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Directional Posterior Feedback Fusion for Query Reformulation in Dense Retrieval

Yiteng Tu, Qingyi Pan, Weihang Su, Jiaxin Mao, Yiqun Liu, Qingyao Ai

arXiv:2610.29464v1cs.IR

TL;DR

Dense retrieval depends on query embeddings capturing users’ needs, but short, ambiguous, or underspecified queries may not do so. DIFF treats feedback as confidence-weighted directional evidence to infer controlled query updates; across BEIR subsets, it delivers the strongest and most consistent retrieval performance across retrievers and feedback sources.

  • Problem

    Dense retrieval depends on query embeddings capturing users’ information needs, yet short, ambiguous, or underspecified queries may only partially describe relevant information.

    Source: §1 · p. 1

  • Method

    DIFF converts feedback vectors into confidence-weighted directional evidence in the original query’s tangent space and jointly fuses it into a controlled update without retraining the retriever.

    Source: §1 · p. 2, §3.1 · p. 3, §3.4 · p. 4

  • Results

    DIFF delivers the strongest and most consistent performance across BEIR subsets, dense retrievers, PRF, and three forms of GRF.

    Source: §5.1 · p. 5

  • Takeaways & Limitations

    Directional constraints let feedback items contribute along supported directions while preserving complementary information from other feedback.

    Source: §1 · p. 2, §7 · p. 9

Abstract

from arXiv · show

Dense retrieval has become a fundamental paradigm for modern information retrieval, supporting efficient semantic search across a wide range of applications. Its effectiveness, however, relies critically on query embeddings that adequately capture users' information needs, which can be difficult for short, ambiguous, or underspecified queries. Feedback-based query reformulation approaches like pseudo-relevance feedback (PRF) alleviate this by incorporating additional evidence from retrieved or generated content. However, existing embedding-space approaches commonly aggregate or interpolate feedback vectors, implicitly treating them as complete target representations. Such pointwise fusion can unnecessarily suppress query corrections along directions not expressed by individual feedback items, even when these directions are supported by complementary evidence. To address this limitation, we propose Directional Posterior Feedback Fusion (DIFF), a query reformulation framework that models feedback as directional evidence rather than target points. DIFF represents feedback as confidence-weighted directional evidence in the tangent space of the original query and jointly infers a controlled query update through prior-regularized posterior fusion, without retraining the underlying dense retriever. Experiments across multiple dense retrievers and different feedback sources show that DIFF achieves the best overall retrieval performance, demonstrating the effectiveness of explicitly modeling how feedback should constrain query updates.

1 Introduction

Short, ambiguous, or underspecified queries may not capture users’ information needs, motivating feedback-based reformulation. DIFF treats feedback as directional evidence and uses prior-regularized fusion to infer a controlled update; experiments report the best overall retrieval performance across diverse settings.

1 Introduction

  • Feedback can come from retrieved documents or LLM-generated content, while the framework focuses on how feedback vectors modify the original query.The paper identifies PRF and generated hypothetical documents, pseudo-answers, and query expansions as feedback sources.Source: §1 · p. 1
  • Pointwise fusion treats each feedback vector as a complete target, potentially discouraging query changes along directions that the feedback does not express.Complementary feedback may support changes along those otherwise unexpressed directions.Source: §1 · p. 1
  • DIFF represents feedback as directional evidence in the original query’s tangent space and jointly fuses it with a prior favoring conservative updates.The approach uses local- and set-level confidence control and has an efficient closed-form solution.Source: §1 · pp. 1–2
  • DIFF achieves the best overall retrieval performance across multiple retrieval benchmarks and dense retrievers.The framework uses fixed pretrained embeddings without retriever updates or an additional reformulation model.Source: §1 · p. 2

2 Preliminary

Dense retrieval represents queries and documents as normalized vectors, making query reformulation a change in direction. The preliminary section motivates replacing point-target fusion with directional constraints so complementary feedback can inform the update without undue suppression.

2.1 Dense Retrieval

  • With L2-normalized query and document embeddings, retrieval uses their inner product, equal to cosine similarity, and query changes primarily alter direction.This geometric view motivates reformulation locally around the original query.Source: §2.1 · p. 2

2.2 Feedback-based Query Reformulation

  • Feedback-based reformulation produces a normalized query representation from feedback that may come from retrieved documents or generated auxiliary texts.The paper focuses on utilizing feedback regardless of how it was obtained; averaging and interpolation are common pointwise approaches.Source: §2.2 · p. 2

2.3 From Point Targets to Directional Evidence

  • Pointwise fusion can shrink evidence from complementary feedback, whereas directional fusion constrains each item only along its supported direction.With K equally confident complementary items, pointwise recovery is τ/(λ + Kτ), while directional recovery is τ_j/(λ + τ_j).Source: §2.3 · p. 3

3 Methodology

DIFF treats feedback as noisy directional constraints in the tangent space of the original query, then fuses them under an original-query prior. Hierarchical confidence and a controlled spherical update regulate the evidence’s influence.

