ELB Decomposition: Theory & Applications

• ELB Decomposition is a framework that systematically splits complex structures into specialized components (e.g., Emotion, Logic, Behavior) applicable across mathematics and NLP.
• It integrates methods from measure theory, stochastic process analysis, convex geometry, and deep learning to yield interpretable and computationally tractable models.
• This technique enhances model interpretability and robustness by isolating key functional components for applications in financial modeling, computational geometry, and clinical NLP.

ELB decomposition refers to a family of methodologies and models across multiple domains—measure theory, stochastic processes, convex geometry, mathematical physics, and most recently NLP—that share the theme of systematically splitting or delineating a complicated object into interpretable or functionally specialized components labeled or denoted as E, L, and B (typically standing for Emotion, Logic, and Behavior in the context of language, or as orthogonal analytic constructs in other fields). The following exposition provides a comprehensive technical treatment of ELB decomposition as applied in recent research, spanning theoretical underpinnings to neural architectures.

1. Mathematical Foundations and Classical Analogues

The concept of decomposition underlies foundational results throughout mathematics and physics. Notable examples include the Lebesgue decomposition of measures, John’s decomposition of the identity for convex bodies, and analytic splittings in stochastic processes.

Measure-Theoretic Decomposition

In Lebesgue measure theory, decomposing a finite measure $v$ with respect to another finite measure $p$ involves splitting $v$ into two mutually singular measures: one absolutely continuous and one singular with respect to $p$. Hilbert space techniques, specifically application of the Riesz orthogonal decomposition theorem, allow precise construction. Let $L^2(p)$ and $L^2(v)$ be associated Hilbert spaces, and $M \subset L^2(v)$ the subspace of limits of $\sigma$-simple function sequences that vanish in $L^2(p)$. The orthogonal projection $P$ onto $M$ yields the decomposition:

$v(E) = \mu_a(E) + \mu_s(E)$

where

$$\mu_a(E) = \int_E (1 - P1) \, dv, \quad \mu_s(E) = \int_E P1 \, dv, \quad \forall E \in \mathcal{R}$$

This geometric perspective streamlines classical proofs and generalizes to additive set functions and operators (Tarcsay, 2014).

Convex Geometry

For a convex body $K$ in $\mathbb{R}^n$ in John position, one finds boundary vectors $u_i$ and weights $c_i$ such that

$$\sum_{i=1}^m c_i u_i u_i^\top = I_n, \qquad \sum_{i=1}^m c_i u_i = 0$$

Functional John ellipsoids extend this identity decomposition to the setting of integrable log-concave functions, constructing analogues via discretized measures and explicit minimization of convex functionals. The functional version involves weighting by density evaluations:

$\sum_{i=1}^m c_i h(u_i)^{1/s} u_i u_i^\top = I_n$

with $h(u_i)$ in correspondence to the normalized log-concave function at $u_i$ (Baêta, 2 Apr 2025). This constructive approach enables effective numerical computation of isotropic positions.

2. Stochastic Process Decomposition and the ELB Paradigm

Decomposition is central to stochastic analysis, where occupation time and additive functionals of Lévy processes are examined under scaling asymptotics.

For a symmetric one-dimensional Lévy process $X_t$ with characteristic exponent $\Psi(x)$, additive functionals take the form:

$$I_n(t) = \frac{1}{a(n)} \int_0^{t^2 a^2(n)} f(X_s) ds$$

where $a(n)$ is an increasing sequence and $f$ satisfies integrability and smoothness conditions. The decomposition result is:

$I_n(t) = I_n^{(1)}(t) + f(0) I_n^{(2)}(t)$

$\,I_n^{(1)}(t)$ vanishes in $L^p$ as $n \to \infty$, whereas $I_n^{(2)}(t)$ is uniformly integrable and dictates the nondegenerate limit law (Valverde, 2013). Moment estimates are analytic, e.g., for integer $k$:

$$E\left[(I_n^{(2)}(t))^{2k}\right] \leq \frac{(2k)!}{k!} t^{2k} \ell^k$$

where $\ell$ is linked to the quadratic behavior of $\Psi(x)$. This analytic decomposition is closely related to, but not identical with, probabilistic ELB decompositions, which more frequently utilize local time and occupation density formulas.

3. ELB Decomposition in Modern NLP Architectures

Recent advances extend the decomposition framework to natural language processing, with a focus on cognitive distortion detection in clinical and therapeutic contexts (Kim et al., 22 Sep 2025). Here, ELB denotes the separation of input text into Emotion, Logic, and Behavior components:

• Emotion: summary of affective state (e.g., "anger," "sadness")
• Logic: formulation of underlying reasoning (e.g., overgeneralization, faulty inference)
• Behavior: actions, intentions, or hypothetical responses

Extraction Methodology

LLMs (LLMs; e.g., GPT-4, Gemini 2.0 Flash) perform zero-shot prompt-based parsing. Each utterance is mapped to three short sentences (one per ELB component) via dedicated extraction prompts.

