DecompScore: Interpretable Score Decomposition

• DecompScore is a framework that decomposes a global score into distinct, locally attributable sub-components for clearer model analysis.
• It employs techniques like local score approximation, denoising score matching, and geometric latent factor extraction to enhance interpretability.
• Applications span dynamical systems, deep generative modeling, multi-source data integration, factual claim assessment, and predictive performance auditing.

DecompScore refers to a family of methods and metrics that decompose a global or aggregate score, support, or factor into interpretable and operational sub-components in a variety of model and data analysis settings. These include data assimilation in dynamical systems, score-based deep generative modeling, multi-source latent factor analysis, factual support in NLP, and predictive performance auditing in forecasting. Despite the diverse domains, the unifying principle is the structured decomposition of a global score or support quantity into distinct, locally attributable elements, enabling more interpretable, modular, or diagnostic assessment of models and algorithms.

1. Score Decomposition in Dynamical Systems and Data Assimilation

In "Score-based Data Assimilation" (Rozet et al., 2023), DecompScore denotes the pseudo-blanket decomposition of the global trajectory score in a stochastic dynamical system into a sum of local, segmentwise score functions. Precisely, for a Markov chain trajectory $x_{1:L}$, the global diffusion score at time $t$ is

$$\nabla_{x_{1:L}(t)} \log p(x_{1:L}(t)) \in \mathbb{R}^{L \times D}$$

The key insight is that due to local Markov structure, each component $\nabla_{x_i(t)} \log p(x_{1:L}(t))$ can be approximated by a local score computed over a sliding window of $2k+1$ states around $i$:

$$\nabla_{x_i(t)} \log p(x_{1:L}(t)) \approx \nabla_{x_i(t)} \log p(x_{i-k:i+k}(t))$$

A single neural network $s_\phi$ is trained to approximate these local scores using denoising score matching, with segments perturbed according to the SDE marginal. Inference is then performed by composing the local scores into the global prior score, and adding a likelihood score component derived from the observation model (decoupled from training). This permits non-autoregressive, zero-shot posterior sampling of arbitrarily long state trajectories. The approach achieves unbiasedness in the decomposition theoretically, and empirical validation is presented on Lorenz-63 and Kolmogorov flow, demonstrating effective posterior inference without simulating gradients through the physical system (Rozet et al., 2023).

2. Functional Decomposition in Deep Score-based Generative Models

In "Divide and Compose with Score Based Generative Models" (Ghimire et al., 2023), DecompScore refers to a methodology for explicitly decomposing the score function of a data distribution into a sum of interpretable component scores, each corresponding to the gradient of a distinct "energy" functional:

$$\begin{aligned} p_0(x) &\propto \prod_{i=1}^K \exp(-E^{(i)}(x)) \ \nabla_x \log p_0(x) &= \sum_{i=1}^K s_i(x), \quad s_i(x) = -\nabla_x E^{(i)}(x) \end{aligned}$$

Practicably, these components are learned using a UNet-style score network conditioned on K learned latent codes per input. Training employs a denoising score matching or continuous-time variational bound, and at generation time, these components can be visualized, composed, or interpolated independently within the reverse SDE. Each component often aligns to a semantic factor (e.g., shape, color, fine structure), enabling structured manipulation and analysis of generative outputs. This framework exposes internal representation disentanglement and provides a toolset for unsupervised learning of compositional features (Ghimire et al., 2023).

3. DecompScore in Multi-source Latent Factor Analysis

The DecompScore framework developed in "Integrative decomposition of multi-source data by identifying partially-joint score subspaces" (Choi et al., 2022) targets the analysis of multi-block or multi-source data matrices. Here, DecompScore refers to a fast geometric algorithm for extracting latent score subspaces that are fully joint, partially shared, or individual to blocks. The method proceeds by:

• Modeling each block as $X_k = U_k V_k^T + E_k$ and stacking all blocks as $t$0 with block-sparse $t$1.
• Sequentially extracting one-dimensional mean directions from overlapping signal-score subspaces by minimizing principal angle distances and deflating the input subspaces.
• Building a direct sum decomposition that uniquely attributes shared and block-specific structural variation to explicit factors under mild independence and orthogonality assumptions.

Empirical benchmarks show this approach outperforms classical JIVE, AJIVE, COBS, and SLIDE in settings with partial associations, with practical applications demonstrated in multi-omics integration for blood-cancer genomics (Choi et al., 2022).

4. DecompScore in Textual Claim Decomposition and Support Assessment

In the context of factual evaluation for NLP, "A Closer Look at Claim Decomposition" (Wanner et al., 2024) introduces DecompScore as an adaptation of FActScore, quantifying the absolute count of atomic subclaims (as generated by a decomposition method $t$2) that are entailed by their parent passage:

$t$3

where $t$4 is an entailment validator, and $t$5 yields subclaims. The metric is diagnostic for decomposition quality, isolating error introduced by the decomposition from the validation model or external references. The work further proposes Russellian/neo-Davidsonian (R-ND) decomposition, yielding high atomicity and subclaim coverage. Empirical analysis shows DecompScore correlates strongly with NLI entailment, and that decomposition sensitivity induces substantial variance in FActScore downstream, motivating the need for precise, well-calibrated decomposition (Wanner et al., 2024).

5. DecompScore for Statistical Inference of Predictive Scores

"Statistical Inference for Score Decompositions" (Dimitriadis et al., 4 Mar 2026) formalizes DecompScore as a decomposition of any strictly consistent scoring function $t$6 (for mean or quantile prediction, etc.) into

$t$7

where MC (miscalibration), DS (discrimination), and UNC (uncertainty) are operationally separable:

• MC: Score improvement from recalibration.
• DS: Discrimination benefit of calibrated forecast over constant reference.
• UNC: Intrinsic unpredictability.

This is achieved via linear recalibration (Mincer–Zarnowitz regression) estimating $t$8 and sample averages, with finite-sample and asymptotic inference procedures for the components—including intersection–union testing for equality of calibration or discrimination between models. Applications in forecasting and financial risk models reveal subtle, previously unidentifiable distinctions between model calibration and predictive resolution, and offer interpretable diagnostics and more powerful hypothesis tests than classical scoring rules alone (Dimitriadis et al., 4 Mar 2026).

6. Comparative Summary Table

DecompScore Context	Decomposition Target	Primary Application Domain
Score-based Data Assimilation (Rozet et al., 2023)	Trajectory score, local window sum	State-space inference, SDEs
Divide & Compose with Generative Models (Ghimire et al., 2023)	Score function into semantic components	Deep generative modeling, disentanglement
Multi-source Factor Analysis (Choi et al., 2022)	Latent scores: joint, partial, individual	Integrative analytics, multi-omics
Textual Claim Decomposition (Wanner et al., 2024)	Atomic subclaims in text	Factual support evaluation, NLP
Score Inference in Forecasting (Dimitriadis et al., 4 Mar 2026)	Calibration, discrimination, uncertainty	Predictive accuracy, forecast auditing

7. Interpretive Context and Implications

Across domains, DecompScore frameworks provide a path from opaque, global or aggregate evaluation quantities towards modular, interpretable components. This permits targeted algorithmic inspection (locally in time or factor space), modular training or inference (as in decoupled likelihood priors), improved diagnostics (by isolating decoherence or misspecification in substructure), and, in statistical settings, valid hypothesis testing on specific properties like calibration or discrimination. A plausible implication is that further refinement of these decomposition techniques will facilitate greater transparency and targeted intervention in complex, high-dimensional, or multi-source inference and decision-making systems.