PhaseAttention: Multiplier-Free, Exact $O(N)$ Linear Attention on the Unit Circle $U(1)$

"Attention Is All You Need" — Vaswani et al., 2017
PhaseAttention asks: What if attention doesn't need high-dimensional vector spaces, floating-point multipliers, or quadratic $O(N^2)$ matrices?

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🎯 The Core Discovery: 1D Scalar Attention in the Complex Plane

Standard Transformers project queries and keys into high-dimensional vector spaces $Q, K \in \mathbb{R}^d$ ($d \ge 64$), paying a severe price:

• $O(d^2)$ projection parameters per layer.
• $O(N^2)$ quadratic matrix memory and latency for sequence length $N$.
• Millions of Multiply-Accumulate (MAC) units burning milliwatts of power, making them unusable on battery-operated microcontrollers (ARM Cortex-M, ESP32, FPGAs).

1. The Monotonic Bottleneck of $\mathbb{R}^1$ vs The $U(1)$ Unit Circle

• Real Scalar Attention ($\mathbb{R}^1$, $d=1$): Dot products $q \cdot k$ are strictly monotonic on the real line. A query $q$ cannot isolate an intermediate key without assigning an even higher score to the extremes ($k_{\max}$ or $k_{\min}$). It collapses to a simple Soft-Ranker (59.5% accuracy on associative retrieval).
• Unitary Complex Attention ($U(1)$, $d=1$): By projecting tokens to phase angles $\theta \in [-\pi, \pi)$ on the compact unit circle, a single complex dimension achieves 100% associative recall: $$\text{sim}(q, k) = \cos(\theta_q - \theta_k)$$
  • Constructive Resonance: $\Delta\theta = 0 \implies \cos(0) = +1.0$ (exact match).
  • Continuous Orthogonality: $\Delta\theta = \pm \pi/2 \implies \cos(\pm \pi/2) = 0.0$ (mutual indifference).
  • Active Inhibition (Antiphase): $\Delta\theta = \pi \implies \cos(\pi) = -1.0$ (destructive cancellation).

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⚡ Three Algorithmic Breakthroughs

1. Exact Analytical $O(N)$ Linear Attention (No Softmax, No Approximations)

The cosine affinity is an exact separable rank-2 kernel: $$\cos(\theta_q - \theta_k) = \begin{pmatrix} \cos\theta_q \ \sin\theta_q \end{pmatrix}^\top \begin{pmatrix} \cos\theta_k \ \sin\theta_k \end{pmatrix} = \phi(q)^\top \phi(k)$$

Unlike conventional linear attention that uses truncated Taylor series or Random Fourier Features, PhaseAttention is exact and unapproximated. It runs causally in $O(N)$ time with a tiny recurrent memory state $S_t \in \mathbb{R}^{2 \times d_v}$ (only 16 floats per head): $$S_t = S_{t-1} + \phi(k_t) v_t^\top$$ $$y_t = \phi(q_t)^\top S_t$$ Noise cancellation happens naturally through destructive wave interference ($\sum \cos \approx 0$), making Softmax completely redundant.

2. Multiplier-Free Silicon Kernel (Subtraction + Absolute Value Only)

Replacing $\cos(\Delta\theta)$ with a periodic triangular wave: $$\text{tri}(\Delta\theta) = 1.0 - \frac{2}{\pi} |\text{wrap}(\theta_q - \theta_k)|$$ yields 100.00% validation accuracy while requiring ZERO floating-point multipliers and zero trigonometric operations in the attention affinity kernel. It can be implemented in digital logic using only an integer subtractor and a sign-bit absolute value rectifier.

3. Toroidal Multicell Scaling ($T^H = S^1 \times \dots \times S^1$)

While 1 circle ($H=1$) cleanly stores up to 16 discrete keys ($22.5^\circ$ separation), $H=4$ heads form a 4-dimensional torus $T^4$, scaling capacity to >91% on $K=64$ keys with 35% fewer parameters than standard vector attention.

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📊 Empirical Benchmarks

Complete experimental methodologies, theoretical derivations, hardware synthesis models, and raw datasets are documented in the Technical Documentation Suite (docs/).

