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REVIEW 4 major objections 5 minor 44 references

Improved Data Encoding for Emerging Computing Paradigms: From Stochastic to Hyperdimensional Computing

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Hardwired Van der Corput low-discrepancy sequences replace pseudo-random sources in stochastic and hyperdimensional computing, delivering higher accuracy and lower power.

desk verdict A compact summary of the authors' own VDC-2^n encoding work, presented as new contributions; the HDC accuracy gain is confounded with dropping position hypervectors. read the letter →

arxiv 2501.02715 v1 pith:UXC652CK submitted 2025-01-06 cs.ET cs.AIcs.LGcs.NE

Mehran Shoushtari Moghadam , Sercan Aygun , M.Hassan Najafi This is my paper · ORCID
classification cs.ETcs.AIcs.LGcs.NE
keywords stochasticcomputinghyperdimensionalVanderCorputsequenceslow-discrepancydataencodingpseudo-randomnessquasi-randomnessunary
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

Stochastic computing and hyperdimensional computing both represent data as long streams of bits, and both usually rely on pseudo-random number generators (typically LFSRs) to produce those streams. This paper claims that a deterministic class of low-discrepancy sequences—Van der Corput sequences with powers-of-two bases (VDC-2^n)—produces better streams: more uniform value coverage and weaker correlation between streams, with a hardware cost of just a hardwired binary counter. The paper reports concrete gains across both paradigms: a sine function in stochastic computing drops mean squared error from 2.256e-4 to 0.523e-4, a hyperdimensional MNIST classifier rises from 81.29% to 87.12% average accuracy at dimension 2K, and an end-to-end unary hypervector generator cuts power by 98%. These results matter because randomness is the main source of error and hardware overhead in both SC and HDC, so replacing it with a simple deterministic generator could make these AI paradigms practical on edge devices.

What carries the argument

The central object is the Van der Corput sequence with base a power of two, written VDC-2^n. The nth term is obtained by writing the integer index in binary (or more generally in base 2^n), reversing its digits, and interpreting the reversed digits as a fraction in [0,1). Because the base is a power of two, the sequence is produced with no arithmetic at all: an ordinary n-bit up-counter (T flip-flops) outputs the sequence when the flip-flop outputs are taken in reverse weight order, and different rewiring schemes yield distinct sequences simultaneously. The low-discrepancy property—points spread evenly rather than clustering—is what does the work: it gives each bitstream a uniform distribution of values and keeps different streams weakly correlated, which is exactly what SC multiplication and HDC orthogonality require. For the HDC encoder, the sequence is quantized and binarized to form hypervectors directly, and a single D-sized sequence is reused with different starting offsets to produce all m hypervectors, removing the need for m independent random generators.

What would settle it

Run the position-free HDC encoder on an image dataset with strong spatial semantics, such as CIFAR-10, and compare against the same encoder with position hypervectors included; if removing position hypervectors drops test accuracy by more than the small margin reported in the paper's MNIST results, the claim that deterministic encoding eliminates the need for position hypervectors is falsified.

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Extended reading notes

Core claim

The central claim is that VDC-2^n sequences—generated by reversing the base-2^n digits of consecutive integers, which in hardware amounts to rewiring the output bits of an n-bit binary counter—can serve as the randomness source for encoding both stochastic bitstreams and hypervectors. In stochastic computing, using VDC-2^n instead of LFSR-generated pseudo-random bits for the sin(x) design lowers the mean squared error from 2.256e-4 to 0.523e-4 while cutting area and energy (Tables I and II). In hyperdimensional computing, generating hypervectors from deterministic low-discrepancy sequences yields 87.12% accuracy on MNIST at D=2K versus an 81.29% average for the LFSR baseline, and because the sequences are naturally well-distributed, the encoding no longer needs orthogonal position hypervectors or the multiplication step that binds them (Fig. 2(b)). The paper also introduces an end-to-end unary design in which level hypervectors are generated deterministically from unary bitstreams rather than randomized, eliminating the traditional random level hypervectors and reducing generator power by 98% while improving learning dynamics on DermaMNIST.

Load-bearing premise

The load-bearing premise is that hardwired VDC-2^n sequences, which are deterministic and periodic, produce bitstreams and hypervectors that are sufficiently uncorrelated and uniform for all input values in a single run, and that position hypervectors can be dropped from HDC encoding without hurting accuracy.

