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Quantifying the Uncertainty of Foundation Models with Singular Value Ensembles

The paper proposes Singular Value Ensemble (SVE), a parameter-efficient method that quantifies epistemic uncertainty in foundation models by training only singular values within a shared basis of singular vectors, achieving calibration comparable to standard ensembles with less than a 1% increase in parameters.

Original authors: Mehmet Ozgur Turkoglu, Dominik J. Mühlematter, Alexander Becker, Konrad Schindler, Helge Aasen

Published 2026-06-01

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CC BY 4.0

Original authors: Mehmet Ozgur Turkoglu, Dominik J. Mühlematter, Alexander Becker, Konrad Schindler, Helge Aasen

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Problem: The Overconfident Expert

Imagine you have a brilliant, world-class expert (a Foundation Model) who has read almost every book in the library. They are amazing at answering questions. However, there is a catch: this expert is dangerously overconfident.

If you ask them a question they know the answer to, they are 100% sure. But if you ask them something completely made up or outside their training (like "What is the capital of a planet that doesn't exist?"), they will still answer with 100% confidence, even though they are completely wrong.

In the real world, this is dangerous. If a doctor's AI says, "I am 100% sure this is a healthy heart," but it's actually sick, the patient might not get treatment. We need a way to know when the expert doesn't know.

The Old Solution: The Committee (Ensembles)

The standard way to fix this is to hire a committee of experts instead of just one.

How it works: You train 5 or 10 different experts from scratch. When they all agree, you trust the answer. When they disagree, you know the answer is uncertain.
The Problem: This is incredibly expensive. Training one giant expert takes a lot of money and electricity. Training 10 of them takes 10 times the money. For the biggest, most powerful AI models, hiring a committee is often impossible for regular people or small companies.

The New Idea: The "Singular Value Ensemble" (SVE)

The authors propose a clever trick called Singular Value Ensemble (SVE). Instead of hiring 10 different experts, they take one expert and create 10 slightly different "personalities" for them using a mathematical shortcut.

Here is how it works, broken down into three steps:

1. The Library of Knowledge (Singular Vectors)

Imagine the expert's brain is a massive library. Inside, the books (knowledge) are organized on shelves.

The Singular Vectors are the shelves. They represent the fundamental, meaningful directions of knowledge the expert learned during pre-training (e.g., "how to talk about cats," "how to do math").
The Singular Values are the volume knobs on those shelves. They decide how loud or important each specific piece of knowledge is.

The paper argues that the shelves (vectors) are perfect and shouldn't be touched. They contain the core wisdom. But the volume knobs (values) can be tweaked to change how the expert thinks.

2. The "Volume Knob" Trick

Instead of training 10 new experts, SVE takes the one expert and creates 10 copies.

The Rule: All 10 copies share the exact same shelves (the frozen knowledge).
The Twist: Each copy gets to turn its own volume knobs up or down slightly differently.
- Copy A might turn up the "math" knowledge and turn down the "poetry" knowledge.
- Copy B might do the opposite.

Because they are adjusting the volume of the same knowledge in different ways, they start to give slightly different answers to the same question. This disagreement tells us: "Hey, we aren't sure about this!"

3. Why It's a Miracle of Efficiency

This is the magic part.

Old Way (Committee): To get 10 experts, you need 10x the brain power and memory.
SVE Way: You only need one brain. You just add a tiny list of numbers (the volume knobs) for each copy.
The Result: The paper shows that this method adds less than 1% to the computer's memory usage. It's like hiring a committee of 10 experts, but you only pay for the salary of one, plus a tiny tip for the rest.

What the Paper Found

The authors tested this on many tasks, from recognizing flowers and pets to answering complex reasoning questions.

