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Derandomizing Dense Binary Hypervector Codebooks for Quantized Scalars

Dmitri Rachkovskij, Evgeny Osipov, Olexander Volkov, Denis Kleyko, Vaclav Snasel

arXiv:2609.38471v1cs.ITcs.AIcs.NEmath.IT

TL;DR

Randomized scalar codebooks can miss their intended similarity laws at finite dimensions, complicating predictable dense-binary HDC representations. This paper analyzes constrained transition-based generators for exponential and linear target profiles. Its finite-dimensional theory and simulations connect distinct generator mechanisms to similarity error and support mechanism-specific design choices.

  • Problem

    Dense binary scalar generators can show finite-dimensional deviations from intended similarity laws, while prior work has largely proposed constructions rather than systematically characterizing those deviations.

    Source: §1 · p. 3

  • Method

    The paper defines a transition-based framework that separates exponential or linear target laws, derandomization constraints, and generator constructions, then derives finite-D similarity and error expressions.

    Source: §1 · p. 3

  • Results

    Simulations across similarity families, dimensions, quantization ranges, reference levels, variants, and constructions validate the finite-dimensional theory and its predicted trends.

    Source: §9 · p. 30

  • Takeaways & Limitations

    Generator and parameter choices can be guided by identifying which finite-dimensional error mechanism dominates for the selected similarity family and regime.

    Source: §9 · p. 30

    The analysis is limited to dense binary scalar codebooks and exponential or linear laws, rather than full vector encodings or task-level behavior.

    Source: §1 · p. 3, §8.3 · p. 29

Abstract

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Hyperdimensional computing and vector symbolic architectures often represent quantized scalar levels by dense binary codebooks whose level-to-level similarity is intended to follow a prescribed function of scalar separation. At finite dimensionality, randomized scalar codebook constructions deviate from this target because of sampling noise, random-start imbalance, update-count fluctuations, component dependence, and finite-capacity effects. We develop a transition-based derandomization framework for dense binary scalar codebooks across two target-similarity families, with similarity decaying exponentially or linearly with level separation. The framework separates the target similarity law, the derandomization variant, and the concrete generator construction, making explicit how initialization, selection, update, and capacity-handling mechanisms shape the induced similarity profile. We formalize derandomization variants that separately constrain initial Hamming weight, update-count variability, and update balance, thereby controlling distinct sources of finite-dimensional error. For each family and variant, we derive the induced mean similarity, identify realization-wise and mean target-matching regimes, and derive exact finite-dimensional expressions for bias, variance, and root-mean-square error. Simulations across dimensions, quantization ranges, reference scalar levels, and generator constructions validate the theory and show how each constraint removes or reduces a specific source of similarity mismatch. The results provide practical guidance for choosing scalar codebook generators that more closely match a desired similarity law under finite-dimensional and hardware-relevant constraints.

1 Introduction

Dense binary scalar codebooks are increasingly used where predictable similarity and efficient implementations matter, but randomized generators can deviate from their target laws at finite dimensionality. This paper develops constrained generators and finite-dimensional analyses for exponential and linear similarity profiles.

Motivation

  • Finite-dimensional randomness can make dense binary scalar codebooks deviate from intended similarity laws, while existing work has largely proposed constructions rather than systematically characterizing those deviations.The motivation is predictable behavior and implementation efficiency in current HDC deployments.Source: §1 · p. 3

Scope and approach

  • The study targets dense binary codebooks with one-density p = 1/2 and exponential or linear similarity dependence on level separation.It constrains initial Hamming weight, transition update counts, and the balance of 1→0 and 0→1 updates.Source: §1 · p. 3

Contributions

  • For both families, the paper specifies generator constructions and derives induced mean similarity, finite-D bias, variance, and RMSE, including conditions for realization-wise or mean target matching.The analysis examines how component selection, reuse, updates, and capacity handling shape similarity.Source: §1 · p. 3
  • Simulations across families, constructions, dimensions, quantization ranges, and reference levels validate the theory and inform mechanism-based generator and parameter selection.The study identifies how derandomization mechanisms affect similarity accuracy.Source: §1 · p. 3

2 Background and Related Work

HDC/VSA maps data to hypervectors whose similarity is designed to reflect input relationships; scalar codebooks provide this mapping for quantized numeric values. Prior constructions span randomized distributed and deterministic localist approaches, with component reuse shaping exponential or linear similarity profiles.

