WARP: Wasserstein-Aligned RAG for Population Opinions

Aman Singh Thakur; Aditya Agrawal; Alwarappan Nakkiran; Alex Karlsson

cs.IRAug 24, 2026

WARP: Wasserstein-Aligned RAG for Population Opinions

Authors: Aman Singh Thakur, Aditya Agrawal, Alwarappan Nakkiran, Alex Karlsson

Abstract

RAG systems are increasingly used to summarize what large collections of documents say. A user asks "What do people think about X?" and receives an answer that reads as consensus. But standard top-k retrieval ranks documents by query similarity, not by how faithfully they represent the population, so minority views quietly disappear. Existing fixes fall short. Diversity re-rankers like MMR and DPP spread retrieved documents apart, but with no target distribution to aim for. Calibration methods based on KL or JS divergence do target one, yet treat opinion bins as unordered: confusing strong positive with strong negative costs no more than an adjacent-bin miss. We introduce WARP, a family of post-retrieval algorithms that calibrate retrieved evidence to the population's opinion distribution. WARP first recovers underrepresented opinions that cosine ranking may bury, then uses Wasserstein-1 distance to select documents whose sentiment-intensity distribution matches the population target, capturing the ordinal structure ignored by KL and JS divergence. We develop three variants for dense, sparse, and variable candidate pools, trading off calibration quality and speed. Across three review domains spanning 35K documents, 156 queries, and 26 entities, WARP's domain-matched variants reduce distributional error by at least 43% with sub-second latency. These gains carry through to generation: a five-judge LLM panel prefers WARP-generated answers in 86% of decided comparisons at k <= 5.

Figures & tables

Figure 1: WARP pipeline. Stage 1 : semantic retrieval returns a Top- $N$ candidate pool. Stage 1.5 : deficit-aware pool expansion recovers under-represented sentiment poles via re-retrieval ( Entity-Gated or Adaptive Expansion ; § 3.3 ; self-bypasses when the pool is already balanced). Stage 2 : $W_{1}$ re-ranking selects the final $k$ documents whose empirical opinion distribution matches the population target $P_{\text{pop}}$ ( $W_{1}$ Minimizer for dense pools, $W_{1}$ -MMR for sparse pools, or WassRank OT as a tuning-free fallback; § 3.4 ). Pool density drives the variant choice (decision matrix in Table 3 ).

Figure 2:

W_{1}

re-rankers (no re-retrieval): domain-matched variants reduce distributional error at least 43% vs. Top-

k

(gray dashed) — 43% Seller Forums (

W_{1}

-MMR), 80% Yelp (

W_{1}

Minimizer), 69% OpinRank (

W_{1}

Minimizer); see Table 1 . Curves converge at

k^{*}{=}3

–

5

documents (black dot; Appendix C.3 ) as most entities’ opinion mass concentrates in 2–3 of 7 bins.

N{=}200

Method	$W_{1}{\downarrow}$	EM%	$W_{1}{\downarrow}$	EM%	$W_{1}{\downarrow}$	EM%	(ms)
	Seller Forums		Yelp Hotels		OpinRank Cars		Latency
Top- $k$ (baseline)	10.33	42.9	13.60	40.1	13.33	27.0	0
MMR *	10.31	43.5	8.67	14.2	9.64	11.3	12
DPP ***	11.69	41.2	5.63	78.8	7.38	69.2	45
OpinionMMR **	10.44	43.5	8.59	12.4	10.02	10.6	12
KL (JS) Minimizer *	8.87	51.4	3.00	93.8	4.54	85.8	9

Table 1: Main results: Diversity baselines and calibrated methods vs WARP re-rankers.

W_{1}{\downarrow}

: distributional distance to the population opinion target. EM%

{\uparrow}

: fraction of selected documents matching the queried entity. EG/AE = pool expansion strategies (Entity-Gated / Adaptive Expansion).

N{=}200

candidates,

k{=}20

output. Paired Wilcoxon vs. Top-

k

{}^{***}p{<}0.001

{}^{**}p{<}0.01

{}^{*}p{<}0.05

. Bold: best per column. † Shared FAISS index across all OpinRank entities (EM=82.9%); entity-specific indices give

W_{1}{=}0.95

with 100% EM (Appendix F.3 ).

