SubQuad: Near-Quadratic-Free Structure Inference with Distribution-Balanced Objectives in Adaptive Receptor framework

Rong Fu; Zijian Zhang; Kun Liu; Jiekai Wu; Xianda Li; Simon Fong

cs.LGFeb 19, 2026

SubQuad: Near-Quadratic-Free Structure Inference with Distribution-Balanced Objectives in Adaptive Receptor framework

Authors: Rong Fu, Zijian Zhang, Kun Liu, Jiekai Wu, Xianda Li, Simon Fong

Organizations: Independent Researcher

Abstract

Comparative analysis of adaptive immune repertoires at population scale is hampered by two practical bottlenecks: the near-quadratic cost of pairwise affinity evaluations and dataset imbalances that obscure clinically important minority clonotypes. We introduce SubQuad, an end-to-end pipeline that addresses these challenges by combining antigen-aware, near-subquadratic retrieval with GPU-accelerated affinity kernels, learned multimodal fusion, and fairness-constrained clustering. The system employs compact MinHash prefiltering to sharply reduce candidate comparisons, a differentiable gating module that adaptively weights complementary alignment and embedding channels on a per-pair basis, and an automated calibration routine that enforces proportional representation of rare antigen-specific subgroups. On large viral and tumor repertoires SubQuad achieves measured gains in throughput and peak memory usage while preserving or improving recall@k, cluster purity, and subgroup equity. By co-designing indexing, similarity fusion, and equity-aware objectives, SubQuad offers a scalable, bias-aware platform for repertoire mining and downstream translational tasks such as vaccine target prioritization and biomarker discovery.

Figures & tables

Figure 1: Overview of the SubQuad framework for near-quadratic-free, equity-aware repertoire inference. Scalable Preprocessing : Raw sequences $\mathcal{S}$ are processed via MinHash-based Indexing to generate a sparse candidate list $\mathcal{CAND}$ and optimized using hardware-aware batching $\mathcal{B}$ . Representation Learning : A Dual-Phase Meta-Encoder utilizes ImmunoBERT -style pretraining followed by MetaNet fine-tuning. The Meta-Controller dynamically adjusts gating weights $\alpha_{m}$ for multi-paradigm fusion. Graph Construction : Multi-channel affinities are integrated via Dynamic Affinity Fusion to produce $\widetilde{a}_{ij}$ . This similarity matrix is refined through RMT-based Thresholding (eigenvalue spectrum analysis) to produce a sparse weighted graph $G=(V,E,W)$ . Fairness-Constrained Clustering : The graph is partitioned into clusters $\mathcal{C}$ by optimizing a joint objective of spatial cohesion and Jensen-Shannon Equity . An Automated Fairness Tuner dynamically calibrates the trade-off weight $\lambda$ to meet target disparity $\delta_{\max}$ . Outputs : The pipeline yields antigen-aware clusters, topological maps (UMAP), and equity heatmaps for clinical interpretation.

Figure 2: Community structure in immune receptor networks. Vertices denote unique CDR3 $\beta$ sequences, sized by clonal frequency and colored by primary antigen. Edges connect receptors with fused similarity above 0.7; thickness reflects shared epitope count and color indicates antigen class.

Tool (Year)	Throughput	Recall	Memory	Purity	Equity Score
	(k seq/s)	(AUC)	(GB)	(%)
SubQuad (Ours)	97.2	0.985	1.4	92	0.91
BertTCR [ 30 ]	84.5	0.970	2.1	87	0.83
TCR-pMHC (PyG) [ 31 ]	60.0	0.920	3.5	82	0.78
ProtBert [ 32 ]	62.3	0.940	3.8	79	0.75
HeteroTCR [ 33 ]	75.0	0.950	1.6	85	0.79

Table 1: Comprehensive Performance Comparison of TCR Analysis Tools (10K Sequences). † All improvements

\geq 3\%

are significant at

p<0.01

under paired bootstrap (10 000 resamples).

Configuration	Throughput (k seq/s)	Memory (GB)	Purity (%)
Full Framework (SubQuad)	97.2	1.4	92
MinHash + GPU Acceleration	84.5	2.1	87
Fairness Constraints Excluded	102.1	1.3	76
Sequence Embeddings Only	45.7	4.2	82
Oncogenic Focus (tumor subset)	89.4	1.6	86

Table 2: Architectural Component Impact Assessment.