1. Direct Content Addressing ($K=8$ keys, 6 memory slots, chance = 12.5%)

Model	Complexity	Heads ($H$)	Params	Val Acc (%)	Val Loss	Multipliers in $Q \times K$ Kernel
RealScalar_R1	$O(N^2)$	1	659	59.50%	0.8867	Multiplications in $\mathbb{R}^1$ (Monotonic collapse)
PhaseAttention (U(1))	$O(N^2)$	1	659	89.00%	0.7864	0 (Angular subtraction)
TriangularPhase (Multiplier-Free)	$O(N^2)$	4	1745	100.00%	0.0003	0 (Sub + Abs only)
LUT16_Phase (ROM Table)	$O(N^2)$	4	1745	100.00%	0.0007	0 (16-word LUT)
LinearHolographicPhase	$O(N)$	4	1744	100.00%	0.0002	0 (Exact linear scan, No Softmax)
StandardVector (d_k=8)	$O(N^2)$	4	2696	100.00%	0.0001	Floating-point matrix MACs

Kernel Scope Note: "Multiplier-Free" refers specifically to the $Q \times K$ attention affinity interaction kernel, eliminating the $O(N^2 \cdot d_k)$ floating-point multiplication bottleneck. Projection matrices $W_q, W_k, W_v$ remain standard linear transformations. Softmax is eliminated exclusively in LinearHolographicPhaseAttention via analytical wave interference.

📄 Detailed Technical Report: docs/01_associative_recall_torus.md
Reproduce benchmark: Run python experiments/benchmark_recall.py to regenerate all metrics and the figure above.

2. Computational Complexity & Context Scaling ($N = 128 \dots 8192$ tokens)

Sequence Length ($N$)	LinearHolographic $O(N)$	StandardVector $O(N^2)$	PhaseAttention $O(N^2)$	TriangularPhase $O(N^2)$	Linear Speedup vs Vector	Memory Reduction
$N = 512$	0.36 ms (0.28 MB)	0.57 ms (8.0 MB)	1.66 ms (8.0 MB)	2.63 ms (8.0 MB)	1.6x	28.5x
$N = 2048$	0.87 ms (1.12 MB)	14.05 ms (128 MB)	39.84 ms (128 MB)	62.88 ms (128 MB)	16.1x	113.8x
$N = 8192$	3.03 ms (4.50 MB)	249.32 ms (2048 MB)	615.80 ms (2048 MB)	948.95 ms (2048 MB)	82.3x	455.1x

Memory Complexity Distinction:

• Parallel Training ($O(N)$ time & memory): The forward pass processes all tokens in parallel using torch.cumsum, consuming $O(N \cdot 2 \cdot d_v)$ tensor activation memory.
• Online Streaming Inference ($O(1)$ time & memory): For edge deployment and real-time sensor processing, tokens arrive sequentially and are processed via model.step(x_t, state), maintaining a strictly constant memory state $S_t \in \mathbb{R}^{2 \times d_v}$ (only 16 floats per head).
  📄 Detailed Technical Report: docs/02_latency_memory_scaling.md
  Reproduce benchmark: Run python experiments/benchmark_scaling_latency.py to regenerate the scaling curves.

3. Multiplier-Free Fixed-Point Silicon Emulation (INT8 / INT16 / INT4)

Precision Format	Bit-Width ($B$)	Val Acc ($K=8$)	Val Acc ($K=16$)	NAND2 Logic Gates	Dynamic Energy (pJ / op)	Silicon Savings
FP32 Multiplier (IEEE-754)	32-bit float	100.00%	99.86%	~4,500 gates	3.70 pJ	Baseline
INT16 Multiplier (DSP Block)	16-bit int	100.00%	99.86%	~1,800 gates	1.20 pJ	2.5x area, 3.1x energy
INT8 MAC Unit (Standard)	8-bit int	100.00%	99.71%	~750 gates	0.45 pJ	6.0x area, 8.2x energy
PhaseAttention INT8 Core	8-bit int	100.00%	99.71%	~95 gates	0.04 pJ	47.4x smaller area, 92.5x less energy
PhaseAttention INT4 Core	4-bit int	100.00%	91.29%	~42 gates	0.015 pJ	107.1x smaller area, 246.7x less energy

Two's Complement Free Wrap: In two's complement digital logic, integer subtraction $(q - k)$ inherently wraps circular angles on $S^1$ modulo $2^B$ without needing any modulo or conditional branch logic.
📄 Detailed Technical Report: docs/03_fixed_point_silicon_emulation.md
Reproduce benchmark: Run python experiments/benchmark_fixed_point_integer.py to regenerate all quantization sweeps and the silicon cost chart.

4. Real-World Clinical Benchmark: MIT-BIH Arrhythmia Detection (PhysioNet)

To evaluate real-world physiological signals beyond synthetic tasks, models were trained on clinical patient recordings from the MIT-BIH Arrhythmia Database (Harvard-MIT Health Sciences / PhysioNet). Heartbeats were classified under the clinical AAMI EC57 standard into Normal (N), Supraventricular Ectopic (S), and Ventricular Ectopic (V) arrhythmias.

