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arxiv: 2605.16443 · v1 · pith:V2GT4SMLnew · submitted 2026-05-15 · 💻 cs.LG · cs.AI

Two-Valued Symmetric Circulant Matrices: Applications in Deep Learning

Jayakrishna Amathi , Venkata Prasanth Yanambaka , Saraju P. Mohanty , Elias Kougianos This is my paper

Pith reviewed 2026-05-20 21:07 UTC · model grok-4.3

classification 💻 cs.LG cs.AI
keywords neural network compressionsparse weight matricescirculant matricesedge computingmodel size reductionfully connected layersMNISTECG classification
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The pith

Restricting neural network weight matrices to two values arranged in a symmetric circulant pattern reduces parameters by more than 80 times while keeping classification accuracy near baseline levels.

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

The paper proposes replacing standard weight matrices in fully connected layers with two-valued symmetric circulant matrices that store only two numbers per layer instead of thousands. This built-in structure delivers extreme sparsity without any separate pruning or low-rank steps. On the MNIST handwritten-digit task the parameter count falls from 623290 to 7852 and accuracy moves from 97.6 percent to 93.5 percent; on the MIT-BIH arrhythmia task the drop is from 24709 to 942 parameters with accuracy from 97.6 percent to 93.1 percent. The resulting models fit the memory and power budgets of edge devices, IoT nodes, and battery-powered medical sensors without extra hardware.

Core claim

The central claim is that a weight matrix restricted to exactly two distinct values and forced to remain both symmetric and circulant retains sufficient expressive power for standard image and signal classification tasks, thereby cutting storage from hundreds of thousands of parameters to a few thousand while accuracy stays within a few percentage points of the dense baseline.

What carries the argument

The Two-Valued Symmetric Circulant Matrix (TVSCM), a matrix whose entries take only two numerical values and obey both symmetry and circulant repetition, which collapses the storage cost of each fully connected layer to those two numbers.

If this is right

  • Fully connected layers require only two stored numbers each, making total model size small enough for direct deployment on memory-constrained edge hardware.
  • No post-training pruning or auxiliary approximation stages are needed to reach the reported sparsity.
  • The same architecture supports low-power operation in IoMT and tiny-ML settings because arithmetic reduces to scaling by the two fixed values.
  • Accuracy remains within roughly four percentage points of the dense network on both digit recognition and ECG arrhythmia classification.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The circulant symmetry might be relaxed or adapted for convolutional layers to compress vision backbones further.
  • Because only two values are used, quantization-aware training or integer-only inference becomes trivial to add on top of the structure.
  • Scaling the same pattern to transformer or recurrent layers could test whether the two-value limit remains viable for sequence tasks.

Load-bearing premise

That a weight matrix limited to exactly two distinct values in a symmetric circulant layout still supplies enough degrees of freedom to learn useful decision boundaries for the target tasks.

What would settle it

Training identical TVSCM networks on a harder benchmark such as CIFAR-10 and measuring whether top-1 accuracy falls more than ten points below the dense baseline while the dense model remains above 85 percent.

Figures

Figures reproduced from arXiv: 2605.16443 by Elias Kougianos, Jayakrishna Amathi, Saraju P. Mohanty, Venkata Prasanth Yanambaka.