3.1 Overview

  • DIFF represents each feedback item as a tangent-space direction, a target projection along that direction, and a confidence-weighted constraint on the query correction.The posterior combines these directional observations with a prior favoring conservative updates around the original query.Source: §3.1 · p. 3

3.2 Directional Evidence Construction

  • Each feedback vector is split into an orthogonal direction and magnitude: the direction proposes where to correct, while the magnitude measures orthogonal novelty.Query-feedback similarity is used to estimate reliability, not to define the proposed correction.Source: §3.2 · pp. 3–4

3.3.1 Local Quality.

  • Local quality uses robust query-relative MAD normalization followed by a sigmoid, giving higher influence to feedback with stronger affinity to that query.This avoids a fixed similarity threshold and reliance on a global similarity scale.Source: §3.3.1 · p. 4

3.3.2 Set-level Confidence.

  • Set-level confidence combines directional coherence with average local quality, so mutually consistent but weak feedback does not alone justify a strong update.Geometric confidence reflects shared signal strength relative to disagreement among directions.Source: §3.3.2 · p. 4

3.4 Posterior Inference and Spherical Update

  • Posterior inference solves a K×K rather than D×D system, with complexity O(K^2D + K^3) when feedback count K is much smaller than embedding dimension D.Set-level confidence also caps the angular update before the tangent correction is mapped onto the unit sphere.Source: §3.4 · p. 4
  • DIFF separates which directions feedback supports, how much each observation is trusted, and how far the query moves; zero correction leaves the query unchanged.The prior and final spherical update keep the reformulation controlled in normalized embedding space.Source: §3.4 · p. 4

4 Experimental Setup

The evaluation uses BEIR subsets, three dense retrievers, and both retrieved and LLM-generated feedback. DIFF is compared with the original retriever and training-free fusion baselines under controlled configurations.

4.1 Datasets and Evaluation Metrics

  • The evaluation covers 14 BEIR subsets and reports NDCG@10 as the primary metric and Recall@100 as complementary retrieval coverage.The subsets span fact verification, question answering, argument retrieval, scientific retrieval, and other scenarios.Source: §4.1 · p. 5

4.2 Retriever and Feedback Construction

  • Across BGE, E5, and Qwen, queries, documents, and feedback use the same embedding model; only query and feedback representations are L2-normalized before reformulation, whose output uses the original index without retriever-parameter updates.Source: §4.2 · p. 5

4.3 Baselines

  • Feedback comes from top-K retrieved documents or LLM-generated HyDE, Q2D, and CoT content; the main results use K=5.Comparisons include the original retriever and fusion-based baselines, with the same encoder, initial ranking, feedback set, and document index within each configuration.Source: §4.2 · p. 5, §4.3 · p. 5

5 Results and Analysis

Across retrievers and feedback sources, DIFF delivers the strongest and most consistent retrieval performance. Its directional fusion and confidence controls support robust reformulation across varying feedback quality and size.

5.1 Main Results

  • Across BEIR subsets, DIFF delivers the strongest and most consistent performance across retrievers and feedback sources.The comparison covers PRF and three forms of GRF; Table 1 reports average NDCG@10 and Recall@100.Source: §5.1 · p. 5, Table 1 · p. 6
  • DIFF remains robust to noisy feedback and outperforms the closest competitor, AnchorQE, without additional retrieval operations.Simple averaging or concatenation can degrade performance, particularly with noisy PRF; AnchorQE requires additional retrieval operations.Source: §5.1 · pp. 5–6

5.2 Ablation Study

  • The complete DIFF performs best in ablations, supporting directional constraints over point targets and reliability control at both item and set levels.Point-target fusion has one of the largest performance drops under both feedback sources; removing confidence components generally degrades results.Source: §5.2 · p. 7

5.3 Analysis

  • As feedback size K increases, DIFF remains stable, while VPRF under PRF deteriorates as lower-ranked documents add noise.GRF performance is generally higher and more stable as K increases; Figure 2 examines the effect of feedback size.Source: §5.3 · p. 7, Figure 2 · p. 8

5.3.2 Case Study: Geometric Behavior of Query Reformulation.

  • In the geometric case study, DIFF controls PRF updates, while confidence control makes updates more conservative for both PRF and HyDE.Without confidence control, the PRF update is noticeably larger; in HyDE, the DIFF variants move in a similar direction.Source: §5.3.2 · p. 7, Figure 3 · p. 8

6 Related Work

Dense retrieval depends on query representations that capture users’ information needs, while query reformulation adds evidence from retrieved or generated feedback. DIFF addresses how that feedback is incorporated by treating it as confidence-weighted directional constraints rather than target representations.