2.1 HDC/VSA: Hypervector Formats and Similarity

  • HDC/VSA uses multiple hypervector formats and similarity measures; for dense binary vectors, similarity is often assessed through Hamming distance or overlap-based dot product.The paper focuses on dense binary hypervectors with target one-density p = 1/2.Source: §2.1 · pp. 3–4

2.2 Scalar Hypervectors in the Compositional Approach

  • In compositional encodings, scalar-level hypervectors are bound to feature identifiers and aggregated, while the scalar codebook contributes similarity structure to the resulting vector representation.This paper isolates the scalar-codebook stage and studies its similarity profiles at practical dimensions and quantization levels.Source: §2.2 · p. 4

2.3 Previous and Related Work

  • Prior scalar generators range from randomized component-flow constructions to localist encodings; with-replacement selection yields exponential profiles, while without-replacement updates support linear decay.The original randomized constructions did not enforce constant Hamming weight or fixed update sizes; deterministic float codes can preserve weight while producing linear similarity.Source: §2.3 · pp. 4–6

3 Methodology and Derandomization Framework

The methodology separates the target similarity law, derandomization constraints, and concrete generator construction. It defines a shared hierarchy for exponential and linear families, along with similarity targets and finite-dimensional error measures.

3 Methodology and Derandomization Framework

  • The framework distinguishes the target law, the derandomization variant controlling finite-D variability, and the generator construction implementing those constraints.The common framework covers both exponential and linear similarity families.Source: §3 · p. 6

3.1 Framework and Derandomization Hierarchy

  • The hierarchy constrains initial weight, transition update count, and update balance; fixing the start weight, fixing update counts, and equal-swap balance control distinct sources of variability.Exponential constructions reuse components across transitions, whereas linear constructions do not; each family realizes the variants through its own generators.Source: §3.1 · p. 7, Figure 1 · p. 8

3.2 Notation, Similarity Measures, and Target Laws

  • The two target laws depend on level separation: the exponential law is parameterized by one-step decay, while the linear law decreases to zero with separation.Some realized similarities and construction-specific induced mean profiles can depend on the anchor level even though the targets do not.Source: §3.2 · p. 10

3.3 Error Metrics and Evaluation Protocol

  • The evaluation defines pointwise similarity mean, variance, bias, and MSE over generator realizations, then summarizes errors across comparison levels and estimates them from independent realizations.The finite-sample identity uses a 1/N normalization for empirical variance.Source: §3.3.1 · p. 10, §3.3.2 · p. 11, §3.3.3 · p. 11

4 Exponential-Similarity Scalar Codebook Generation

Exponential-family generators reuse components across level transitions, producing a mean similarity profile governed by level separation and a one-step decay factor. The section compares Subtract–Add and Single–Commit updates, then instantiates the derandomization variants for Single–Commit.

4.1 Common Exponential Template

  • Exponential-family constructions retain component eligibility across transitions, so mean similarity depends on level separation and follows an exponential law governed by a one-step decay factor.The update rule is level-homogeneous and permutation-invariant, and fixed-count variants may induce within-transition dependence.Source: §4.1 · pp. 11–12

4.2 Subtract–Add and Single–Commit Update Schemes

  • Subtract–Add performs Degrade and Refill substeps, whereas Single–Commit flips components at most once per transition; both preserve marginal one-density at 1/2.Single–Commit uses one Bernoulli mask and XOR; Subtract–Add uses two masks with AND and OR operations.Source: §4.2 · p. 12