Algorithm	Complexity	Parameters	Time (s)
$\gamma$ -Ward Clustering	$O(n^{1+1/\gamma})$	$\gamma=8$	1.87
3SUM-Optimized Routine	$O(n^{2}/\mathrm{polylog}\,n)$	$w=64$	2.94
Optimized LSH Pipeline	$O(n^{1+1/\gamma})$	$\gamma=2$	0.68
HNSW Approximate NN (our deployment)	$O(n\log n)$	ef=200	0.71

Table 3: Algorithmic Complexity and Empirical Time Comparison.

Metric	SARS-CoV-2	CMV	EBV	Tumor	Autoimmune
Epitope Identification	89%	85%	82%	84%	80%
Cluster Homogeneity	92%	88%	86%	86%	83%
JS Disparity	9%	11%	13%	12%	15%
DP Disparity	7%	9%	10%	8%	12%
EO Disparity	8%	10%	12%	9%	14%
Processing Duration	38 min	42 min	45 min	41 min	48 min

Table 4: Immunological Performance Across Disease Contexts.

Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 3: Latency scaling of HNSW retrieval under $10^{7}$ sequences. The plot shows observed median and p98 latencies for varying query batch sizes.

Model	Kernel throughput (k seq/s)	Peak memory (GB)	AUC
xTrimoPGLM [ 38 ]	72.1	8.3	0.91
SubQuad (affinity kernel only)	89.4	1.6	0.98

Appendix

Table 5: Component-level throughput and memory efficiency at

10^{7}

sequence scale ( extrapolated ).

Method	Recall@100	Purity	JS-Disparity (%)
SubQuad	0.96 $\pm$ 0.01	0.91 $\pm$ 0.01	8 $\pm$ 1
MinHash-GPU baseline	0.89 $\pm$ 0.02	0.84 $\pm$ 0.02	19 $\pm$ 2

Appendix

Table 6: Quality metrics on 1 M-sequence cross-repertoire mix (10 donors, mean ± std,

n

=3).

Figure 4: UMAP projection of ImmunoBERT embeddings showing conserved antigen clusters.

Figure 5: F1 Score Heatmap for MinHash Parameter Selection

Figure 6: Parameter optimization landscape for MinHash configurations.

Figure 7: Performance enhancement across computational domains.

Figure 8: Topological community organization in immune receptor network. Node size indicates TCR frequency, node color indicates primary antigen, edge thickness represents the number of shared antigens, and edge color denotes shared antigen type. Key pathways are highlighted for CMV-specific TCR convergence (orange), EBV-associated immunotherapy targets (green), and cross-reactive pathways (blue).

Figure 9: Feature distributions of immune receptor sequences across different antigens. Each subplot compares two features among frequency, CDR3 length, centrality, community, and eigen centrality. Colors indicate antigen specificity.

Figure 10: Scale sensitivity of fairness metrics. Normalized disparity ( $\Psi=\Delta_{\mathrm{fair}}/\Delta_{\mathrm{baseline}}$ ) is plotted for JS divergence, demographic parity, and equalized odds across sequence scales. The green shaded region highlights the stability of equalized odds for large-scale settings.

Figure 11: Clinical decision support dashboard with human-AI collaboration.

Model	Antigen Macro-F1	Rare Recall@100	AUPRC
ESM-2-150M [ 41 ]	$0.627\pm 0.011$	$0.481\pm 0.018$	$0.601$
ProtST-ESM-1B [ 42 ]	$0.645\pm 0.009$	$0.503\pm 0.015$	$0.618$
SubQuad (Ours)	$\mathbf{0.712\pm 0.006}$	$\mathbf{0.594\pm 0.010}$	$\mathbf{0.681}$

Appendix

Table 7: Performance comparison on antigen classification and rare clonotype retrieval tasks.

Parameter	Configuration	Source
Receptor Sequences	TCR $\beta$ -chain repertoires	VDJdb [ 39 ]
CDR3 Variants	2.65K (compact), 1.2M (extended), 1M (scalability)	McPAS-TCR [ 40 ]
Oncogenic Targets	48.7K tumor neoantigens	NEPdb [ 43 ]
Cross-Domain Tooling	PyTorch-Geometric, Fairlearn	ML Commons
Biological Interactions	25.1K pairwise associations	COVIDSeq
Computational Infrastructure	NVIDIA A100 (80GB), Dual Xeon 6348	-

Appendix

Table 8: Experimental configuration summary.

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