Editorial extensions

If this is right

  • If the deterministic encoding holds up beyond the reported benchmarks, stochastic circuits for functions like sin(x), division, and other nonlinear blocks can be built with lower error and smaller area, making SC competitive for approximate AI and signal processing on chips where LFSRs were the bottleneck.
  • Hyperdimensional encoders can drop position hypervectors and the binding multiplication entirely, simplifying the memory hierarchy and reducing latency for image classification and other structured-data tasks.
  • The single-source dynamic unary generator enables online or continuous learning of new classes, since hypervectors can be generated on the fly at negligible power.
  • A unified VDC-2^n generator block could serve both SC and HDC modules in a single chip, replacing separate RNGs and cutting design and verification cost.

Reading between the lines

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

  • A natural next test is whether VDC-2^n also beats other low-discrepancy sources like Sobol or Halton under the same HDC settings, since those are also deterministic but require more hardware; if it does, the specific structural simplicity of VDC-2^n, not just any low-discrepancy property, is the key virtue.
  • Because the generator is deterministic and seed-free, SC and HDC results become exactly repeatable across runs and chips, which could ease verification and make numerical results in these paradigms more trustworthy than the averaged-over-seeds numbers common in the literature.
  • The approach may extend to other probabilistic computing schemes such as Bayesian neural networks or in-memory computing, where replacing random sources with hardwired low-discrepancy sequences could offer similar gains—though the paper does not test those settings.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper proposes using powers-of-2 bases of Van der Corput (VDC-2^n) low-discrepancy sequences as deterministic random sources for both stochastic computing (SC) bitstream generation and hyperdimensional computing (HDC) hypervector generation. It reports that this approach improves accuracy and reduces hardware cost compared with LFSR-based pseudo-random baselines, presenting results for an SC sin(x) design, MNIST HDC classification, several other image-classification datasets, and a DermaMNIST end-to-end unary HDC design. The manuscript is written largely as a consolidation of the authors' prior publications, with Figures and Tables summarizing those results and citing the earlier papers for design details.

Significance. If the central claim were established, the work would be valuable: a hardwired counter-based deterministic RNG that generates high-quality SC bitstreams and HDC hypervectors could reduce area, power, and run-to-run variance in resource-constrained systems, and the combination of unary processing with HDC is a promising direction. The paper's strengths are its clear identification of a hardware-friendly VDC-based RNG and its concrete numerical comparison with LFSR-based baselines in Tables I-IV. However, the manuscript in its current form does not provide the experimental methodology, ablations, or statistical support needed to attribute the observed gains to VDC-2^n encoding, and one headline efficiency number is contradicted by the paper's own table. The significance is therefore conditional on substantial additional evidence being supplied.

major comments (4)
  1. [Section IV, Tables I/II and Fig. 3] The text in Section IV states that the proposed design 'improves the accuracy and reduces energy consumption by up to 77% and 92%, respectively,' but Table II shows proposed sin(x) energy of 365.9 pJ versus baseline 936.8 pJ, which is a 61% reduction, and power of 812.2 μW versus 2178.2 μW, which is a 63% reduction; the 92% figure is not supported by the table. The 98% power reduction claimed for the DermaMNIST end-to-end unary design in Fig. 3 is also presented without any measurement details or supporting table. These unsupported numbers undermine the efficiency claims that are central to the paper.
  2. [Section III, Fig. 2(b) ➏ and Table III] The proposed HDC design ➏ differs from the baseline ➎ in two respects: the source of randomness is changed from LFSR to Sobol+VDC-2^n, and position hypervectors and their binding operations are removed. Because no ablation separates these two changes, the improved MNIST accuracy in Table III cannot be attributed specifically to VDC-2^n encoding; it may stem from eliminating the position-binding step or from a different classification protocol. Moreover, Table III labels the proposed method 'Sobol+VDC-2^n', not VDC-2^n alone, so the contribution of VDC-2^n is not isolated.
  3. [Section IV, Tables III and IV and Fig. 3] The manuscript provides no experimental methodology: there is no statement of the number of runs, no error bars or standard deviations for the proposed methods, no dataset split or training epochs/retraining procedure, and no synthesis technology or tool flow for the hardware results. Many reported differences are small (e.g., Breast MNIST 68.59 versus 68.47 and CIFAR-10 41.97 versus 41.71 at D=8K in Table IV), so without variance measures the claim of 'significantly improves' is not supported. This missing methodology is load-bearing because the baselines are stochastic LFSR-based designs whose performance varies with the random seed.
  4. [Section III, Fig. 2(b) ➏] The assertion that deterministic VDC-2^n sequences generate sufficiently orthogonal hypervectors so that position hypervectors can be dropped is supported only by citations to the authors' prior works [30]–[35]. No derivation, correlation measurement, or ablation appears in this manuscript. Because the success of HDC encoding hinges on hypervector orthogonality, the paper needs to provide direct evidence (e.g., cosine-similarity distributions, or classification accuracy with and without position hypervectors) rather than relying on self-citations.
minor comments (5)
  1. [Sections II and III, Table I] The notation is inconsistent: the text refers to 'VDC-2^n' while Table I reports 'VDC4' and 'VDC128,256,512'; a brief mapping between base notation and sequence names would improve clarity.
  2. [Fig. 2] Figure 2 is very dense, with markers ➊–➑ distributed between two panels; the reader must search across the figure to connect each marker to the corresponding text. Splitting the SC and HDC panels into separate figures or adding a marker-to-design table would substantially improve readability.
  3. [References and reproducibility] Several key references are the authors' own unpublished or under-review works, patents, or workshop papers (e.g., [11], [26], [29], [31], [34]); no code or data availability statement is provided, which limits reproducibility.
  4. [Table III] The 'SOTA HDCs accuracy' comparisons in Table III are reported at different dimensions (D=2K versus D=10K) and with different retraining conditions (w/ versus w/o retraining); the table should state these conditions explicitly so that the comparison is not misleading.
  5. [Section IV] The phrase 'improves the accuracy and reduces energy consumption by up to 77% and 92%, respectively' is ambiguous; the accuracy improvement should be reported as an MSE reduction (from 2.256e-4 to 0.523e-4) rather than a percentage, which may be confused with classification accuracy.