4.3 Single–Commit Derandomization Variants and Generator Constructions

  • Single–Commit variants differ in initialization and update-set constraints, with Baseline, Fixed-F, and Fixed-FpD controlling initialization and update counts in different combinations.Fixed-kpD adds equal-swap balance, selecting k current 1-components and k current 0-components per transition.Source: §4.3 · p. 13
  • Equal-swap updates preserve |x_q| = D/2 at every level and make the total flip count nonrandom.For fixed-count updates, the marginal per-component flip probability is r = F/D; for equal-swap updates it is r = 2k/D.Source: §4.3 · p. 13

5 Linear-Similarity Scalar Codebook Generation

Linear-family generators use without-replacement component updates to produce similarity that decays linearly with level separation. Fixed-count and balanced-update constraints determine whether this match holds in expectation or in every realization.

5.1 Basic Idea

  • The linear family uses without-replacement selection across levels, unlike the exponential family’s with-replacement reuse, changing the dependence of similarity on separation.Each component can be flipped in at most one transition in the linear constructions.Source: §5.1 · p. 14
  • Choosing F = D/Q flips every component once across Q transitions, exhausts the remaining pool at q = Q, and yields Tlin(Δ) = 1 − Δ/Q.Other F values are not considered; larger fixed counts exhaust the pool early, while smaller counts leave components unflipped.Source: §5.1 · p. 14

5.2 Analysis

  • A remaining-pool identity expresses expected similarity through expected pool depletion, and applies across without-replacement selection mechanisms.Expected overlap similarity equals the probability that a uniformly chosen component has the same value at both levels.Source: §5.2 · p. 15
  • Random-count updates acquire finite-capacity corrections near pool exhaustion, with Perm-Scan mean overlap slightly above the linear target near terminal levels.Pool-Bernoulli follows the target closely away from exhaustion; empirical fixed-order variants show slightly larger terminal bias than Pool-Bernoulli.Source: §5.2 · p. 16
  • Fixed-count updates with F = D/Q match the linear target in mean overlap and exact normalized Hamming similarity; Fixed-kpD matches both similarities in every realization.Fixed-count updates need not preserve Hamming weight unless the 1→0 and 0→1 flip counts are balanced.Source: §5.2.3 · p. 16

5.3 Linear-Similarity Derandomization Variants and Generator Constructions

  • The five linear-family variants use remaining-pool or deterministic constructions; Fixed-FpD combines fixed counts with half-one initialization, while Fixed-kpD also balances flips in both directions.Block-Swap uses successive disjoint balanced blocks, and Float uses cyclic shifts of a contiguous half-one vector.Source: §5.3 · p. 16

6 Accuracy of Approximating the Target Similarity Laws

The analysis derives finite-dimensional accuracy for both target-similarity families. Exponential variants have zero theoretical bias, while linear-family random-count variants acquire terminal bias and additional variance from capacity truncation.

6.1 Finite-D Accuracy for Exponential Similarity Family

  • Exponential-family variance depends on pairwise component behavior, and fixed-Hamming-weight initialization induces dependence across components.Fixed-kpD is treated separately because its equal-swap update does not fit the common variance template.Source: §6.1 · p. 17, §6.1.1 · p. 18, §6.1.2 · p. 18
  • For the exponential family, Baseline, Fixed-pD, Fixed-F, and Fixed-FpD have mean similarity Texp(Δ) and zero theoretical bias.Fixed-kpD also has zero bias while preserving Hamming weight D/2 at every level.Source: §6.1.1 · p. 18, §6.1.2 · p. 18

6.2 Finite-D Accuracy for Linear Similarity Family

  • Fixed-count linear variants match Tlin(Δ) in mean and have zero bias; Fixed-FpD has zero theoretical bias and variance in every realization.The fixed-count results apply to Pool-Uniform and Perm-Scan when F = D/Q is an integer.Source: §6.2 · p. 19
  • In linear random-count variants, capacity truncation makes the cumulative distinct flip count random, producing terminal bias and an additional variance term.The additional variance term accounts for update-count variability.Source: §6.2 · p. 19, §6.2.2 · p. 19

7 Experimental Investigation

Simulations show increasingly regular realized profiles under stronger derandomization, while theoretical RMSE predictions closely track experiments across both similarity families. RMSE rankings and anchor effects vary by family and variant, with Fixed-kpD matching the linear target realization-wise.