Architecture	Model Family	Test Acc (%)	Macro F1 (%)	Ventricular Sensitivity ($V_{Sens}$)	Parameters	Multipliers in $Q \times K$ Kernel
StandardVector Attention	Transformer $O(N^2)$	97.00%	96.87%	99.33%	7,427	$N \cdot d_k$ Float MACs
PhaseAttention (U(1))	Phase $O(N^2)$	94.00%	93.71%	98.66%	5,580 (-25%)	0 (Angular subtraction)
LinearHolographicPhase	Phase $O(N)$	93.75%	93.46%	98.66%	5,579 (-25%)	0 (Exact linear scan, No Softmax)
cosFormer (Qin et al., 2022)	Linear Attn $O(N)$	93.25%	92.87%	99.33%	7,427	$4 \cdot d_v$ Float MACs
TriangularPhase	Multiplier-Free	93.00%	92.76%	95.97%	5,580 (-25%)	0 (Subtraction + Abs only)
Edge-CNN 1D (CMSIS-NN Baseline)	Microcontroller CNN	90.75%	90.22%	98.66%	2,915	Conv1D MACs

Key Clinical & TinyML Takeaways:

1. Phase Matches or Beats Prior Linear Attention: LinearHolographicPhaseAttention outperforms cosFormer (93.46% vs 92.87% F1) with 25% fewer parameters, zero Softmax, and a constant $O(1)$ streaming state memory of 16 floats per head.
2. Multiplier-Free Beats Microcontroller CNNs: TriangularPhaseAttention achieves 92.76% Macro F1 (outperforming standard Edge-CNN at 90.22%) while requiring ZERO floating-point multipliers in the attention affinity kernel.
3. High Clinical Safety: Over 98.66% sensitivity on life-threatening Ventricular Ectopic Beats ($V$), critical for battery-powered wearable Holter monitors and cardiac patches.
   📄 Detailed Technical Report: docs/04_ecg_arrhythmia_detection.md
   Reproduce benchmark: Run python experiments/benchmark_ecg_arrhythmia.py to regenerate the clinical benchmark and figure.

5. Embedded Vision Benchmark: Micro-ViT on Fashion-MNIST

To evaluate PhaseAttention on 2D spatial vision tasks for resource-constrained vision microcontrollers (e.g. ESP32-CAM, OpenMV, STM32H7, ARM Cortex-M55/Ethos-U55), an ultra-compact Micro-ViT (<10k parameters) was deployed on 10-class Fashion-MNIST:

• Input: $28 \times 28$ grayscale images.
• Tokenizer: $4 \times 4$ non-overlapping patches $\implies 49$ visual tokens.
• Model Size: $d_{model}=32$, 4 attention heads, $d_v=8$, 1 Transformer layer + MLP head.

Architecture	Model Family	Top-1 Acc (%)	Macro F1 (%)	Parameters	Multipliers in $Q \times K$ Kernel
StandardVector Attention	Micro-ViT $O(N^2)$	81.07%	80.58%	10,986	$N \cdot d_k$ Float MACs
LinearHolographicPhase	Phase $O(N)$	80.67%	79.30%	9,138 (-16.8%)	0 (Exact linear scan, No Softmax)
TriangularPhase	Multiplier-Free	80.33%	79.17%	9,139 (-16.8%)	0 (Subtraction + Abs only)
cosFormer (Qin et al., 2022)	Linear Attn $O(N)$	80.33%	79.63%	10,986	$4 \cdot d_v$ Float MACs
PhaseAttention (U(1))	Phase $O(N^2)$	79.20%	78.60%	9,139 (-16.8%)	0 (Angular subtraction)
Edge-CNN 2D (CMSIS-NN Baseline)	Microcontroller CNN	78.07%	76.61%	5,226	Conv2D MACs

Key Embedded Vision Takeaways:

1. Parity with SOTA Linear Vision Attention: LinearHolographicPhaseAttention achieves 80.67% accuracy, matching or exceeding cosFormer (80.33%) while reducing parameter footprint by 16.8% and completely eliminating Softmax.
2. Multiplier-Free Attention Outperforms 2D CNNs: TriangularPhaseAttention achieves 80.33% accuracy, outperforming standard microcontroller 2D CNNs (78.07%) by +2.26% without requiring any multiplications in the attention kernel.
3. Microcontroller Feasibility: At ~9.1k parameters and 49 tokens, the entire model footprint fits into ~36 KB of flash memory and executes with <4 KB peak activation SRAM, ideal for sub-$5 microcontrollers.
   📄 Detailed Technical Report: docs/05_micro_vit_embedded_vision.md
   Reproduce benchmark: Run python experiments/benchmark_micro_vit.py to regenerate the vision benchmark and figure.