Approach	W/L/T	Fair%	Win%
Random	16/36/104	31%	10%
MMR ∗	39/8/109	83%	25%
DPP ∗	54/7/95	89%	35%
WassRank OT ∗	66/28/62	70%	42%
W 1 -MMR ∗	44/10/102	81%	28%
W 1 Minimizer ∗	51/8/97	86%	33%

Table 2: Generation evaluation (

k{=}5

): 5-judge majority vote, position-controlled blind pairwise vs. Top-

k

(

N{=}156

queries). Fair% = W/(W+L). Win% = W/

N

. All approaches significant at

p{<}0.001

(sign test) except Random.

k{=}10

in Appendix G.1.3 .

Pool Condition	Method	ms	Complexity
Dense (EM ${\geq}$ 85%)	$W_{1}$ Minimizer	9	$O(\|C_{e}\|\cdot k\cdot m)$
Sparse (baseline EM ${\sim}$ 40%)	$W_{1}$ -MMR	154	$O(N\cdot k\cdot m)$
Variable / unknown	EG + $W_{1}$ Min	13	$O(\|C_{e}\|\cdot k\cdot m)$

Table 3: Deployment decision matrix. Pool density determines algorithm choice; all methods SLA-compliant (

N{=}200

k{=}20

m{=}7

bins).

|C_{e}|

= entity-matched candidates. Noise tolerance details in Appendix F.1 .

Appendix figures & tables26 assets

Supplementary material from the paper’s appendix.

Appendix

Sentiment	Intensity	SI Score
Positive	High / Med / Low	+30 / +20 / +10
Neutral	Any	0
Mixed	—	0
Negative	High / Med / Low	$-30$ / $-20$ / $-10$

Appendix

Table 4: Sentiment-Intensity (SI) mapping to the 7-bin ordinal scale. Mixed labels (praise-and-criticism co-occurring in one review) collapse to the neutral bin; they account for 4–8% of extractions across domains and

W_{1}

Minimizer never crosses the Top-

k

baseline on Yelp even at 50% label corruption (§ F.1 ), so this pooling is bounded in impact. Multi-issue axes require multi-marginal transport — outside our current scope (§Limitations).

Dataset	Domain	Source	Size	Entities	Queries	Search	Embedding	Chunking	Labeling
Seller Forums	E-comm. seller	Public	${\sim}$ 8K	6	36	Hybrid	Titan V2 ( Amazon Web Services, 2024a ) (1024d)	300 tok / 20% overlap	LLM-enriched metadata
Yelp Hotels	Hospitality	Yelp Open	14K	10	60	Semantic	MiniLM-L6 (384d)	Whole review	LLM-extracted (Claude)
OpinRank Cars	Automotive	OpinRank	13K	10	60	Semantic	MiniLM-L6 (384d)	Whole review	LLM-extracted

Appendix

Table 5: Dataset summary across three evaluation domains.

Type	Idx	Seller Forums	Yelp Hotels	OpinRank Cars
breadth	0	What do sellers think about {entity}?	What do guests think about {entity}?	What do owners think about {entity}?
breadth	1	Summarize seller opinions on {entity}.	Summarize guest opinions on {entity}.	Summarize owner opinions on {entity}.
polar	0	What are the biggest complaints about {entity}?	What are the biggest complaints about {entity}?	What are the biggest complaints about {entity}?
polar	1	What positive experiences have sellers had with {entity}?	What positive experiences have guests had with {entity}?	What do owners praise most about {entity}?
segment	0	How do small sellers vs large sellers feel about {entity}?	How do business travelers vs leisure travelers feel about {entity}?	How do commuters vs enthusiasts feel about {entity}?
segment	1	Do new sellers and experienced sellers differ in their views of {entity}?	Do solo guests and families differ in their views of {entity}?	Do new buyers and long-term owners differ in their views of {entity}?

Appendix

Table 6: Query templates per domain. Each template is instantiated with every selected entity (6 for Seller Forums, 10 for Yelp/OpinRank), producing 6 questions per entity (2 breadth, 2 polar, 2 segment).