7.1 Examples of Similarity Profiles

  • Across three realizations, exponential profiles generally become smoother with increasing derandomization, though visual smoothness does not exactly track RMSE ordering.Figure 4 shows representative overlap-similarity profiles for the five variants.Source: §7.1 · p. 19, Figure 4 · p. 20
  • Across three realizations, linear profiles become increasingly regular, and Fixed-kpD reproduces the target realization-wise by construction.Figure 5 compares the five constructions at anchors 0, 33, 67, and 100.Source: §7.1 · p. 19, Figure 5 · p. 21

7.2 RMSE of Similarity: Experiments versus Theory

  • Across exponential variants and flip rates, theoretical predictions closely match experimental RMSE in magnitude and anchor dependence, with mean RMSE ordered Baseline ≈ Fixed-F > Fixed-pD > Fixed-FpD > Fixed-kpD.Fixed-kpD gives the lowest RMSE; constant-weight variants reduce error by removing random initial Hamming-weight imbalance.Source: §7.2 · p. 23
  • For the exponential family, anchor-averaged RMSE follows Baseline ≈ Fixed-F > Fixed-pD > Fixed-FpD > Fixed-kpD, while anchor effects differ across variants.Baseline and Fixed-F have slightly larger RMSE at interior anchors; Fixed-pD and Fixed-FpD show effects associated with constant-weight initialization.Source: §7.2 · p. 23
  • For Fixed-kpD in the linear family, larger-separation error raises endpoint-anchor RMSE above interior-anchor RMSE, while anchor-to-anchor spread generally narrows as r increases.Constant-weight initialization also lowers Fixed-pD's overall RMSE relative to Baseline.Source: §7.2 · pp. 23, 25
  • For the linear family, theoretical RMSE closely matches experiments, and mean RMSE decreases Baseline > Fixed-pD > Fixed-F > Fixed-FpD > Fixed-kpD.Fixed-count variants are essentially unbiased, while random-count variants show nonzero terminal bias; Fixed-kpD matches the linear target with zero bias and variance.Source: §7.2 · p. 24

8 General Discussion

The discussion explains finite-dimensional accuracy through distinct sources of variability and the generator mechanisms that shape them. It recommends selecting constraints for the dominant error source while recognizing the analysis's scalar-codebook scope and an undeveloped variance theory for one random-count construction.

8.1 Mechanisms of Derandomization

  • Constant-weight initialization removes random-start imbalance, fixed update counts remove update-count variability, and equal-swap updates additionally balance transitions and preserve Hamming weight.These constraints are complementary, and their accuracy effects depend on the similarity family and operating regime.Source: §8.1 · p. 25
  • Selection, update, and capacity-handling rules shape residual dependence and finite-D behavior; in the linear family, pool depletion can produce terminal deviations and bias.Remaining-pool Bernoulli thinning and fixed-order proposed-count selection can differ near pool exhaustion despite both using random-count without-replacement updates.Source: §8.1 · p. 26

