Uncertainty-driven Embedding Convolution (UEC)

Updated 4 July 2026

The paper introduces UEC as a novel post-hoc framework that converts deterministic text embeddings into Gaussian representations using a diagonal Laplace approximation.
It applies adaptive uncertainty-weighted fusion across multiple pretrained models to produce input-specific ensemble weights and uncertainty-aware similarity scores.
Empirical evaluations on retrieval, classification, and STS tasks show that UEC significantly boosts performance with minimal added computational overhead.

Searching arXiv for the primary UEC paper and closely related uncertainty-aware embedding work. Uncertainty-driven Embedding Convolution (UEC) is a post-hoc framework for combining multiple pretrained text embedding models by first converting each deterministic embedding into a probabilistic embedding, then computing adaptive ensemble weights from embedding uncertainty, and finally evaluating similarity with an uncertainty-aware score (Lim et al., 28 Jul 2025). In the formulation introduced in "Uncertainty-driven Embedding Convolution" (Lim et al., 28 Jul 2025), the method is designed for settings in which no single embedding model consistently performs best across domains, languages, or tasks, and it treats uncertainty as a first-class signal for both fusion and retrieval-time scoring rather than as a purely auxiliary diagnostic.

1. Definition, scope, and terminological placement

UEC was introduced for text embeddings as an uncertainty-aware ensemble method over multiple pretrained encoders such as BGE, E5, and GTE (Lim et al., 28 Jul 2025). Its motivation is that embedding models are often complementary, but conventional ensemble methods such as uniform averaging, manually weighted averaging, and deterministic model merging treat embeddings as equally reliable and therefore cannot down-weight a model that is unreliable on a particular input, express confidence in the fused representation, or calibrate similarity scoring by uncertainty (Lim et al., 28 Jul 2025).

The framework takes, for an input text $x$ , $K$ pretrained embedding models and converts each model output into a Gaussian embedding

$z_k(x) \sim \mathcal{N}(\mu_k(x), \Sigma_k(x)).$

It then produces a fused probabilistic embedding

$z(x) \sim \mathcal{N}(\mu(x), \Sigma(x)),$

together with an uncertainty-aware scalar similarity score between two inputs (Lim et al., 28 Jul 2025).

The term “convolution” in UEC refers to Gaussian convolution in embedding space, not to a literal CNN-style convolution. Concretely, the method defines the fused embedding as a weighted linear combination of independent Gaussian embeddings, with exact uncertainty propagation under independence (Lim et al., 28 Jul 2025). This distinguishes UEC from architectures in which uncertainty modulates spatial kernels or graph message-passing operators.

A useful contextual distinction is that other papers use closely related terminology differently. "Rectified Iterative Disparity for Stereo Matching" (Xiao et al., 2024) defines UEC as Cost volume-based disparity Uncertainty Estimation, not Uncertainty-driven Embedding Convolution. Conversely, earlier work such as "Convolutional Gaussian Embeddings for Personalized Recommendation with Uncertainty" (Jiang et al., 2020) can be read as a precursor in the broader sense of uncertainty-aware embedding plus convolutional interaction modeling, but it does not use the term UEC. This suggests that UEC, in the strict sense, is best reserved for the text-embedding framework of (Lim et al., 28 Jul 2025).

2. Probabilistic embedding construction

UEC begins by transforming deterministic embeddings into probabilistic ones using Laplace approximation (LA) applied only to the last layer (Lim et al., 28 Jul 2025). With training data

$\mathcal{D} = \{(x,y)\},$

Bayes’ theorem is written as

$p(w \mid \mathcal{D}) = \frac{p(\mathcal{D}\mid w)p(w)}{p(\mathcal{D})}.$

Because the exact posterior is intractable, the method approximates it with a Gaussian centered at the MAP estimate $\hat w$ , using the second-order expansion

$-\log p(w\mid\mathcal{D}) \approx - \log p(\hat w\mid\mathcal{D}) + \frac{1}{2}(w-\hat w)^\top H_{\hat w}(w-\hat w),$

with Hessian

$H_{\hat w} = \nabla^2\big[-\log p(\mathcal{D}\mid w)-\log p(w)\big]\Big|_{w=\hat w},$

so that

$p(w\mid\mathcal{D}) \approx \mathcal{N}(\hat w, H_{\hat w}^{-1}).$

In practice, the method uses diagonal Laplace approximation on the final layer rather than a full-covariance posterior over all parameters (Lim et al., 28 Jul 2025).

