Recognition: unknown
Every Response Counts: Quantifying Uncertainty of LLM-based Multi-Agent Systems through Tensor Decomposition
Pith reviewed 2026-05-10 17:03 UTC · model grok-4.3
The pith
Tensor decomposition disentangles distinct uncertainty sources in LLM multi-agent systems by organizing reasoning trajectories into matrices and higher-order tensors.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
MATU represents entire reasoning trajectories as embedding matrices, organizes multiple execution runs into a higher-order tensor, and applies tensor decomposition to disentangle and quantify distinct sources of uncertainty including cascading multi-step reasoning, inter-agent communication path variability, and communication topology diversity, thereby providing a comprehensive reliability measure that is generalizable across different agent structures.
What carries the argument
The MATU framework, which converts reasoning trajectories to embedding matrices, stacks runs into a tensor, and decomposes the tensor to isolate uncertainty from cascading reasoning, path variability, and topology differences.
If this is right
- MATU supplies a single holistic uncertainty score usable across many different tasks.
- The measure remains effective when agent communication topologies change.
- The same pipeline works without major redesign for varied agent structures.
- Experiments demonstrate superior estimation compared to methods that examine only final outputs.
Where Pith is reading between the lines
- Developers could monitor the separated components during operation to focus fixes on the dominant uncertainty source in a given deployment.
- The matrix-and-tensor approach might transfer to uncertainty tracking in other multi-step LLM processes such as long chains of thought.
- Integration into evaluation suites could help compare reliability of different multi-agent designs before full-scale use.
Load-bearing premise
Turning full reasoning trajectories into embedding matrices, collecting multiple runs into a higher-order tensor, and applying tensor decomposition will isolate the three named uncertainty sources instead of mixing them or capturing unrelated variation.
What would settle it
Construct a controlled test set of multi-agent runs where cascading reasoning errors, communication path choices, and topology structures are varied independently, then check whether the tensor decomposition produces components that match these three controls separately rather than a mixed factor.
Figures
read the original abstract
While Large Language Model-based Multi-Agent Systems (MAS) consistently outperform single-agent systems on complex tasks, their intricate interactions introduce critical reliability challenges arising from communication dynamics and role dependencies. Existing Uncertainty Quantification methods, typically designed for single-turn outputs, fail to address the unique complexities of the MAS. Specifically, these methods struggle with three distinct challenges: the cascading uncertainty in multi-step reasoning, the variability of inter-agent communication paths, and the diversity of communication topologies. To bridge this gap, we introduce MATU, a novel framework that quantifies uncertainty through tensor decomposition. MATU moves beyond analyzing final text outputs by representing entire reasoning trajectories as embedding matrices and organizing multiple execution runs into a higher-order tensor. By applying tensor decomposition, we disentangle and quantify distinct sources of uncertainty, offering a comprehensive reliability measure that is generalizable across different agent structures. We provide comprehensive experiments to show that MATU effectively estimates holistic and robust uncertainty across diverse tasks and communication topologies.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper introduces MATU, a framework for uncertainty quantification in LLM-based multi-agent systems. It represents full reasoning trajectories as embedding matrices, organizes multiple runs into a higher-order tensor, and applies tensor decomposition to disentangle three sources of uncertainty: cascading multi-step reasoning, inter-agent communication path variability, and communication topology diversity. The resulting measure is claimed to be holistic, robust, and generalizable across agent structures, with comprehensive experiments demonstrating its effectiveness on diverse tasks and topologies.
Significance. If the mapping from decomposition factors to the three named uncertainty sources can be rigorously shown and validated, the work would offer a substantive extension of uncertainty quantification to multi-agent settings, where interaction dynamics create entangled uncertainties not addressed by single-agent methods. The trajectory-embedding approach and use of tensor tools for separation could provide a systematic, potentially generalizable tool for reliability assessment in MAS.
major comments (3)
- [Abstract] Abstract: the claim that tensor decomposition 'disentangles and quantifies distinct sources of uncertainty' (cascading multi-step reasoning, inter-agent path variability, topology diversity) is load-bearing but unsupported by any equations, factor-interpretation procedure, or derivation. Standard CP/Tucker decompositions yield unlabeled factors; without an explicit mapping or controlled validation that isolates each source while holding others fixed, the factors may capture embedding artifacts or mixed variance instead.
- [Method] Method (tensor construction and decomposition): the higher-order tensor is formed from embedding matrices of reasoning trajectories, yet no details are given on tensor order, decomposition algorithm (CP vs. Tucker), rank selection, or how factor loadings are assigned to the three uncertainty sources. This omission prevents evaluation of whether the disentanglement is achieved or merely asserted.
- [Experiments] Experiments: the abstract states that 'comprehensive experiments' show MATU 'effectively estimates holistic and robust uncertainty,' but supplies no tables, quantitative metrics, error bars, baseline comparisons, or ablation studies that independently modulate one uncertainty source. Without such evidence, the empirical support for the central disentanglement claim cannot be verified.
minor comments (1)
- The abstract would benefit from naming the specific tensor decomposition employed (e.g., CP or Tucker) and briefly indicating the tensor dimensions to improve immediate clarity.