8.2 Accuracy and Design Implications

  • Exponential SC variants can share a mean decay profile while retaining different finite-D variability, and deterministic equal-swap constructions match both linear similarities realization-wise.For linear fixed-count constructions, mean overlap similarity matches the target when F = D/Q; normalized Hamming similarity matches realization-wise.Source: §8.2 · p. 26
  • Finite-D accuracy depends on bias and variance as well as mean matching, with reference-level RMSE reflecting separation distributions and, for linear codes, exposure to the terminal boundary.Finite-capacity corrections supplement the reference-level effects in the linear family.Source: §8.2 · pp. 26–27
  • The design implication is to choose the least restrictive construction that controls the dominant error source in the intended operating regime.Constant-weight initialization targets start imbalance, fixed counts target update-count variability, and equal-swap updates eliminate Hamming-weight drift.Source: §8.2 · p. 27

8.3 Limitations and Future Directions

  • The analysis covers dense binary scalar codebooks and exponential and linear laws, but does not develop finite-D variance and RMSE theory for adaptive Pool-Bernoulli.Its exact random-count variance theory applies to fixed-order Perm-Scan; propagation of scalar-level effects through the full HDC pipeline remains future work.Source: §8.3 · p. 29

9 Conclusions

The paper presents a unified transition-based framework for dense binary scalar codebooks that separates target laws, derandomization constraints, and generator constructions. Its finite-dimensional analysis and simulations support choosing constraints according to the dominant error source rather than treating derandomization as a uniform reduction in error.

9 Conclusions

  • The framework analyzes exponential and linear target laws by separating similarity targets, derandomization constraints, and concrete generator mechanisms.It connects these mechanisms to induced mean similarity, bias, variance, and RMSE, with simulations validating trends across tested settings.Source: §9 · pp. 29–30
  • Generator choice should target the dominant finite-dimensional error source, including start imbalance, update-count variability, Hamming-weight drift, dependence, or capacity effects.The conclusions characterize derandomization as mechanism-specific error control, subject to implementation constraints.Source: §9 · p. 30

1 Supplementary Note 1: Exponential Similarity Family

These notes characterize exponential mean-similarity decay for Subtract–Add and Single–Commit constructions, deriving decay factors through overlap recursions or component-level Markov-chain eigenvalues, including fixed-count Single–Commit variants. They distinguish average-case overlap behavior from worst-case convergence to stationarity and describe how the schemes relate under equal expected executed-flip cost versus fixed net-flip probability.

1.1 Subtract–Add

  • Subtract–Add gives T_SA(Δ) = 1/2 + (1/2)ρ_SA^Δ, with ρ_SA = (1 − r)/(1 + r) for 0 ≤ r ≤ 1.The derivation uses Degrade–Refill with α = r/(1 + r), yielding the overlap recursion and exponential decay law.Source: §1.1.2 · p. 37, §1.1.1 · p. 37

1.2 Single–Commit

  • For the Single–Commit (SC) update, each component flips with probability r per level transition, and induced mean similarity decays monotonically with separation Δ.Source: §1.2.1 · p. 38

1.3 Comparison of Subtract–Add and Single–Commit

  • For 0 < r ≤ 1/2, Single–Commit decorrelates faster than Subtract–Add at equal expected executed-flip cost.At fixed net-flip probability u, both have decay ρ = 1 − 2u and similarity T(Δ) = 1/2 + (1/2)(1 − 2u)^Δ; Subtract–Add requires extra executed flips because a component can flip twice in one transition.Source: §1.3 · p. 39

1.4 Markov Chain and Eigenvalue Perspective

  • For exponential-family updates, the decay parameter ρ is the nontrivial eigenvalue of the associated 2×2 single-component Markov kernel.With a symmetric equal-flip kernel, the mean similarity is E[S(a,b)] = 1/2 + (1/2)ρ^Δ; the SA kernel has eigenvalues 1 and (1 − r)/(1 + r).Source: §1.4 · p. 40, §1.4.1 · pp. 40–41
  • For component-wise SC and SA, E[S(a,b)] = T(Δ) = 1/2 + (1/2)ρ^Δ, with ρ_SC = 1 − 2r and ρ_SA = (1 − r)/(1 + r).Source: §1.4.2 · p. 41
  • At p = 1/2, fixed-count SC variants retain the mean law 1/2 + (1/2)ρ^Δ, but constrained per-step flips change finite-D fluctuations.These variants couple component selections within a step; equal-swap also preserves Hamming weight.Source: §1.4.2 · p. 41
  • Unlike worst-case mixing analyses, the paper studies average-case similarity and its variance/RMSE; this low-dimensional overlap can relax smoothly even while worst-case distance to stationarity remains large.The mean profile uses the overlap-associated decay eigenvalue rather than the full transition operator’s many eigenvalues.Source: §1.4.2 · p. 42

