neuron via different dendrites. Because the voltage across every branch of a parallel circuit is equal, the same signal is…
Figure 60
Figure 60. Figure 60: Cognitive performance as a response to a neuromodulation factor that corresponds to stochastic subsampling in memory retrieval. 17.8 The signature of sparse holographic represen￾tations In our computational framework, hyperedges are bundles of sparse holographic repre…
Figure 61
Figure 61. Figure 61: Footprint of auto-associative memory partitioned in 4 equal-size blocks. 126 [PITH_FULL_IMAGE:figures/full_fig_p126_61.png]

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Reference graph

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