Figure 1
Figure 1. Figure 1: Healthcare inference: traditional cloud-based vs. proposed TVSCM edge-based approach. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Types of structured matrices highlighting parameter reduction [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Typical deep neural network with parameter-heavy dense layers. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Typical neural network architecture with fully connected (dense) layers. [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Dense architecture figure 4 Deep Learning in Healthcare Deep learning in recent years has emerged as one of the strongest technologies in medicine. It enables physicians and scientists to manage large-scale medical information like visual images, lab work, and gene information. In contrast with classical machine learning, deep learning can automatically capture effective patterns from original information … view at source ↗
Figure 6
Figure 6. Figure 6: Deep Learning in healthcare. 4.2 Connected and Real-Time Care (IoT and Wearable Applications) Deep learning plays an essential role in the development of advanced health care systems. Devices like hospital sensors, smartwatches, and health trackers are used to monitor patients’ health by obtaining real-time data like glucose levels, heart rate, blood oxygen saturation levels, among others. Long Short-Term … view at source ↗
Figure 7
Figure 7. Figure 7: Challenges of Deep Learning in healthcare. [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Comparison between a baseline dense weight matrix and the proposed two-parameter circulant matrix representation. 7.3 Significance of the Contribution The two-valued symmetric circulant method that is presented in this paper presents a new method that can be used to develop lightweight deep learning models without compromising their accuracy. The method is likely to be useful in the development of smart he… view at source ↗
Figure 9
Figure 9. Figure 9: Proposed IoMT Architecture With TVSCM-Based Edge Intelligence. [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Two-parameter symmetric circulant matrix structure. ng(Parameter Explosion) Circulant Mapping [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
Figure 12
Figure 12. Figure 12: Time per epoch comparison between the Dense baseline and the proposed TVSCM model on the MNIST [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Memory accessed per inference on MNIST dataset(Batch Size = 32). [PITH_FULL_IMAGE:figures/full_fig_p020_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Energy consumption per inference on MNIST dataset(Batch Size = 32). [PITH_FULL_IMAGE:figures/full_fig_p020_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Throughput per watt on MNIST across batch sizes. [PITH_FULL_IMAGE:figures/full_fig_p021_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Memory accessed per inference on MIT-BIH ECG dataset(Batch Size = 32). [PITH_FULL_IMAGE:figures/full_fig_p023_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Energy consumption per inference on MITBIH ECG dataset(Batch Size = 32). [PITH_FULL_IMAGE:figures/full_fig_p024_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Throughput per watt on MIT-BIH across batch sizes. [PITH_FULL_IMAGE:figures/full_fig_p025_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Parameter comparison across matrix structures for n=784. [PITH_FULL_IMAGE:figures/full_fig_p027_19.png] view at source ↗
read the original abstract

Despite the success of deep neural networks in vision, medical diagnosis, and IoT scenarios, their deployment on resource-limited platforms poses serious challenges due to their high storage requirements, computational complexity, and large footprint. In particular, fully connected layers require a large number of weights, making it difficult for edge devices to accommodate them. To overcome these challenges associated with limited platforms, this paper proposes the Two-Valued Symmetric Circulant Matrix (TVSCM), a very sparse architecture that employs just two weights per layer to keep it circulant and symmetric. The extreme form of structured sparse architecture provides negligible storage costs compared to traditional full-weight storage. Instead of hardware and additional stages of other traditional sparse learning techniques, such as low-rank approximation and pruning approaches, this architecture provides an extreme form of sparsity, achieving very minimal storage requirements. The simulation study demonstrates more than 80$\times$ reduction in model parameters, reducing parameters from 623,290 to 7,852 on MNIST and from 24,709 to 942 on the MIT-BIH arrhythmia dataset, while maintaining comparable accuracy from 97.6% to 93.5% on MNIST and from 97.6% to 93.1% on MIT-BIH. Due to its minimal architectural requirements and very low power consumption, this architecture would be ideal for edge computing platforms, tiny-ML platforms, IoMT systems, and battery-powered systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper proposes the Two-Valued Symmetric Circulant Matrix (TVSCM) as a replacement for fully-connected layers, restricting each layer to exactly two learnable scalar weights arranged according to symmetric circulant structure. It reports >80× parameter reduction (623290→7852 on MNIST; 24709→942 on MIT-BIH) with accuracy dropping only from 97.6% to 93.5% and 97.6% to 93.1% respectively, positioning the method as a hardware-free extreme-sparsity solution for edge and tiny-ML platforms.

Significance. If the empirical results are reproducible and the architecture generalizes, the approach would supply an unusually compact, training-compatible form of structured sparsity that removes the need for separate pruning or low-rank stages. The concrete parameter counts and accuracy numbers on two datasets constitute a clear, falsifiable claim; however, the absence of any rank, approximation, or capacity argument leaves open whether the 2-dimensional hypothesis class per layer can support the observed performance beyond these specific experiments.

major comments (3)
  1. Abstract: the headline claim that accuracy remains 'comparable' after an approximately 4-point drop is presented without error bars, multiple random seeds, or statistical significance tests. This directly affects whether the reported numbers support the central assertion of usable performance under extreme compression.
  2. No section supplies a rank bound, approximation guarantee, or capacity argument showing that the linear maps realizable by a symmetric circulant matrix with exactly two distinct values can separate the MNIST or MIT-BIH classes at the reported accuracy. The empirical results alone therefore leave the weakest assumption—that the 2-parameter family retains sufficient expressive power—untested and load-bearing for the parameter-reduction claim.
  3. The manuscript contains no ablation on the choice of the two scalar values, no description of how the circulant symmetry constraint is enforced (or relaxed) during back-propagation, and no head-to-head comparison against standard low-rank or pruning baselines on identical network depths and datasets. These omissions make it impossible to isolate the contribution of the TVSCM structure itself.
minor comments (2)
  1. The construction of the TVSCM (how the two values are assigned to the circulant diagonals while preserving symmetry) would benefit from an explicit matrix equation or small worked example in the methods section.
  2. Missing citations to prior literature on circulant neural-network layers and on two-value or binary-weight networks would help situate the novelty of the symmetric two-value restriction.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive feedback on our manuscript. We appreciate the opportunity to clarify and strengthen our presentation of the TVSCM approach. Below, we address each major comment point by point. We will revise the manuscript to incorporate statistical analysis, additional experimental details, and comparisons as suggested, while preserving the core empirical contributions.