2 Supplementary Note 2: Linear Similarity Family Mean Laws and Finite-Capacity Effects

For the linear family, the mean similarity follows expected depletion of the unused component pool, with finite-capacity corrections near exhaustion. Fixed-count sampling without replacement matches the linear target when the nominal transition count divides the dimension.

2.1 Proof of Lemma 5.1

  • Under without-replacement updates, a component differs between two levels exactly when it leaves the nested unused pool, linking expected Hamming distance to expected pool depletion.With complement-invariant initialization at p = 1/2, agreement probability also gives the overlap-normalized similarity.Source: §2.1 · p. 44

2.2 Random-Count Remaining-Pool Constructions

  • Finite-capacity correction terms make the mean similarity depart from the linear target as the probability of falling below the nominal pool size increases.The terms are negligible away from exhaustion and become appreciable near the terminal levels.Source: §2.2 · p. 45
  • One-sided truncation of proposed counts clips overshoots but leaves undershoots unflipped, producing upward terminal bias near pool exhaustion.At the terminal boundary, this can leave a positive residual pool although the linear target is zero.Source: §2.2 · p. 45
  • Fixed-order proposed-count variants show slightly larger terminal bias than adaptive Bernoulli-thinning variants because proposed counts can overshoot the remaining pool.Adaptive thinning empties the pool in the next transition when fewer than F components remain.Source: §2.2 · pp. 45–46
  • When each transition flips exactly F previously unused components and F = D/Q is an integer, mean similarity matches the linear target for every anchor and separation.In this regime, Hamming distance is Δ·F.Source: §2.2 · p. 46

3 Supplementary Note 3: Accuracy of Approximating the Target Similarity Laws

This supplementary note derives exact finite-dimensional accuracy expressions for exponential and linear target similarity laws, including mean, bias, variance, and RMSE. It analyzes how derandomization variants and generator mechanisms—including flip-count variability, initialization dependence, and capacity capping—affect target matching and accuracy.

3 Supplementary Note 3: Accuracy of Approximating the Target Similarity Laws

  • The note derives finite-dimensional mean, bias, and variance for both similarity families and all five derandomization variants, then obtains RMSE under the paper’s evaluation protocol.The theoretical RMSE is compared with experiments reported in section 7.2.Source: §3 · p. 47

3.2 Finite-D Accuracy for Exponential Similarity Family

  • For four exponential-family variants, finite-dimensional variance is characterized by one-step parameters ρstep and γstep; Fixed-kpD instead uses an overlap-moment recursion.The one-step parameters summarize the mean sign multiplier and the mean product of two distinct components’ multipliers within one transition.Source: §3.2 · pp. 47–48
  • Bernoulli-start variants retain independent components at each level, whereas constant-weight initialization induces same-level component correlations and cross-level covariance.For Bernoulli starts, the sign-sum variance is D at every level; the constant-weight covariance depends on γstep.Source: §3.2.2 · p. 49, §3.2.3 · p. 50
  • For the exponential family, Baseline, Fixed-F, Fixed-pD, Fixed-FpD, and Fixed-kpD match the target mean exactly, giving zero theoretical bias.Source: §3.2.2 · p. 49, §3.2.3 · p. 50, §3.2.4 · p. 51