Instance Construction and Multiple-Instance Learning (MIL)

Each ELB-enriched utterance is further processed by LLMs to extract cognitive distortion instances, encoded as triplets $(\text{type}_i, \text{text}_i, s_i)$:

$$B = \{ x_i = (\text{type}_i, \text{text}_i, s_i) \}_{i=1}^N$$

where $s_i$ is a salience score assigned by the LLM to quantify the perceived relevance of instance $i$.

Normalization ensures comparative weighting:

$\hat{p}_i = \frac{s_i}{\sum_{j=1}^N s_j}$

Multi-View Gated Attention Aggregation

A Multi-View Gated Attention network integrates instance embeddings. For each instance:

$$h_i = \sigma(W_g x_i) \cdot \tanh(W_f x_i) \cdot s_i$$

$\,W_g,W_f$ are learned matrices, $\sigma$ is the sigmoid function, $\tanh$ is hyperbolic tangent.

Multiple independent attention views $h^{(k)}$ are averaged:

$$h_{\text{multi}} = \frac{1}{K} \sum_{k=1}^K h^{(k)}$$

Global context ($z$) is projected and concatenated with $h_{\text{multi}}$:

$$z' = \tanh(W_z z), \qquad v = \text{ReLU}(W_c [h_{\text{multi}}, z'])$$

Final classification is performed via a softmax layer.

Significance for Interpretability and Detection

• ELB decomposition aligns closely with psychological theory (CBT’s cognitive triangle).
• It enables fine-grained attribution: predictions are explainable in terms of which psychological axis (Emotion, Logic, Behavior) triggered a classification.
• It enhances robustness by reducing the omission rate in composite and ambiguous distortion cases ("Emotional Reasoning", "Labeling").
• Salience scores from LLMs offer a "soft evidence" mechanism for focusing attention on diagnostically important segments of text.

4. Connections and Comparative Structure

ELB decomposition is conceptually related to classical analytic and geometric decompositions by:

Context	Decomposed Elements	Methodology
Measure Theory	Absolutely continuous, singular	Orthogonal projection
Convex Geometry	Rank-one matrices $u_i u_i^\top$	Lagrangian minimization
Lévy Process Functionals	Leading term, error term	Fourier analysis
NLP Cognitive Distortion	Emotion, Logic, Behavior	Prompted LLM extraction

Common themes include separation into negligible vs. main terms, orthogonality (literal or functional), and weighted aggregation to recover original structural identities.

5. Applications and Theoretical Implications

The constructive and interpretable nature of ELB decomposition—whether in functional analytic, probabilistic, or neural frameworks—opens avenues for:

• Mathematical Analysis: Quantification of weak limits, isotropic positions, and variational extremality.
• Statistical Modeling: Refined inference under constraints (e.g., ELB in interest rate models (Ikeda et al., 2020) for financial economics).
• Machine Learning/NLP: High-precision clinical psychology tools, aggregating multiple interpretable diagnostic signals for robust and generalizable mental health inference (Kim et al., 22 Sep 2025).
• Computational Geometry: Efficient approximations to log-concave functions, embedding optimization via explicit convex functional minimization.

6. Summary of Key Formulations

Some canonical mathematical forms appearing in ELB decomposition include:

$$\sum_{i=1}^m c_i u_i u_i^\top = I_n, \qquad \sum_{i=1}^m c_i h(u_i)^{1/s} u_i u_i^\top = I_n$$

$I_n(t) = I_n^{(1)}(t) + f(0)I_n^{(2)}(t)$

$$B = \{ x_i = (\text{type}_i, \text{text}_i, s_i) \}, \qquad h_i = \sigma(W_g x_i) \cdot \tanh(W_f x_i) \cdot s_i$$

These exemplify the central technical mechanism: decomposition into interpretable, analyzable, or computationally tractable parts that, when suitably aggregated, reconstruct the functional or analytic identity of the original object.

7. Perspectives and Future Directions

Current trends suggest further exploration of ELB decomposition across:

• Heterogeneous LLMs, extending ELB to multilingual and multimodal inference
• High-dimensional geometric analysis, leveraging functional decompositions in random matrix theory and information geometry
• Probabilistic limit theorems, refining analytic decompositions for occupation times in increasingly complex stochastic models

A plausible implication is that ELB-style frameworks will continue to unify disparate approaches to decomposition in mathematics, theoretical physics, statistics, and artificial intelligence, particularly where interpretability remains a core requirement for application.