Let $K$ 0 be an embedding model with $K$ 1 layers, $K$ 2 the final-layer parameters, and $K$ 3 the penultimate representation. In the deterministic setting,

$K$ 4

Under the Laplace approximation, the final-layer posterior becomes

$K$ 5

which induces a Gaussian output embedding

$K$ 6

Thus each encoder produces a mean $K$ 7 and covariance $K$ 8 without retraining the full model (Lim et al., 28 Jul 2025).

The uncertainty represented here is at the embedding/model output level, is input-dependent through $K$ 9, and is represented in practice by a diagonal / per-dimension covariance approximation, not by a full covariance matrix (Lim et al., 28 Jul 2025). The paper explicitly notes that this is post-hoc but not fully data-free, because a small labeled dataset is still needed to fit the diagonal Laplace approximation. For MIRACL, the setup uses binary-labeled query-passage pairs with 50 labeled examples per selected language/model setting. For MMTEB, LA is fitted using subsets of MS MARCO and SNLI, specifically 3,983 query-passage pairs from MS MARCO and 3,775 sentence pairs from SNLI; the implementation uses the laplace-redux library (Lim et al., 28 Jul 2025).

3. Gaussian convolution and uncertainty-driven weighting

Given $z_k(x) \sim \mathcal{N}(\mu_k(x), \Sigma_k(x)).$ 0 independent embedding models,

$z_k(x) \sim \mathcal{N}(\mu_k(x), \Sigma_k(x)).$ 1

UEC defines the fused embedding as

$z_k(x) \sim \mathcal{N}(\mu_k(x), \Sigma_k(x)).$ 2

Under the independence assumption, the fused representation remains Gaussian: $z_k(x) \sim \mathcal{N}(\mu_k(x), \Sigma_k(x)).$ 3 This is the paper’s formal notion of Gaussian convolution (Lim et al., 28 Jul 2025).

The weighting rule is motivated through a surrogate to contrastive learning. For a positive pair $z_k(x) \sim \mathcal{N}(\mu_k(x), \Sigma_k(x)).$ 4, the method defines

$z_k(x) \sim \mathcal{N}(\mu_k(x), \Sigma_k(x)).$ 5

which expands to

$z_k(x) \sim \mathcal{N}(\mu_k(x), \Sigma_k(x)).$ 6

The paper interprets the mean-distance term as fidelity and the trace terms as uncertainty (Lim et al., 28 Jul 2025).

For retrieval, because document embeddings are precomputed offline, the method retains only query-dependent terms and solves

$z_k(x) \sim \mathcal{N}(\mu_k(x), \Sigma_k(x)).$ 7

The appendix defines

$z_k(x) \sim \mathcal{N}(\mu_k(x), \Sigma_k(x)).$ 8

and derives the optimal coefficients as

$z_k(x) \sim \mathcal{N}(\mu_k(x), \Sigma_k(x)).$ 9

Since the embeddings are typically $z(x) \sim \mathcal{N}(\mu(x), \Sigma(x)),$ 0-normalized so that $z(x) \sim \mathcal{N}(\mu(x), \Sigma(x)),$ 1, the paper simplifies this to inverse-uncertainty weighting: $z(x) \sim \mathcal{N}(\mu(x), \Sigma(x)),$ 2 The weighting is therefore analogous to precision-weighted model averaging, and the paper characterizes it as Bayes-optimal under the surrogate loss and the stated assumptions (Lim et al., 28 Jul 2025).

This weighting is data-wise rather than global. A heatmap analysis shows that Arabic inputs receive higher weight from Arabic-specialized embeddings, while Chinese inputs favor Chinese embeddings (Lim et al., 28 Jul 2025). This suggests that UEC is not merely an uncertainty-aware ensemble in the average-case sense; it is an input-conditional mixture governed by posterior variance estimates.

4. Uncertainty-aware similarity and retrieval-time scoring

Once two inputs $z(x) \sim \mathcal{N}(\mu(x), \Sigma(x)),$ 3 and $z(x) \sim \mathcal{N}(\mu(x), \Sigma(x)),$ 4 have fused Gaussian embeddings,

$z(x) \sim \mathcal{N}(\mu(x), \Sigma(x)),$ 5

UEC treats similarity as a random variable. Rather than using KL divergence between Gaussians, which the paper states is too expensive for large-scale retrieval because it requires matrix inverses and log-determinants, the method extends cosine similarity through a Gaussian approximation of the dot product (Lim et al., 28 Jul 2025).

