Geometric Inductive Biases for Semi-Supervised Equalization: The Constellation-Aware Transformer

Avi Caciularu

cs.LGSep 27, 2026

Geometric Inductive Biases for Semi-Supervised Equalization: The Constellation-Aware Transformer

Authors: Avi Caciularu

Organizations: Google Research

Abstract

Decoding signals over unknown channels with minimal pilot overhead is a critical challenge in next-generation communications. Existing deep learning approaches typically rely on generic encoders that struggle to model long-range temporal dependencies or efficiently capture the channel's physical properties from scarce data. We argue that standard architectures suffer from agnostic estimation gaps, as they must implicitly learn the constellation geometry that is already known. We introduce the Constellation-Aware Transformer (CAT), a novel architecture that explicitly injects geometric inductive biases into the equalization process. CAT is composed of a stack of custom TransFIRmer blocks, which use an "early interaction" paradigm to co-process received signals and ideal constellation symbols. Each block features a split Feed-Forward Network that applies a Finite Impulse Response (FIR)-inspired filter for deconvolution and a parallel MLP for geometric refinement. We show that this design is structurally aligned with the optimal linear (MIMO Wiener) receiver: its attention can implement a matched-filter bank, and its bidirectional FIR branch provides the non-causal filtering that block MMSE equalization requires. In the semi-supervised setting, CAT needs fewer pilots than VAE and standard Transformer baselines: on two of our three ISI channels, it reaches a lower SER with 64 pilots than they do with 128.

Figures & tables

Modulation	$K$	Attn. Overhead	SER Gain vs. Vanilla Trans.
16-QAM	16	$\sim$ 13%	3.8 dB
64-QAM	64	$\sim$ 56%	3.0 dB
256-QAM	256	$\sim$ 300%	2.2 dB

Table 1 : Scalability across constellation orders (

h^{(1)}

channel,

N\!=\!256

, SNR 16–24 dB, gains at target SER

10^{-2}

Channel	Pilots	VAE-CNN	Trans.	CAT (Ours)	Optimal
$h^{(1)}$ (L=5)	16	0.2900	0.3392	0.3580	0.0121
	32	0.1251	0.1192	0.0842
	64	0.0523	0.0610	0.0198
	128	0.0494	0.0290	0.0156
$h^{(2)}$ (L=4)	16	0.3447	0.3563	0.3330	0.0101
	32	0.1843	0.1593	0.1372

Table 2 : SER on channels with memory (

E_{x}/N_{0}=17

dB, payload 256). CAT vs. VAE-CNN and Vanilla Transformer.

Method	14 dB	17 dB	20 dB	24 dB	26 dB
VAE-CNN	0.1842	0.0523	0.0215	0.0068	0.0031
Vanilla Trans.	0.1763	0.0610	0.0248	0.0091	0.0042
CAT (Ours)	0.1205	0.0198	0.0072	0.0019	0.0008

Table 3 : SER vs. SNR on

h^{(1)}

(

N_{p}=64

, payload 256). The 17 dB column repeats Table 2 .

Our Full Method
Model Variant	Memoryless	ISI ( $h^{(1)}$ )
CAT (built with TransFIRmer blocks)	0.0599 $\pm$ 0.0005	0.0198
Architecture & Prior Ablations
CAT without Inverse FIR Filter	0.0608 $\pm$ 0.0005	0.0287
CAT with MLP-FFN (No FIR)	0.0615 $\pm$ 0.0006	0.0351
No Prior + Bidirectional FIR	–	0.0438

Table 4 : Ablation study. Memoryless:

N_{p}=16

, SNR

=22

dB, 64 payload symbols (as in Figure 2(b) ), SER

\pm

95% CI. ISI:

h^{(1)}

E_{x}/N_{0}=17

dB,

N_{p}=64

, payload 256. 4 4 4 “Vanilla Trans” = self-attention on signals only; “Self-Only” = signals and constellations attend within their own group; “w/o Inverse FIR” = forward conv only; “No Prior + Bidirectional FIR” = signal-only attention with the TransFIRmer FFN; “Pos. Emb. on Constellation” = positional embeddings also added to the constellation tokens.