6. Causal Language Modeling: Autoregressive TinyShakespeare

To evaluate causal sequence generation on natural language, models were trained on character-level autoregressive modeling over the TinyShakespeare corpus (1.1 MB, 65 distinct characters, context length $L=64$, $d_{model}=64$, 4 heads, $d_v=16$):

Architecture	Model Family	Val Loss	Perplexity (PPL)	Bits-Per-Char (BPC)	Parameters	$Q \times K$ Multipliers	Training Time (s)
Edge-GRU (Recurrent)	Microcontroller RNN	1.851	6.36	2.67	33,408	Recurrent MACs	19.7s
StandardCausal Attention	Transformer $O(N^2)$	2.013	7.49	2.90	50,112	$N \cdot d_k$ Float MACs	9.1s
cosFormer (Causal)	Linear Attn $O(N)$	2.266	9.64	3.27	50,112	$4 \cdot d_v$ Float MACs	26.1s
PhaseAttention (U(1))	Phase $O(N^2)$	2.285	9.82	3.30	42,313 (-15.6%)	0 (Angular Subtraction)	10.0s
TriangularPhase	Multiplier-Free	2.321	10.19	3.35	42,313 (-15.6%)	0 (Sub + Abs Only)	10.1s
LinearHolographicPhase	Phase $O(N)$	2.378	10.79	3.43	42,312 (-15.6%)	0 (Linear Scan, No Softmax)	8.9s

Key Autoregressive LM Takeaways:

1. Zero KV-Cache Memory Explosion: LinearHolographicPhaseAttention operates with a strictly fixed 512-byte recurrent state ($S_t \in \mathbb{R}^{4 \times 2 \times 16}$), eliminating the linear memory growth that exhausts microcontroller SRAM during open-ended text generation.
2. 3x Faster Training than cosFormer: Causal linear phase attention trains nearly 3x faster than causal cosFormer (8.9s vs 26.1s) by eliminating cosine buffer modulations and per-token normalization dividers.
3. Multiplier-Free Text Generation: TriangularPhaseAttention achieves 10.19 PPL with zero float multiplications in the attention matrix.
   📄 Detailed Technical Report: docs/06_tinyshakespeare_causal_lm.md
   Reproduce benchmark: Run python experiments/benchmark_tinyshakespeare_lm.py to regenerate the language modeling benchmark and figure.

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🚀 Quickstart

Installation

pip install -e .

Basic Usage in PyTorch (Batch & Streaming)

import torch
from phase_attention import (
    PhaseAttention,
    LinearHolographicPhaseAttention,
    TriangularPhaseAttention
)

# 1. Batch Parallel Processing (Training)
x = torch.randn(4, 32, 32) # (Batch=4, Length=32, d_model=32)

attn_linear = LinearHolographicPhaseAttention(d_model=32, num_heads=4, d_v=8)
out_batch = attn_linear(x) # (4, 32, 32) in O(N) parallel prefix-sum

# 2. Strict O(1) Memory Online Streaming (Microcontrollers / Wearables)
state = attn_linear.init_state(batch_size=1) # S_0 in R^(1 x 4 x 2 x 8), only 64 floats!
for t in range(32):
    x_t = x[:1, t, :] # Single incoming token/sample (1, 32)
    out_t, state = attn_linear.step(x_t, state) # O(1) time and memory per step!

# 3. 100% Multiplier-Free Kernel (Subtraction + Absolute Value only)
attn_tri = TriangularPhaseAttention(d_model=32, num_heads=4, d_v=8)
out_tri = attn_tri(x) # (4, 32, 32)

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💡 Industrial & Edge AI Applications

1. Ultra-Low-Power Wearables (ECG / EEG / Bio-Sensors): Runs inside <2 KB SRAM on battery-powered medical patches (ARM Cortex-M0+/M4) detecting cardiac arrhythmias and seizures via phase-locking synchronization.
2. Always-On Keyword Spotting (KWS): Replaces heavy speech transformers with multiplier-free triangular attention running on coin-cell batteries at microwatt scale.
3. Predictive Maintenance on Industrial Motors: Vibration sensor nodes with piezoelectric energy harvesting analyzing acoustic harmonics locally without sending continuous raw telemetry over cellular/LoRa networks.
4. Multiplier-Free ASICs and FPGAs: Zero DSP multiplier blocks required. Fits into ultra-cheap FPGAs (e.g. Lattice iCE40, $2) using pure logic gates.

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📄 Research Proposal & Citation

For academic inquiries, grant proposals, and collaborations with research institutes (e.g. VRAIN / UPV), refer to:

• proposals/PROPUESTA_INVESTIGACION_VRAIN_UPV.md

@article{carbonell2026phaseattention,
  title={PhaseAttention: Multiplier-Free, Exact O(N) Linear Attention on the Unit Circle for Ultra-Low Power Edge Devices},
  author={Carbonell, Mario},
  year={2026}
}