Symbol	Meaning
$d$	A candidate document
$e$	Target entity
$k$	Output size (documents returned)
$N$	Candidate pool size
$SI(d)$	Sentiment-intensity label of $d$
$P_{\text{pop}}$	Ground-truth population distribution

Appendix

Table 7: Notation reference.

† Only fires when pool expansion (Entity-Gated / Adaptive) detects a deficit.
Component	Local FAISS	Managed Bedrock Knowledge Base
Retrieval	$<$ 50 ms	${\sim}$ 1.5 s
Re-retrieval †	$<$ 5 ms	${\sim}$ 100 ms
Re-ranking (p99)	$<$ 310 ms	$<$ 310 ms
End-to-end	$<$ 365 ms	${\sim}$ 1.9 s

Appendix

Table 8: Per-component latency breakdown. Re-ranking is hardware-only (no model inference); retrieval and re-retrieval depend on the vector-store backend. p99 for re-ranking; typical/median for the rest.

Domain	Method	Queries	p50 (ms)	p95 (ms)	p99 (ms)	Max (ms)
Seller Forums	$W_{1}$ Minimizer	36	0.1	38.7	45.5	48.0
Seller Forums	$W_{1}$ -MMR	36	156.8	163.1	166.2	167.4
Seller Forums	EG + $W_{1}$ Min	36	0.6	211.0	216.2	218.8
Seller Forums	DPP	36	47.8	49.1	51.9	53.0
Yelp Hotels	$W_{1}$ Minimizer	60	65.0	270.8	289.9	295.3
Yelp Hotels	$W_{1}$ -MMR	60	81.6	284.7	301.6	303.5

Appendix

Table 9: Per-query re-ranking latency percentiles (

N{=}96

queries: 36 Seller Forums + 60 Yelp Hotels, single-threaded on the hardware above). p99 drives production SLAs; max reported for tail auditing. All WARP variants stay under 310 ms at p99.

	Yelp		OpinRank
Method	$W_{1}$	EM%	$W_{1}$	EM%
KL (JS)-MMR	2.51	38.2%	3.32	36.1%
$W_{1}$ -MMR	4.02	19.4%	3.41	21.8%
KL (JS) Minimizer	3.00	93.8%	4.54	85.8%
$W_{1}$ Minimizer	2.76	93.8%	4.14	85.8%

Appendix

Table 10: Entity-match confound in hybrid variants. KL(JS)-MMR retrieves

\sim

1.7–2

\times

more entity-matched documents than

W_{1}

-MMR on both Yelp and OpinRank. In the controlled Minimizer comparison (same entity filter, EM equalized)

W_{1}

wins on both domains — the KL(JS)-MMR

W_{1}

advantage in the uncontrolled comparison traces to the EM gap, not to metric superiority.

Approach	EM%	$W_{1}$	$\Delta$ vs Top-K	$\Delta$ vs EF+Top-K
Top K (no filter)	40.1	13.60	—	—
EntityFilter + Top-K	100.0	7.38	$-45.7\%$	—
EntityFilter + MMR	100.0	6.99	$-48.6\%$	$-5.3\%$
EntityFilter + DPP	100.0	5.64	$-58.5\%$	$-23.6\%$
$W_{1}$ Minimizer	93.8	2.76	$-79.7\%$	$-62.6\%$
$W_{1}$ -MMR	19.4	4.02	$-70.4\%$	$-45.5\%$

Appendix

Table 11: Entity-filter control experiment (Yelp Hotels,

k{=}20

). Entity filtering alone reduces

W_{1}

by 45.7%;

W_{1}

Minimizer achieves 62.6% additional reduction beyond the entity-filter control.

Figure 3: $W_{1}$ Minimizer convergence vs. output size $k$ . Median convergence point $k^{*}$ (where $\Delta W_{1}<0.5$ per additional document): Seller Forums $k^{*}{=}3$ , Yelp $k^{*}{=}4$ , OpinRank $k^{*}{=}5$ .

Entity	Top- $k$ $W_{1}$	$W_{1}$ Min	$W_{1}$ -MMR	WassRank OT
Booking Experience	18.30	1.95	3.78	1.58
Breakfast	7.55	1.48	2.87	1.42
Breakfast Quality	5.78	0.87	3.79	0.65
Business Center	24.51	9.76	6.72	9.26
Casino	14.79	0.86	4.07	0.62
Loyalty Program	16.31	4.07	4.49	2.95

Appendix

Table 12: Per-entity breakdown (Yelp Hotels, 10 aspects).