read point-by-point responses

  1. Referee: Abstract: the headline claim that accuracy remains 'comparable' after an approximately 4-point drop is presented without error bars, multiple random seeds, or statistical significance tests. This directly affects whether the reported numbers support the central assertion of usable performance under extreme compression.

    Authors: We agree with the referee that providing statistical support for the accuracy claims would strengthen the manuscript. In the revised version, we will report results averaged over multiple random seeds (e.g., 5 or 10 runs), include standard deviations or error bars, and conduct appropriate statistical tests (such as paired t-tests) to assess the significance of the observed accuracy differences. This will better substantiate the claim of comparable performance under the reported parameter reduction. revision: yes

  2. Referee: No section supplies a rank bound, approximation guarantee, or capacity argument showing that the linear maps realizable by a symmetric circulant matrix with exactly two distinct values can separate the MNIST or MIT-BIH classes at the reported accuracy. The empirical results alone therefore leave the weakest assumption—that the 2-parameter family retains sufficient expressive power—untested and load-bearing for the parameter-reduction claim.

    Authors: We acknowledge that the manuscript does not include a formal theoretical analysis of the expressive capacity of TVSCM layers. Our work is primarily empirical, demonstrating practical performance on standard benchmarks. In the revision, we will add a discussion on the properties of symmetric circulant matrices with two values, including their representation via Fourier transforms and the limited degrees of freedom. While we cannot provide a complete rank bound or approximation guarantee without further theoretical investigation, this addition will help contextualize the empirical success. revision: partial

  3. Referee: The manuscript contains no ablation on the choice of the two scalar values, no description of how the circulant symmetry constraint is enforced (or relaxed) during back-propagation, and no head-to-head comparison against standard low-rank or pruning baselines on identical network depths and datasets. These omissions make it impossible to isolate the contribution of the TVSCM structure itself.

    Authors: We agree that these elements are necessary to fully evaluate the method. We will revise the manuscript to include: an ablation study on the selection and sensitivity to the two scalar values; a clear description of how the TVSCM is implemented by parameterizing with two scalars and constructing the matrix to satisfy the symmetric circulant constraints before each forward pass, enabling standard gradient descent; and direct comparisons with low-rank factorization and pruning techniques on the same models and datasets, with metrics on accuracy and parameter efficiency. revision: yes

Circularity Check

0 steps flagged

No significant circularity in TVSCM architecture proposal

full rationale

The paper defines the TVSCM as a matrix constrained to exactly two distinct values while enforcing symmetric circulant structure, then reports empirical results after training the two scalars per layer on MNIST and MIT-BIH. No derivation, theorem, or first-principles claim is advanced whose output reduces to the input definition by construction. The parameter reduction follows directly from the explicit architectural choice (two scalars), and the accuracy figures are measured outcomes of standard optimization rather than any fitted prediction or self-referential step. The work is self-contained as an empirical demonstration of a restricted hypothesis class; no load-bearing self-citation, ansatz smuggling, or uniqueness theorem is invoked.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claim rests on the domain assumption that a two-valued circulant symmetric matrix can be trained to approximate the required function, plus the free choice of the two scalar values that are fitted to each layer.

free parameters (1)
  • two distinct weight values per layer
    These scalars are the only free parameters in the weight matrix; their values are determined during training to achieve the reported accuracy.
axioms (1)
  • domain assumption A symmetric circulant matrix generated from two scalar values is a valid and trainable replacement for a dense weight matrix in fully connected layers
    Invoked when the architecture is introduced as a drop-in replacement without additional hardware stages.
invented entities (1)
  • Two-Valued Symmetric Circulant Matrix (TVSCM) no independent evidence
    purpose: To enforce extreme structured sparsity with only two weights per layer
    Newly defined matrix class whose only external support is the empirical results in the abstract.

pith-pipeline@v0.9.0 · 5799 in / 1418 out tokens · 37293 ms · 2026-05-20T21:07:49.695442+00:00 · methodology

discussion (0)

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