3.3 Finite-D Accuracy for Linear Similarity Family

  • Fixed-kpD Block-Swap and Float match T_lin exactly, giving zero theoretical bias and variance.These deterministic equal-swap constructions are exceptions to the general linear-variant derivations.Source: §3.2.4 · p. 52
  • Fixed-F has mean similarity 1 − Δ/Q and variance (1 − Δ·F/D)/D, with F = D/Q.The derivation uses K = Δ·F distinct flips for Δ ≤ Q; thus the mean matches the linear target without bias.Source: §3.2.4 · pp. 52–53
  • Fixed-FpD retains Fixed-F’s mean 1 − Δ/Q and zero bias, but its fixed-weight initialization makes coordinates dependent and changes the finite-D variance.The derivation attributes this variance distinction to |x_0| = D/2.Source: §3.2.4 · p. 53
  • Fixed-pD’s random requested flip counts yield mean similarity 1 − E[K]/D; its variance includes both conditional similarity variation and Var(K)/D^2.The number of distinct flips is capacity-capped, K = |B_b| − |B_a|, so the count moments account for capping.Source: §3.2.4 · pp. 54–55
  • Baseline’s bias is Δ/Q − E[K]/D, and its variance is (1 − E[K]/D)/D + Var(K)/D^2.Unlike Fixed-pD, Baseline uses i.i.d. initialization, yielding the stated conditional variance before averaging over K.Source: §3.2.4 · p. 55

4 Supplementary Note 4: Experiments

Supplementary profiles show how update schedules shape Hamming-similarity curves, while RMSE experiments broadly agree with theory and reveal different error patterns for exponential and linear targets.

4.1 Supplementary Similarity Profiles

  • Exponential-family profiles show identical per-seed Hamming curves for Baseline and Fixed-pD, and for Fixed-F and Fixed-FpD.Fixed-F and Fixed-FpD show smaller target deviations because each step flips exactly F components; Baseline and Fixed-pD have random per-step flip counts.Source: §4.1 · p. 56
  • For linear similarity, Fixed-F, Fixed-FpD, and Fixed-kpD share exact linear Hamming profiles because each transition flips exactly F previously unused components..Source: §4.1 · p. 56

4.2 RMSE of Similarity: Experiments versus Theory

  • Exponential-family RMSE decreases Baseline ≈ Fixed-F > Fixed-pD > Fixed-FpD > Fixed-kpD; linear-family RMSE decreases Baseline > Fixed-pD > Fixed-F > Fixed-FpD > Fixed-kpD.The theoretical predictions closely match experimental RMSE values and anchor-dependent trends.Source: §4.2.2 · p. 65, §4.2.3 · p. 68
  • For exponential similarity, random initialization increases variance across realizations, keeping Fixed-F close to Baseline despite its fixed per-step flip count.Constant-weight initialization in Fixed-pD reduces overall RMSE relative to Baseline.Source: §4.2.2 · p. 65
  • Exponential-family anchor effects depend on each generator’s separation-dependent error; their spread narrows as r increases and errors depend less on separation.Fixed-kpD has larger endpoint than interior RMSE, whereas Baseline and Fixed-F show slightly larger interior RMSE.Source: §4.2.2 · p. 66

4.2.3 Linear Similarity Family

  • For linear similarity, random-count variants show nonzero bias near level Q because finite capacity limits new distinct flips below the ideal linear count.The remaining-pool constraint also reduces run-to-run variance near pool exhaustion; Fixed-pD’s constant-weight initialization lowers overall variance versus Baseline.Source: §4.2.3 · pp. 68–69
  • For linear similarity, Fixed-kpD matches the target exactly, with zero bias and variance up to numerical precision.The construction is deterministic by design.Source: §4.2.3 · p. 68
  • For linear similarity, Fixed-F and Fixed-FpD variance falls with separation, while constant-weight initialization makes Fixed-FpD’s endpoint RMSE reduction stronger.Interior anchors contain more small-separation comparisons.Source: §4.2.3 · p. 69
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