Method	BLER Gain	ECE	Calibration Improvement
VAE-CNN	—	0.085	—
Vanilla Transformer	0 dB (reference)	0.089	—
CAT (Ours)	+2.5 dB	0.012	$7.4\times$

Table 5 : Coded BLER with 3GPP 5G NR LDPC (Rate 1/2, CW=1024) on the

h^{(1)}

channel. Gains at target BLER

=10^{-2}

Profile	Delay spread	Vanilla Trans.	VAE-CNN	CAT (Ours)	Gain
TDL-A (NLOS)	30 ns	0.0385	0.0421	0.0195	$1.97\times$
TDL-D (LOS)	30 ns	0.0198	0.0231	0.0112	$1.77\times$

Table 6 : SER on 3GPP TDL channels (16-QAM,

N=128

, 500 realizations per profile). Gain is the SER of the stronger baseline divided by that of CAT.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Metric	VAE-CNN	Vanilla Trans.	CAT (Ours)
Parameter Count	$\sim 150$ k	$\sim 9$ k	$\sim 10$ k
Self-Attn Ops	0	$\mathcal{O}(N^{2}d)$	$\mathcal{O}((N+K)^{2}d)$
Pilot Efficiency	Low	Medium	High

Appendix

Table 7 : Parameter and complexity comparison (approximate for

N=128

d=10

Channel	SNR	CAT	CAT w/ MLP-FFN	FIR gain
$h^{(1)}$ (L=5)	14 dB	0.1205	0.1284	7%
	17 dB	0.0198	0.0351	77%
	24 dB	0.0019	0.0025	32%
$h^{(2)}$ (L=4)	14 dB	0.0921	0.1195	30%
	17 dB	0.0340	0.0489	44%
	24 dB	0.0024	0.0041	71%

Appendix

Table 8 : CAT vs. CAT with MLP-FFN on ISI channels (

N_{p}=64

, payload 256). The CAT entries for

h^{(1)}

are those of Table 3 . FIR gain is the relative SER increase when the FIR-FFN is replaced by an MLP.

Steps	Method	SER	SER increase
5,000	Cold-start	$1.56\times 10^{-2}$	—
200	Cold-start	$1.63\times 10^{-2}$	$+4\%$
50	Cold-start	$2.30\times 10^{-2}$	$+47\%$
50	CAVIA-init	$1.59\times 10^{-2}$	$+2\%$
10	CAVIA-init	$1.82\times 10^{-2}$	$+17\%$

Appendix

Table 9 : Convergence speed under different adaptation strategies (

h^{(1)}

E_{x}/N_{0}=17

dB,

N_{p}=128

pilots, payload 256). The 5,000-step cold-start run is the

N_{p}=128

result of Table 2 ; the last column is the SER increase relative to it.

Setting	VAE-CNN	Vanilla Trans.	CAT	Overhead
$N$ =128, 16-QAM	11.2 $\mu$ s	12.5 $\mu$ s	16.0 $\mu$ s	+28%
$N$ =256, 16-QAM	18.1 $\mu$ s	21.0 $\mu$ s	24.5 $\mu$ s	+17%
$N$ =128, 64-QAM	11.2 $\mu$ s	12.5 $\mu$ s	24.8 $\mu$ s	+98%

Appendix

Table 10 : Wall-clock time of one training step (forward pass, backward pass and optimizer update; single RTX 4090, batch size 1).

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Learning the Channel Gain from Anywhere to Anywhere via Cross-environment Transformer Estimators

Prasenjit Dhara, Daniel Romero

Dept. of Information and Communication Technology, University of Agder, Grimstad, Norway.

Channel Modeling Spatio-Temporal Transformers Transformer Architectures Meta-Learning Cone-Beam Computed Tomography

Open paper AskPDF arXiv

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Abstract

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