It Works: The "Volume Knob" method was just as good at spotting uncertainty as hiring a full committee of 10 experts.
It's Accurate: The experts didn't just get confused; they actually got better at answering questions correctly compared to just using one expert.
It Scales: It worked on small models and massive models (like the 7-billion-parameter LLaMA-2).
It's Fast: Because it doesn't require training 10 separate giant brains, it saves a massive amount of time and energy.

The Bottom Line

The paper introduces a way to make powerful AI models "humble." Instead of making them overconfident, this method lets them say, "I'm not sure," when they encounter something new. It does this by creating a "committee" of one expert with different volume settings, saving huge amounts of money and computing power while keeping the AI safe and reliable.

Technical Summary: Quantifying the Uncertainty of Foundation Models with Singular Value Ensembles

1. Problem Statement

Foundation models have achieved remarkable performance across diverse tasks through large-scale pretraining, yet they frequently produce overconfident and uncalibrated predictions. In high-stakes domains such as medical diagnosis, autonomous driving, and agricultural decision support, a model's ability to recognize its own ignorance (epistemic uncertainty) is as critical as its predictive accuracy. Overconfident misdiagnoses or incorrect risk assessments can lead to severe consequences.

The standard approach to quantifying epistemic uncertainty involves training an ensemble of multiple independently initialized models (Deep Ensembles). While effective, this method suffers from a fundamental limitation: computational and memory costs scale linearly with the number of ensemble members ( $M$ ). For foundation models with billions of parameters, even a modest ensemble (e.g., $M=4$ ) becomes computationally prohibitive, rendering principled uncertainty estimation inaccessible for practitioners with limited resources or those relying on parameter-efficient fine-tuning (PEFT).

Existing implicit ensembling methods (e.g., MC-Dropout, BatchEnsemble, LoRA-Ensemble) attempt to reduce overhead but often struggle to match the calibration performance of explicit ensembles or still require learning significant new parameters for each member.

2. Methodology: Singular Value Ensemble (SVE)

The authors propose Singular Value Ensemble (SVE), a parameter-efficient implicit ensembling method grounded in the insight that knowledge in foundation models is organized along linear subspaces defined by the singular vectors of weight matrices.

Core Assumption

The method relies on the hypothesis that the singular vectors ( $U$ and $V$ ) of pre-trained weight matrices correspond to meaningful, orthogonal "knowledge directions" in the representation space. The singular values ( $\Sigma$ ) encode the relative importance or strength of these directions. Therefore, an ensemble can be constructed by preserving the shared knowledge basis (singular vectors) while modulating the contribution of each direction via member-specific singular values.

Technical Implementation

SVD Decomposition: For each target weight matrix $W \in \mathbb{R}^{m \times n}$ in a pre-trained model, the Singular Value Decomposition is computed: $W = U\Sigma V^\top$ .
Parameter Freezing: The singular vectors $U$ and $V$ are frozen, preserving the pre-trained knowledge basis.
Member-Specific Learning: Instead of learning new weight matrices, the method trains $M$ copies of the singular values, denoted $\{\Sigma^{(m)}\}_{m=1}^M$ , one for each ensemble member. Additionally, a per-member bias vector $b^{(m)}$ is trained.
Reconstruction: During inference, the weight matrix for member $m$ is reconstructed as $W^{(m)} = U\Sigma^{(m)}V^\top$ .
Initialization and Diversity: To ensure diversity among members without destroying the pre-trained structure, singular values are initialized with multiplicative perturbations: $\Sigma^{(m)} = \Sigma \odot (1 + \epsilon^{(m)})$ , where $\epsilon^{(m)} \sim \mathcal{N}(0, \sigma_{init}^2 I)$ . Stochastic mini-batch sampling during joint training further drives members to converge to distinct combinations of the shared subspaces.