W_{1}

Minimizer achieves

<

2.0 on 7/10 entities.

Entity	Count	H	Min%	Top- $k$	$W_{1}$ Min	$W_{1}$ -MMR	WassRank OT
Oil Change Interval	144	1.49	21%	3.96	1.08	2.95	0.78
Dealer Experience	425	1.47	30%	23.07	11.71	3.40	10.28
Vehicle Size	42	1.37	19%	0.00	8.31	6.19	8.31
Sales Experience	776	1.33	28%	15.66	3.01	3.19	2.57
Fuel Tank Size	269	1.32	36%	13.67	3.06	3.25	3.05
Resale Value	427	1.30	39%	12.19	2.24	2.40	2.33

Appendix

Table 13: Per-entity breakdown (OpinRank, LLM-extracted entities). Count = opinion mentions, H = Shannon entropy, Min% = minority sentiment fraction.

Domain	Approach	$N$	Reduction	$p_{\text{adj}}$
Yelp	$W_{1}$ Minimizer ∗∗∗	57	$+81.4\%$	$<$ 0.000001
Yelp	$W_{1}$ -MMR ∗∗∗	57	$+70.2\%$	$<$ 0.000001
Yelp	EG + $W_{1}$ Min ∗∗∗	57	$+89.8\%$	$<$ 0.000001
Yelp	AE + $W_{1}$ Min ∗∗∗	57	$+89.1\%$	$<$ 0.000001
Yelp	WassRank OT ∗∗∗	57	$+83.3\%$	$<$ 0.000001
OpinRank	$W_{1}$ Minimizer ∗∗∗	54	$+72.4\%$	$<$ 0.000001

Appendix

Table 14: Benjamini–Hochberg FDR correction (

\alpha{=}0.05

) on paired-Wilcoxon

p

-values for the

W_{1}

family. Every

W_{1}

-family algorithm survives on every domain (14/14). Reduction is per-query paired reduction relative to Top-

k

; aggregate reductions in Table 1 weight queries differently.

{}^{***}p_{\text{adj}}{<}10^{-3}

{}^{**}p_{\text{adj}}{<}10^{-2}

{}^{*}p_{\text{adj}}{<}0.05

Dataset	Method	#Entities	Mean $\Delta W_{1}$	95% CI
Yelp	$W_{1}$ Minimizer	10	10.85	$[8.26,13.37]$
Yelp	EG + $W_{1}$ Min	10	12.09	$[8.72,15.37]$
Yelp	$W_{1}$ -MMR	10	9.58	$[6.40,12.80]$
OpinRank	$W_{1}$ Minimizer	9	9.65	$[6.92,12.34]$
OpinRank	$W_{1}$ -MMR	9	10.22	$[6.48,14.17]$
Seller Forums	$W_{1}$ Minimizer	4	5.32	$[0.67,10.38]$

Appendix

Table 15: Entity-clustered bootstrap (10,000 resamples). Mean improvement in

W_{1}

relative to Top-

k

; 95% CIs computed by resampling entities with replacement. All CIs exclude zero.

Entity	$N$ reviews	$W_{1}(\text{star-}P_{\text{pop}},\text{LLM-}P_{\text{pop}})$
Booking Experience	1,372	3.66
Breakfast	788	1.44
Breakfast Quality	637	4.15
Business Center	142	3.23
Casino	208	1.36
Loyalty Program	607	1.57

Appendix

Table 16: Star-derived vs. LLM-derived

P_{\text{pop}}

per Yelp entity. Mean

W_{1}

gap: 2.88. Bounds:

W_{1}(\text{WARP},\text{star-}P_{\text{pop}})\leq 5.64

W_{1}(\text{Top-}k,\text{star-}P_{\text{pop}})\geq 10.72

; worst-case reduction

\geq 47.4\%

Figure 4: Dirichlet perturbation robustness across all three domains. Dashed red line = Top- $k$ baseline. At $\varepsilon{=}0.2$ , $W_{1}$ Minimizer remains 72% better than Top- $k$ on Yelp.