Circularity Check

2 steps flagged · score 7.0 of 10

The central HDC improvement is attributed to VDC-2^n sequences, but the load-bearing design, orthogonality assumption, and headline efficiency numbers are imported from the authors' own prior papers via self-citations, with no ablation or independent validation presented here.

  1. self citation load bearing [Section III, Fig. 2(b), HDC encoding paragraph]
    "Fig. 2(b) exhibits the process of applying our method to the encoding stage of HDC. while the baseline methods (➎) incorporate and bind (using element-wise multiplication) both Position and Level HVs for encoding images, employing quasi-random sequences to generate HVs eliminates the need for Position HVs and subsequent multiplication operations ( ➏) [30]–[32]."

    The paper's central HDC accuracy claim (Table III) is credited to VDC-2^n/Sobol encoding, but design ➏ differs from baseline ➎ in two ways: the sequence source and the removal of Position HVs. The sentence justifies the removal only by citing refs [30]–[32], all authored by the same group (e.g., [30] is 'No-multiplication deterministic hyperdimensional encoding' by Moghadam, Aygun, and Najafi). No derivation or measurement of HV orthogonality and no ablation separating the two changes are provided in this manuscript. The load-bearing premise therefore reduces to a self-citation, and the claimed accuracy gain cannot be assigned to VDC encoding from the presented evidence.

  2. ansatz smuggled in via citation [Section III, end-to-end unary generator paragraph; Section IV, Fig. 3]
    "Another key contribution of this design is a lightweight hardware used to generate Level HVs. For the first time in the literature, we generate Level HVs not randomly but deterministically using our unary generator, eliminating the need for randomness."

    The 'first time in the literature' contribution is not derived or demonstrated in this manuscript; it is the prior result of the authors' ISLPED'24 paper [35], which is also the source cited for the end-to-end unary design in Fig. 3. The asserted optimal randomness and orthogonality of deterministic unary hypervectors is an ansatz imported through that self-citation. The subsequent claims of 98% power reduction and 15% energy improvement in Section IV are likewise presented as outcomes of this same self-authored prior work, so the efficiency conclusion rests on a self-citation chain rather than on evidence contained in this paper.

full rationale

The SC sin(x) portion is not inherently circular: the polynomial approximation is standard, the baseline LFSR comparison is external, and Tables I and II report directly measurable MSE and hardware numbers. The circularity problem is concentrated in the HDC portion. The central premise that quasi-random VDC-2^n/Sobol sequences eliminate the need for Position hypervectors and improve accuracy is imported from refs [30]–[32], all by the same authors, without derivation of correlation properties or an ablation separating the source change from the removal of position binding. The end-to-end unary generator and its headline efficiency gains are similarly attributed to the authors' own ISLPED'24 paper [35]. Because the load-bearing support for the HDC results is a chain of self-citations rather than machine-checked proofs, public code, or independent benchmarks, the central claim reduces to the prior papers' own assertions. I score 7 rather than 8–10 because the experiments are potentially reproducible and the SC results are externally benchmarked, so the paper is not wholly derivative by construction; however, the HDC causal claim is not established by the evidence presented here.