After $z(x) \sim \mathcal{N}(\mu(x), \Sigma(x)),$ 6-normalization, the similarity is

$z(x) \sim \mathcal{N}(\mu(x), \Sigma(x)),$ 7

and is approximated by

$z(x) \sim \mathcal{N}(\mu(x), \Sigma(x)),$ 8

with

$z(x) \sim \mathcal{N}(\mu(x), \Sigma(x)),$ 9

and

$\mathcal{D} = \{(x,y)\},$ 0

The first term is the ordinary mean alignment, while the remaining terms quantify uncertainty projected along the matched directions and the covariance interaction term (Lim et al., 28 Jul 2025).

The final uncertainty-aware similarity is then defined by the probit-style approximation

$\mathcal{D} = \{(x,y)\},$ 1

Compared with ordinary cosine similarity, which would use only $\mathcal{D} = \{(x,y)\},$ 2, UEC divides the mean similarity by a quantity that grows with uncertainty. This makes uncertainty a direct confidence penalty: if uncertainty is low, $\mathcal{D} = \{(x,y)\},$ 3; if uncertainty is high, the score is downscaled (Lim et al., 28 Jul 2025).

The method therefore uses uncertainty in two separate but coupled places: first in the ensemble coefficients $\mathcal{D} = \{(x,y)\},$ 4, and second in the similarity score itself through $\mathcal{D} = \{(x,y)\},$ 5 (Lim et al., 28 Jul 2025). The ablation results indicate that both parts matter. On the MIRACL subset, the full UEC system yields NDCG@10 = 59.65, Recall@10 = 80.07, and AUC@10 = 91.04. Removing uncertainty-aware similarity gives 58.72 / 78.13 / 82.48; removing uncertainty-driven convolution gives 48.45 / 66.69 / 10.30; removing both gives 46.78 / 62.66 / 4.01 (Lim et al., 28 Jul 2025). The especially large degradation in AUC@10 when uncertainty-driven convolution is removed supports the paper’s claim that adaptive uncertainty-weighted fusion is the more critical component.

5. Empirical evaluation, efficiency, and operating regime

UEC is evaluated on retrieval, classification, and Semantic Textual Similarity (STS), with appendix experiments additionally covering Bitext Mining, Clustering, and Reranking (Lim et al., 28 Jul 2025). The retrieval benchmarks include a MIRACL multilingual subset with English, Russian, Chinese, and Arabic, each with 25 query-passage pairs per language, and an MMTEB retrieval subset containing SCIDOCS, LegalBenchCorporateLobbying, BelebeleRetrieval, WikipediaRetrievalMultilingual, and StackOverflowQA. STS is evaluated on ten datasets including STSBenchmark, FinParaSTS, SICK-R, STS22.v2, SemRel24STS, and STS12–STS17. Classification uses FinancialPhrasebankClassification, SwissJudgementClassification, PoemSentimentClassification, MassiveIntentClassification, and TweetTopicSingleClassification (Lim et al., 28 Jul 2025).

The reported metrics are NDCG@10, Recall@100 or Recall@10, and AUC@10 for retrieval; Accuracy and F1 for classification; Spearman correlation for STS; F1 for bitext mining; V-measure for clustering; and MAP for reranking (Lim et al., 28 Jul 2025). In this setup, AUC@10 is used as an uncertainty metric via abstention.

The main quantitative findings are consistent across tasks. On the MIRACL multilingual subset, UEC reports average NDCG@10 = 59.65, Recall@10 = 80.07, and AUC@10 = 91.04, compared with 48.83, 65.65, and 30.87 for the weighted ensemble baseline (Lim et al., 28 Jul 2025). On the MMTEB retrieval subset, UEC achieves Avg NDCG@10 = 78.86, Avg Recall@100 = 90.59, and Avg AUC@10 = 66.05. On STS, it reports Avg Spearman = 76.39 and ranks first on 8 of 10 datasets. On classification, it reports Avg Accuracy = 68.89 and Avg F1 = 61.04 (Lim et al., 28 Jul 2025). Appendix averages are also strongest for Bitext mining with Avg F1 = 32.39, Clustering with Avg V-measure = 42.79, and Reranking with Avg MAP = 53.33 (Lim et al., 28 Jul 2025).

The framework is also designed to be lightweight. The paper states that it preserves time complexity $\mathcal{D} = \{(x,y)\},$ 6 and memory complexity $\mathcal{D} = \{(x,y)\},$ 7, where $\mathcal{D} = \{(x,y)\},$ 8 is the number of models and $\mathcal{D} = \{(x,y)\},$ 9 is the embedding dimension, and reports only 0.6% additional runtime for similarity estimation (Lim et al., 28 Jul 2025). In the efficiency table, uniform and weighted baselines are normalized to 1.000 runtime, whereas UEC reports 1.006. This is important because the method adds sample-specific weighting and uncertainty-aware scoring without retraining base encoders.