6.1 Dense Retrieval

  • Dense retrieval encodes queries and documents as vectors for similarity-based retrieval, but short or underspecified queries may not fully express the information need.The paper positions its query-refinement approach as complementary to retriever training improvements and keeps the pretrained retriever fixed.Source: §6.1 · p. 8

6.2 Query Reformulation

  • PRF and generative relevance feedback broaden available evidence, while DIFF models feedback as confidence-weighted directional constraints rather than target points.Existing embedding-space methods often aggregate or interpolate feedback representations; DIFF works across feedback sources without retraining the retriever.Source: §6.2 · p. 8

7 Conclusions

DIFF models feedback as directional evidence in the original query’s tangent space, enabling selective corrections while retaining complementary information. Experiments demonstrate its effectiveness and robustness, and the paper identifies broader geometric modeling and richer reliability estimation as future directions.

7 Conclusions

  • DIFF lets each feedback item constrain only its supported correction direction, preserving complementary information from other feedback.The authors identify extending beyond local tangent-space modeling, richer reliability estimation, and joint use of heterogeneous feedback sources as future work.Source: §7 · pp. 8–9
  • Future work includes modeling more complex embedding-space geometry and richer feedback reliability or heterogeneous-source mechanisms.Source: §7 · p. 9

Algorithm 1 Directional Posterior Feedback Fusion (DIFF)

Algorithm 1 takes feedback and update controls and outputs a reformulated query through directional evidence, hierarchical confidence, posterior inference, and a spherical update. The accompanying decomposition explains why pointwise fitting can suppress complementary corrections.

Algorithm 1 Directional Posterior Feedback Fusion (DIFF)

  • Algorithm 1 organizes DIFF into directional-evidence construction, hierarchical confidence, posterior inference, and a spherical update to produce q′.Its inputs include prior strength and maximum update angle; the posterior step forms a direction matrix and diagonal weights from feedback precisions.Source: Algorithm 1 Directional Posterior… · p. 10

A.1 Decomposition of Pointwise Feedback Fitting

  • Pointwise fitting penalizes correction components orthogonal to each feedback direction, potentially suppressing complementary corrections supported by other items.Source: §A.1 · p. 10

A.2 Proof of Theorem 2.1

The theorem establishes conditions on when the directional and pointwise objectives agree, and the DIFF objective can be solved efficiently using a system sized by the number of feedback items. The algorithm then uses directional evidence, confidence estimation, posterior inference, and a spherical update.

A.2 Proof of Theorem 2.1

  • Under the theorem assumptions, the directional and pointwise objectives have unique tangent-space minimizers that coincide precisely when the supported directions are collinear.The proof also characterizes the objectives’ relationship through a nonnegative penalty on components orthogonal to supported directions.Source: §A.2 · pp. 10–11

B Efficient Solution to the DIFF Objective

  • The efficient solution replaces a D×D system with a K×K system, yielding complexity O(K^2D + K^3) when K≪D.Here, K is the number of feedback items and D is the embedding dimension.Source: B Efficient Solution to the DIFF… · p. 11

C DIFF Algorithm

  • DIFF’s workflow comprises directional evidence calculation, hierarchical confidence calculation, posterior inference, and a spherical update.The posterior inference step solves the optimization objective.Source: C DIFF Algorithm · p. 11

D Implementation Details

The experiments report subset-level NDCG@10 results and analyze query-level changes and DIFF’s hyperparameters. The reported change patterns differ by feedback source, as do preferred update bounds.

D Implementation Details

  • The implementation uses batch size 256 and FP16 inference, selects hyperparameters by grid search, and averages experiments over five runs on one NVIDIA A800 GPU.For DIFF, the search covers λ∈{0.5, 1, 2} and θup∈{π/12, π/6, π/4}.Source: D Implementation Details · p. 12

E.1 NDCG@10 on Subsets

  • Table 3 reports detailed NDCG@10 results for all subsets.Source: §E.1 · p. 12

E.2 NDCG@10 Changes (ΔNDCG@10)

  • Under PRF, VPRF degrades more queries than it improves, while DIFF leaves most unchanged and has nearly balanced improvements and degradations; under HyDE, DIFF produces more improvements than degradations.DIFF’s ΔNDCG@10 distribution is described as tightly concentrated around zero under PRF; under HyDE, it preserves a large fraction of queries.Source: §E.2 · p. 12

E.3 Hyperparameter Analysis (λ and θup)

  • A smaller update bound performs consistently well under PRF, whereas HyDE has competitive performance with larger bounds; increasing λ tends to mitigate the effect of a larger bound.The analysis links λ to regularizing the inferred correction and θup to bounding the final angular displacement.Source: §E.3 · p. 12

E.4 Efficiency Analysis

DIFF’s execution time is compared with other query reformulation methods on 10 queries. It is comparable to lightweight baselines, while AnchorQE takes approximately 3.67 times as long.

E.4 Efficiency Analysis

  • AnchorQE takes approximately 3.67 times DIFF’s execution time, attributed to its additional retrieval operations for estimating feedback contributions.Source: §E.4 · p. 13
  • DIFF has comparable efficiency to the lightweight baselines VPRF, Concat-P, and DIME.Source: §E.4 · p. 13
  • The efficiency comparison measures execution time on 10 queries.Source: §E.4 · p. 13

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