Assumptions & free parameters 3 free parameters · 4 assumptions · 2 invented entities

The central claim depends on four unproved or lightly cited assumptions, including the key domain assumption that counter-generated VDC streams are uncorrelated enough for SC and HDC. Three design parameters (VDC bases, bitstream length, hypervector dimension) are chosen by hand or scanned to produce the reported numbers.

free parameters (3)
  • VDC base per bitstream stage (VDC-2^n) = VDC4, VDC128, VDC256, VDC512 (Table I)
    Each stage of the sin(x) SC design uses a different base; no selection rule is given, so these are hand-chosen tuning values that affect the reported MSE.
  • Bitstream length N = 1024, 512, 256
    Lengths are set per stage in Table I; the MSE depends on N, and the choice is not derived in the manuscript.
  • Hypervector dimension D = 1K, 2K, 8K
    Accuracy is scanned over D in Tables III and IV; D is a hyperparameter rather than a fixed constant, and the reported gains vary with it.
assumptions (4)
  • standard math VDC-2^n sequences have low discrepancy and uniform coverage of [0,1).
    Invoked in Section III to justify replacing LFSR randomness with deterministic counter-based sequences; no proof is given in this paper.
  • standard math A hardwired n-bit binary counter with digit reversal produces the VDC-2^n sequence with negligible hardware.
    The hardware cost claims in Tables II and the energy numbers rest on this implementation assumption described in Section III.
  • domain assumption Bitstreams and hypervectors produced by VDC-2^n comparisons are sufficiently uncorrelated or orthogonal for the SC and HDC operations used.
    Central to the accuracy claims; stated in Section III as improved correlation properties and supported only by citations to the authors' own prior work.
  • domain assumption Removing position hypervectors and binding operations from HDC encoding does not hurt accuracy when level hypervectors are VDC-generated.
    The designs in Fig. 2(b) parts 6 and 7 drop position hypervectors; Table III compares to a baseline that keeps them, so the paper assumes they are unnecessary.
invented entities (2)
  • Single-source dynamic HV generator (end-to-end unary structure)
    purpose: Generates all needed hypervectors from one deterministic VDC/unary sequence, avoiding multiple random sequences and position hypervectors.
    Introduced in Section III (Fig. 2(b) part 8) and evaluated only through the authors' prior publications [33], [35]; no independent falsifiable handle is provided in this manuscript.
  • Unary-based correlated level HV generator
    purpose: Deterministically creates level hypervectors with a left shifter, up-counter, and comparator instead of random generation.
    Presented in Section III (Fig. 2(b) part 7); claims of eliminating randomness are supported only by self-citations to [33], [34].

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Cite this review

Pith. "Pith review of Improved Data Encoding for Emerging Computing Paradigms: From Stochastic to Hyperdimensional Computing." pith.science (2026). https://pith.science/paper/UXC652CK

@misc{pith2026250102715,
  author       = {Pith},
  title        = {Pith review of: Improved Data Encoding for Emerging Computing Paradigms: From Stochastic to Hyperdimensional Computing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UXC652CK}},
  note         = {Machine review of arXiv:2501.02715}
}
read the original abstract

Data encoding is a fundamental step in emerging computing paradigms, particularly in stochastic computing (SC) and hyperdimensional computing (HDC), where it plays a crucial role in determining the overall system performance and hardware cost efficiency. This study presents an advanced encoding strategy that leverages a hardware-friendly class of low-discrepancy (LD) sequences, specifically powers-of-2 bases of Van der Corput (VDC) sequences (VDC-2^n), as sources for random number generation. Our approach significantly enhances the accuracy and efficiency of SC and HDC systems by addressing challenges associated with randomness. By employing LD sequences, we improve correlation properties and reduce hardware complexity. Experimental results demonstrate significant improvements in accuracy and energy savings for SC and HDC systems. Our solution provides a robust framework for integrating SC and HDC in resource-constrained environments, paving the way for efficient and scalable AI implementations.

Figures

Figures reproduced from arXiv: 2501.02715 by the authors.

Figure 1
Figure 1. Pseudo-randomness vs. Quasi-randomness in random sequences. [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Applying our Deterministic approach to the encoding part of SC and HDC models. (a) Implementing non-linear functions and division operation using the proposed VDC-2nsequence generator in SC and (b) Improving HDC performance by generating high-quality HVs using deterministic sequences. TABLE I ACCURACY EVALUATION OF SC SIN(X) DESIGNS (FIG. 2(A)➊,➋). Func. Design Approach Polyn. order N ✝ Sequence type Total # of Dela… view at source ↗
Figure 3
Figure 3. Performance evaluation of the end-to-end Unary structure for [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.