Generalized Discrimination Value (GDV)

• Generalized Discrimination Value (GDV) is a quantitative, parameter-free measure that rigorously defines class separability in high-dimensional data.
• It unifies classical divergence measures through a tunable parameter, leveraging properties like convexity, monotonicity, and invariance to enhance statistical decision frameworks.
• GDV is applied across quantum information, deep neural network layer analysis, and pattern recognition to optimize classification and infer structural insights.

The Generalized Discrimination Value (GDV) is a parameter-free quantitative measure introduced to rigorously characterize the separability of classes or distinct quantities in high-dimensional data representations. Originally developed to unify divergences between different means and to derive refined inequalities among standard measures, GDV quickly found wide application as an analytic tool in statistical decision theory, quantum information, machine learning classification, and the analysis of deep neural network internal representations.

1. Formal Definition and Mathematical Formulation

The GDV arises from the generalized triangular discrimination measure, denoted $L_t(a, b)$, a parametric family that extends and unifies classic triangular discrimination and symmetric divergence forms:

$$L_t(a, b) = \frac{2^t (\sqrt{ab})^t (a - b)^2}{(a + b)^{t + 1}}$$

where $a, b > 0$ (or probability values), and $t \in \mathbb{Z}$ controls the measure’s flavor. For $t = 0$, $L_0$ recovers the Jain–Srivastava symmetric discrimination; for $t = 1$, $L_1$ is the classical triangular discrimination.

More generally, the measure can be written as $L_t(a, b) = b \cdot f_{L_t}(a/b)$, with

$$f_{L_t}(x) = \frac{2^t (x - 1)^2 x^{t/2}}{(x + 1)^{t + 1}}$$

This generator function $f_{L_t}(x)$ vanishes at $x = 1$, i.e., when $a = b$ as required for a divergence.

When used to assess clustering or class separability in vector spaces, GDV is frequently defined as the rescaled difference:

$$\text{GDV} = \frac{1}{\sqrt{D}} \left( \frac{1}{L} \sum_l \bar{d}(C_l) - \frac{2}{L(L - 1)} \sum_{l < m} \bar{d}(C_l, C_m) \right)$$

where $\bar{d}(C_l)$ is the mean intra-class distance for class $C_l$, $\bar{d}(C_l, C_m)$ is the mean inter-class distance, $D$ is the embedding dimension, and $L$ is the number of classes. Inputs are z-scored along dimensions for invariance.

2. Convexity, Monotonicity, and Theoretical Properties

GDV possesses several crucial mathematical properties:

• Convexity: The generator $f_{L_t}(x)$ is convex for nonnegative integer $t$ (specifically for $t \in \{-1, 0, 1, 2\}$, proved by evaluating its second derivative). Convexity is vital for statistical inference and optimization, ensuring the corresponding divergence function is robust and well-behaved.
• Monotonicity: $\partial f_{L_t}(x)/\partial t > 0$ for $x \neq 1$, i.e., tuning $t$ yields monotonic families of discrimination values. This "dial" can be used to adjust the sensitivity of the GDV in practical applications.
• Invariance: The GDV is designed to be invariant under translation and scaling (via z-scoring) and insensitive to permutation of dimensions, ensuring its reliability in high-dimensional tasks.
• Null Property: $f_{L_t}(1) = 0$ implies that identical quantities yield zero discrimination.

3. Relationship to Classical Means and Divergences

The GDV framework unifies the seven classical means: harmonic, geometric, arithmetic, Heronian, contra-harmonic, root-mean-square, and centroidal.

Inequalities among pairwise differences of these means are expressed proportionally in terms of triangular discrimination and its generalizations, e.g.:

• $3 D_{CR} = 2 D_{AH} = 4 (\Delta)$
• $3 D_{AN} = D_{AG} = D_{NG}$

This enables GDV not only to serve as a "distance" between two values or distributions but also as a tool to derive and refine foundational inequalities in mathematical statistics. The GDV thus acts as a master measure, encompassing known divergences as special cases.

4. Generating (Exponential) Representations and Extensions

Generalized discrimination measures can be extended and embedded into more versatile forms via series expansions and exponential representations, yielding closed-form or generating-function divergences:

$$\Delta_t(a, b) = \frac{(a - b)^2}{a + b} \exp\left( \frac{(a - b)^2}{\sqrt{ab}} \right)$$

$$E_{L_t}(a, b) = (a + b) f_{L_t}(a/b) = (a + b) \exp(\text{series in}~(\sqrt{ab} - ...))$$

These forms facilitate the derivation of further inequalities, theoretical analysis, and practical exploitation in pattern recognition and information theory.

5. Applications in Statistical Decision Theory and Optimization

The convex and tunable nature of GDV makes it appealing for optimization, information theory, and robust statistical inference:

• It underpins generalized divergence measures used in decision-theoretic frameworks, where the ability to quantify class separation impacts strategies in hypothesis testing and classification.
• The flexibility in $t$ and generating coefficients allows tailoring GDV to the structure of the modeled problem, adapting to the needs of information measures, statistical modeling, or signal discrimination.

6. Connections to Quantum Information, Machine Learning, and Neural Representations

GDV has broad utility:

• Quantum State Discrimination: In quantum detection, GDV is interpreted as the optimal figure of merit for distinguishing quantum states within convex programming frameworks, characterizing the maximum achievable discrimination value for measurement operators under various constraints (Nakahira et al., 2015).
• Deep Learning and Hidden Layer Analysis: In deep neural networks, GDV quantifies layer-wise class separability, revealing the structure of data transformation through training, identifying energy barriers and "master curves" in layerwise representations, and supporting automatic model selection (Schilling et al., 2018).
• Forecast Verification: For probabilistic and quantile forecasts, GDV underlies RUC curve analyses and measures economic forecast value, relating discrimination strengths to skill scores and cost–loss tradeoffs (Bouallegue et al., 2015).
• Pattern Recognition: Incorporation of GDV-like metrics allows for generalized distance weighted discrimination and advanced linear discriminant analyses that outperform classical approaches in accuracy and robustness (Lam et al., 2016, Liu et al., 2023), especially in high-dimensional, low-sample-size settings.

7. Implications and Future Directions

GDV serves as a robust, flexible, and unifying measure for class separability, divergence, and inequality derivation in both theoretical and applied contexts:

• Its adaptability and parameterization allow for fine-tuning in specific applications, such as probabilistic modeling, hypothesis tests, embedding space analysis, and quantum information processing.
• Its connection to convexity, monotonicity, and normalization ensures statistical soundness and computational efficiency.
• As digital models of cognition and perception evolve, GDV is increasingly used to bridge the gap between statistical, geometric, and information-theoretic characterizations of discrimination, offering deep insights into model structure and data representation.

GDV continues to inform the development of discrimination metrics, optimization algorithms, and analytic tools across domains ranging from mathematical statistics and quantum state discrimination to neural LLMing, deep learning interpretability, and advanced pattern recognition.