Parameter Efficiency

For a transformer with hidden dimension $d$ , SVE adds only $6d$ singular values and $9d$ bias entries per layer per member (for attention and MLP projections). The total parameter overhead relative to the base model is approximately:
$\text{Overhead} \approx \frac{5(M-1)}{4d}$
For $d=768$ and $M=8$ , this results in an overhead of $<1\%$ . For larger models like LLaMA-2-7B ( $d=4096$ ), the overhead drops to $\approx 0.2\%$ . This is significantly more efficient than LoRA-Ensemble (which adds rank- $r$ matrices) and orders of magnitude more efficient than Deep Ensembles.

3. Key Contributions

Introduction of SVE: A novel, parameter-efficient implicit ensemble method that creates diverse ensemble members by learning per-member singular values while sharing pre-trained singular vectors.
New Baseline for Uncertainty: Establishes SVE as a strong baseline for uncertainty quantification in foundation models, achieving state-of-the-art calibration while maintaining predictive accuracy.
Extensive Validation: Validated across Natural Language Processing (NLP) and Vision tasks using backbones ranging from 22M to 7B parameters (BERT, LLaMA-2-7B, DINOv1/v2). The results demonstrate generalization across model sizes and task types.

4. Experimental Results

The authors evaluated SVE on NLP tasks (ARC-Easy, SST-2) and Vision tasks (Flowers102, CIFAR-100, DTD, Oxford Pets, OOD detection, and CIFAR-100-C corruptions).

Calibration: SVE consistently achieves superior or competitive Expected Calibration Error (ECE) compared to Deep Ensembles, LoRA-Ensemble, and Bayesian methods (e.g., Bayes-LoRA, BLoB). Notably, on ARC-Easy with LLaMA-2-7B, SVE ( $M=16$ ) achieved an ECE of 3.8%, significantly outperforming Deep Ensembles (9.9%) and LoRA-Ensemble (9.0%), approaching the calibration of Bayesian methods while being simpler to implement.
Accuracy: SVE maintains predictive accuracy comparable to explicit ensembles and other implicit methods. On CIFAR-100, it achieved 85.6% accuracy, outperforming Deep Ensembles (85.4%) and single models.
Robustness: Under distribution shifts (CIFAR-100-C) and Out-of-Distribution (OOD) detection (CIFAR-100 $\to$ CIFAR-10), SVE demonstrated strong robustness, matching or exceeding the performance of LoRA-Ensemble and Deep Ensembles in terms of AUROC and AUPRC, despite using far fewer parameters.
Backbone Dependency: The benefits of SVE scale with the quality of the pre-trained representations. On models with random initialization, SVE underperformed; however, with strong backbones (DINOv2), it outperformed all baselines, validating the hypothesis that meaningful singular vector bases are crucial for the method's success.
Computational Cost: SVE reduces parameter and memory overhead by a factor of 10 compared to LoRA-Ensemble and 700 compared to Deep Ensembles. Inference time is comparable to LoRA-Ensemble and significantly faster than Bayesian sampling methods.

5. Significance and Claims

The paper claims that SVE democratizes principled uncertainty estimation for resource-constrained settings. By leveraging the linear subspace structure of foundation models, SVE enables the creation of diverse ensembles with minimal parameter overhead ( $<1\%$ ).

The authors position SVE as a balanced approach that:

Achieves accuracy comparable to LoRA-Ensemble.
Achieves calibration comparable to complex Bayesian methods.
Operates with the computational efficiency of parameter-efficient fine-tuning.

The work suggests that as foundation models improve, the "knowledge basis" encoded in their singular vectors becomes increasingly valuable. SVE leverages this by preserving these representations while introducing diversity through lightweight adjustments, making reliable uncertainty estimation accessible for large-scale models where traditional ensembling is infeasible.

Limitations Acknowledged:
The authors note that inference still requires one forward pass per member (similar to other implicit ensembles), so FLOPs remain comparable to Deep Ensembles. Integration with quantized backbones is non-trivial due to the need for dequantization before SVD. Furthermore, because SVE preserves the pre-trained basis, it may inherit biases or spurious correlations present in the backbone.