The operating hyperparameters reported for the experimental protocol include a temperature parameter for sharpening ensemble coefficients and a parameter $p(w \mid \mathcal{D}) = \frac{p(\mathcal{D}\mid w)p(w)}{p(\mathcal{D})}.$ 0 to scale the influence of $p(w \mid \mathcal{D}) = \frac{p(\mathcal{D}\mid w)p(w)}{p(\mathcal{D})}.$ 1 in uncertainty-aware similarity. For MIRACL, the temperature is fixed at 1.5, with $p(w \mid \mathcal{D}) = \frac{p(\mathcal{D}\mid w)p(w)}{p(\mathcal{D})}.$ 2; for MMTEB, the temperature is fixed at 1.8 with the same $p(w \mid \mathcal{D}) = \frac{p(\mathcal{D}\mid w)p(w)}{p(\mathcal{D})}.$ 3 search range (Lim et al., 28 Jul 2025). The exact modified formula using temperature or $p(w \mid \mathcal{D}) = \frac{p(\mathcal{D}\mid w)p(w)}{p(\mathcal{D})}.$ 4 is not explicitly written in the main equations.

UEC belongs to a broader family of uncertainty-aware embedding methods, but its mechanism is distinct. "Exploring Uncertainty Measures for Image-Caption Embedding-and-Retrieval Task" (Hama et al., 2019) also uses uncertainty for model averaging and reliability assessment, and shows that posterior uncertainty is better than feature uncertainty for query rejection in retrieval. However, that work does not define a Gaussian fusion rule or an uncertainty-aware similarity of the UEC form. "Convolutional Gaussian Embeddings for Personalized Recommendation with Uncertainty" (Jiang et al., 2020) represents users and items as Gaussian embeddings and applies a CNN over a sampled interaction cube, making it a plausible precursor in the broader sense of uncertainty-aware embedding plus convolution. Yet its uncertainty enters through Monte Carlo sampling over user-item latent pairs rather than through post-hoc Laplace approximation, closed-form uncertainty-driven coefficients, and uncertainty-penalized similarity (Jiang et al., 2020).

Other neighboring works illuminate what UEC is not. "REGE: A Method for Incorporating Uncertainty in Graph Embeddings" (Shafi et al., 2024) is an uncertainty-aware graph embedding wrapper that augments node embeddings with a scalar radius and uses uncertainty-conditioned perturbation of latent states plus curriculum over graph views. It is explicitly not a new graph convolution primitive. Similarly, "U3-xi: Pushing the Boundaries of Speaker Recognition via Incorporating Uncertainty" (Li et al., 22 Jan 2026) is best interpreted as uncertainty-aware pooling and uncertainty-weighted embedding aggregation rather than uncertainty-driven convolution. These comparisons suggest that UEC is most precisely characterized as a probabilistic embedding fusion and scoring framework, not as a generic label for all uncertainty-aware embedding models.

A terminological caveat is necessary because the acronym is overloaded. In stereo matching, UEC denotes Cost volume-based disparity Uncertainty Estimation (Xiao et al., 2024). That method also operates on learned matching representations and uses a lightweight residual head, but it is not Uncertainty-driven Embedding Convolution. The semantic overlap is limited to the idea that uncertainty should be extracted from learned representations and then used to improve downstream refinement.

The principal limitations stated for UEC are also structurally informative (Lim et al., 28 Jul 2025). The method is validated mainly on text embeddings; it uses diagonal LA, so richer uncertainty estimators may exist; it requires that all embeddings have the same dimensionality; it still needs a small labeled dataset to fit LA, so it is post-hoc but not fully data-free; and its derivations assume independence among model outputs. A plausible implication is that performance may degrade when ensemble members are highly correlated, when diagonal LA provides poor variance estimates, or when the calibration data are mismatched. The paper’s Bayes-optimality claim is correspondingly restricted: it is optimal under the specified surrogate loss and assumptions, not for arbitrary downstream task objectives (Lim et al., 28 Jul 2025).

In its most concise technical form, UEC can therefore be summarized by three coupled operations: post-hoc conversion of deterministic embeddings into Gaussian embeddings by diagonal Laplace approximation, uncertainty-driven Gaussian convolution with coefficients

$p(w \mid \mathcal{D}) = \frac{p(\mathcal{D}\mid w)p(w)}{p(\mathcal{D})}.$ 5