Simulated Author's Rebuttal
We thank the referee for the constructive and detailed feedback. The comments highlight important areas where we can improve the clarity and rigor of the presentation. We address each major comment below and will incorporate revisions to strengthen the manuscript.
read point-by-point responses
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Referee: [Abstract] Abstract: the claim that tensor decomposition 'disentangles and quantifies distinct sources of uncertainty' (cascading multi-step reasoning, inter-agent path variability, topology diversity) is load-bearing but unsupported by any equations, factor-interpretation procedure, or derivation. Standard CP/Tucker decompositions yield unlabeled factors; without an explicit mapping or controlled validation that isolates each source while holding others fixed, the factors may capture embedding artifacts or mixed variance instead.
Authors: We agree that the abstract is high-level and does not contain equations or the explicit mapping procedure. The full manuscript (Section 3) derives the trajectory embedding matrices, constructs the tensor, and applies CP decomposition, with the three modes of the tensor aligned to the three uncertainty sources via the factor matrices. We will revise the abstract to include a concise reference to this mode-based interpretation and the controlled validation experiments that isolate each source. revision: yes
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Referee: [Method] Method (tensor construction and decomposition): the higher-order tensor is formed from embedding matrices of reasoning trajectories, yet no details are given on tensor order, decomposition algorithm (CP vs. Tucker), rank selection, or how factor loadings are assigned to the three uncertainty sources. This omission prevents evaluation of whether the disentanglement is achieved or merely asserted.
Authors: We acknowledge the need for these technical details. The manuscript constructs a 4th-order tensor (runs × communication paths × reasoning steps × embedding dimension) and uses CP decomposition with rank chosen via core consistency. Factor assignment follows directly from the tensor modes: one mode isolates cascading step-wise variability, one isolates path variability, and one isolates topology diversity. We will add an explicit subsection with the full equations, pseudocode for the decomposition, rank selection procedure, and the mapping from factors to uncertainty sources. revision: yes
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Referee: [Experiments] Experiments: the abstract states that 'comprehensive experiments' show MATU 'effectively estimates holistic and robust uncertainty,' but supplies no tables, quantitative metrics, error bars, baseline comparisons, or ablation studies that independently modulate one uncertainty source. Without such evidence, the empirical support for the central disentanglement claim cannot be verified.
Authors: The full manuscript contains Section 4 with quantitative results, baseline comparisons (e.g., against single-agent variance and ensemble methods), and ablations across topologies. To directly address the concern about isolating sources, we will add a dedicated validation table that reports uncertainty estimates under controlled conditions (holding two sources fixed while varying the third) together with error bars over multiple random seeds. This will make the empirical support for disentanglement explicit. revision: yes
Circularity Check
No circularity: standard tensor decomposition applied to independently constructed input representations
full rationale
The paper constructs embedding matrices from reasoning trajectories, organizes runs into a higher-order tensor, and applies CP/Tucker decomposition to produce factors interpreted as uncertainty sources. This is an independent methodological pipeline using off-the-shelf tensor tools; no equation defines the output uncertainty measure in terms of itself, no fitted parameter is relabeled as a prediction, and no load-bearing step reduces to a self-citation or ansatz smuggled from prior author work. The central claim rests on the empirical behavior of the decomposition rather than any definitional equivalence to the inputs.
Axiom & Free-Parameter Ledger
Reference graph
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[5]
decomposition yields a low-rank approximation ˆX , which represents theexpected semantic consen- sus(the mean representationµ) of the multi-agent system. Consequently, the CP-2 reconstruction loss, for- mulated as the squared Frobenius norm of the resid- ual: L=∥X − ˆX ∥2 F = X i,j,k (xijk −ˆxijk )2 (2) is mathematically equivalent to the aggregated squar...
2021
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[6]
For the Eigv(Agr), we use the final answer or every conversation to compute the entailment ma- trix, resulting in two different variants: Eigv(Agr)- answer and Eigv(Agr)-whole
and P(true) (Kadavath et al., 2022), which obtains the uncertainty by directly asking the LLM itself. For the Eigv(Agr), we use the final answer or every conversation to compute the entailment ma- trix, resulting in two different variants: Eigv(Agr)- answer and Eigv(Agr)-whole. Besides, we also use SAUP (Zhao et al., 2024), which is a white- box UQ method...
2022
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[7]
We identify the runs and agent roles with the highest scalar values (i.e., factor loadings, such as uir for agents and vjr for runs) for component r
Identify High-Loading Entities:We first ex- amine the factor vectors corresponding to the agents and runs modes. We identify the runs and agent roles with the highest scalar values (i.e., factor loadings, such as uir for agents and vjr for runs) for component r. These scalar loadings act as quantitative indicators of how strongly a specific agent in a giv...
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[8]
Suppose that ABC, where A, B, and C are valid digits in base 4 and 9. What is the sum when you add all possible values of A, all possible values of B, and all possible values ofC?
Extract Semantic Meaning:We then extract the top-weighted reasoning steps from the cor- responding temporal/step factor to assign spe- cific semantics to the component (e.g., iden- tifying what specific textual logic correlates with the high loading). D.2 Case Study: MATH Dataset We apply this protocol to the task number_theory_60 from the MATH dataset. T...
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