Figure 5: Mislabel sensitivity: $W_{1}$ vs. label error rate. Dashed red line = Top- $k$ baseline. $W_{1}$ Minimizer never crosses Top- $k$ even at 50% error on Yelp.

Method	Seller $W_{1}$	Yelp $W_{1}$	OpinRank $W_{1}$
Top- $k$ (baseline)	10.33	13.60	13.33
Top- $k$ + 5% Noise	10.07	13.00	12.97
Reduction	2.5%	4.4%	2.7%
Significance	ns ( $p>0.05$ , all domains)
$W_{1}$ Minimizer	8.84	2.76	4.14
Reduction	14.4%	79.7%	69.0%

Appendix

Table 17: Noise control: 5% Gaussian perturbation of retrieval scores yields negligible

W_{1}

improvement (2–5%,

p>0.05

), confirming that our 43–89% distributional improvements (

W_{1}

-MMR on sparse pools;

W_{1}

Minimizer / EG+

W_{1}

Min on dense) represent genuine distributional optimization.

Figure 6: $\lambda$ -sensitivity for $W_{1}$ -MMR (left) and OpinionMMR (right). $W_{1}$ -MMR is stable at $\lambda\leq 0.5$ and degrades at $\lambda{=}0.9$ . OpinionMMR shows higher $W_{1}$ across all domains with limited sensitivity to $\lambda$ .

Figure 7: $W_{1}$ vs. pool size $N$ ( $k{=}20$ ). $N{=}100$ suffices for dense pools; $W_{1}$ -MMR benefits from larger pools on sparse data.

Figure 8:

W_{1}(\hat{P}_{\text{pop}},P^{*}_{\text{pop}})

vs. entity-matched documents sampled. With 50–100 documents per entity, estimation error is comparable to or below the re-ranking improvement magnitude itself.

Judge	Consensus Tie Rate ( $k{=}5$ )	Consensus Tie Rate ( $k{=}10$ )
Claude Sonnet 4	46%	46%
Mistral Large 3	48%	61%
DeepSeek v3.2	63%	71%
Llama 3.3 70B	73%	86%
Amazon Nova Pro	79%	86%

Appendix

Table 18: Consensus tie rates across the 5-judge panel. Higher tie rates indicate stronger positional bias (position-swap converts inconsistent preferences to ties).

Judge (vs. others)	$k{=}5$ $\kappa_{d}$	$k{=}5$ Agree%	$k{=}10$ $\kappa_{d}$	$k{=}10$ Agree%
Claude vs. others	0.68	88.2%	0.86	93.5%
Llama 3.3 vs. others	0.89	96.0%	0.97	98.8%
Mistral L3 vs. others	0.89	95.8%	0.95	97.8%
DeepSeek vs. others	0.93	97.4%	0.95	97.9%
Nova Pro vs. others	0.88	95.3%	0.94	97.3%

Appendix

Table 19: Mean directional

\kappa_{d}

per judge vs. all others (fairness dimension). Non-Anthropic judges agree at

\kappa_{d}{\geq}0.95

k{=}10

; at

k{=}5

\kappa_{d}{=}0.88

–

0.93

	$k{=}5$			$k{=}10$			$k{=}20$
Approach	W/L/T	Fair%	Win%	W/L/T	Fair%	Win%	W/L/T	Fair%	Win%
Random	16/36/104	31%	10%	12/40/104	23%	8%	3/26/127	10%	2%
MMR ∗∗	39/8/109	83%	25%	21/9/126	70%	13%	5/12/139	29%	3%
OpinionMMR ∗∗	50/11/95	82%	32%	25/8/123	76%	16%	7/14/135	33%	4%
DPP ∗∗	54/7/95	89%	35%	42/6/108	88%	27%	25/3/128	89%	16%
WassRank OT ∗∗	66/28/62	70%	42%	75/28/53	73%	48%	33/3/120	92%	21%

Appendix

Table 20: 5-judge majority-vote generation results (cross-domain, 156 questions) at

k{\in}\{5,10,20\}

. All calibration methods achieve

p<0.001

k{=}5

; retain significance at

k{=}10

(

p\leq 0.021

) and

k{=}20

(

p\leq 0.004

). Random fails at all

k

values. At

k{=}20

, MMR/OpinionMMR Fair% collapse (29%/33%) as semantic diversity saturates; distributional methods maintain or grow their advantage; WassRank OT rises 70%

\to

73%

\to

92%. Fair% = W/(W+L). Win% = W/

N

	$k{=}5$		$k{=}10$
Approach	Prop. (A)	Cov. (B)	Prop. (A)	Cov. (B)
Stratified (oracle)	35%	—	—	—
$W_{1}$ Minimizer	34%	38%	30%	27%
DPP	26%	35%	24%	25%
$W_{1}$ -MMR	23%	36%	18%	17%
OpinionMMR	21%	30%	13%	19%

Appendix

Table 21: Judge prompt ablation: fairness win rate (%) under proportional vs. coverage definitions (3 domains, 156 questions per approach).

W_{1}

Minimizer dominates under proportional accuracy; the gap narrows under coverage. Stratified sampling (oracle) confirms the Prompt A criterion tracks the ground-truth distribution, not any specific retrieval strategy —

W_{1}

Minimizer matches oracle Fair% within 1 pp at

k{=}5

Approach	Info% (A)	Info% (B)	Grnd% (A)	Grnd% (B)
$W_{1}$ Minimizer	44%	40%	28%	24%
DPP	38%	40%	27%	24%
Random	19%	18%	15%	8%

Appendix

Table 22: Control dimensions remain stable across prompt variants (

k{=}5

\Delta

= Prompt B

-

Prompt A). Shifts

\leq

4 pp confirm the fairness prompt change does not contaminate other evaluation axes.

Domain	Method	$N$	W/L/T	Fair%	Win%
Seller Forums	$W_{1}$ Minimizer	36	10/9/17	53%	28%
	KL (JS) Minimizer	36	7/10/19	41%	19%
Yelp	$W_{1}$ Minimizer	60	39/4/17	91%	65%
	KL (JS) Minimizer	60	31/8/21	79%	52%
OpinRank	$W_{1}$ Minimizer	60	29/8/23	78%	48%
	KL (JS) Minimizer	60	35/10/15	78%	58%

Appendix

Table 23:

W_{1}

Minimizer vs. KL(JS) Minimizer, generation evaluation at

k{=}5

(156 queries, single-judge Claude Sonnet 4, position-controlled, blind pairwise vs. Top-

k

W_{1}

leads KL(JS) by +7 pp cross-domain; both significantly beat Top-

k

(

p{<}0.001

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Retrieval-Augmented Generation Must Move Beyond Factual Grounding to Represent Diverse Opinions

Aditya Agrawal, Alwarappan Nakkiran, Darshan Fofadiya, Alex Karlsson+2

Agentic Retrieval-Augmented Generation Systems Search-Augmented Reasoning Enrichment Analysis Dynamic Epistemic Logic

Open paper AskPDF arXiv

This position paper argues that Retrieval-Augmented Generation (RAG) systems exhibit a factual bias-optimizing for epistemic uncertainty reduction while ignoring the aleatoric uncertainty inherent in opinion-rich content. This misalignment demands a paradigm shift in RAG system design. A survey of 34 major RAG benchmarks reveals that only one addresses opinion synthesis, confirming that the bias is structural and embedded in datasets, retrieval-generation objectives, and evaluation metrics alike. Beyond technical limitations, this bias poses risks to transparent and accountable AI. Namely, echo chamber effects that amplify dominant viewpoints, which can lead to opinion manipulation and under-representation of minority voices. We formalize the problem through the lens of uncertainty quantification, showing that factual queries should minimize posterior entropy while opinion queries must preserve it. We derive a unified objective over coverage, fidelity, and fairness using the Wasserstein distance. As an existence proof, we present Opinion-Aware RAG (O-RAG), an architecture featuring LLM-based opinion extraction and entity-linked opinion metadata. We evaluate it across two domains -- e-commerce seller forums and public hotel reviews. Experiments demonstrate 18-48% reduction in Wasserstein distance to corpus-level sentiment distributions, +26.8% sentiment diversity, and +42.7% entity match rate. Human evaluators preferred opinion-enriched generation 79.2% of the time. We propose a research agenda and argue that as RAG systems increasingly mediate access to information, their ability to represent diverse perspectives is of the essence.

Abstract

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