Feature Map and Diversity Preservation

• Feature map and diversity preservation is a strategy to extract, organize, and maintain distinct, domain-specific features that capture non-redundant aspects of data representations.
• Efficient diversity sampling techniques like random sampling, Latin Hypercube Sampling, and hill climbing are deployed to maximize Feature Space Hypercube Coverage while mitigating infeasible samples.
• These methods drive robust applications in test data generation, representation learning, and search optimization by integrating statistical feedback and domain-specific mapping.

Feature map and diversity preservation refer to a set of structural, algorithmic, and statistical principles guiding the extraction, organization, and maintenance of a wide range of informative, non-redundant structural or semantic attributes within data representations. The goal is to ensure that a model, search process, or algorithm efficiently explores or retains distinct regions of a feature space—thus enhancing robustness, generalizability, or utility, particularly in settings such as test data generation, representation learning, and search-based optimization. Methods for diversity preservation range from direct regularization on learned features to algorithmic strategies that operate on explicitly constructed feature maps—each with measurable impacts on coverage, efficiency, and the ability to capture domain-specific phenomena.

1. Feature Identification and Domain-Specific Feature Spaces

A foundational insight is that diversity in generated data or model behavior is best measured and achieved relative to a set of carefully selected domain-specific features. These features can be numeric (e.g., string length, number of digits in an input), categorical, or otherwise descriptive of the aspects most relevant to the problem domain. The selection of features determines the dimensionality and scope of the “feature space”—a conceptual or actual space in which each axis corresponds to a specific feature and each point represents an individual test case, model, or data sample.

In the context of test data generation, feature functions are mappings from an input (such as a string representing an arithmetic expression) to a vector of feature values, for example:

• Length: Number of characters in the string
• NumDigits: Number of digit characters present

The preferred region of the feature space is typically defined as a hypercube, specifying bounds for each feature (e.g., string lengths from 3 to 50, number of digits from 2 to 25). By focusing on these curated features, one gains direct control over the types of diversity manifested in the generated data and avoids ambiguities or irrelevance common to generic, information-theoretic measures such as normalized compression distance (Feldt et al., 2017).

2. Diversity Sampling and Search Algorithms

To populate the feature space efficiently and produce maximal feature diversity, several algorithmic strategies are evaluated:

• Random Sampling Variants:
  • rand-once: Fixed model parameters; produces limited coverage (≈39.6% Feature Space Hypercube Coverage (FSHC)).
  • rand-freqN/rand-mfreqN: Regularly resamples the stochastic model parameters during sampling—improving coverage through stochasticity and, in the case of rand-mfreqN, detecting and responding to infeasible samples.
  • Latin Hypercube Sampling (LHS): Stratifies parameter space to guarantee broader spread.
• Nested Monte Carlo Search (NMCS):
  • Implements branching evaluation of generation decisions (e.g., number or subexpression) by running simulations at choice points.
  • Underperforms in FSHC (typically 44–46%) due to dependence on the coverage properties of the underlying “default” model.
• Hill Climbing in Parameter Space:
  • Operates on the parameters of a stochastic choice model, proposing candidate variations via Gaussian perturbations.
  • Candidate acceptance is governed by statistical comparison (Mann–Whitney U test, p < 0.2) of densities in the feature space, contingent on not producing excessive infeasible or non-preferred samples (rates above 33–50% are rejected).
  • Achieves highest FSHC (≈52.7% on average) and does so efficiently (around 236 seconds average search time).

Each approach trades off between speed, coverage, and the risk of model-induced bias or local optima.

Method	FSHC (%)	Time (s)	Notes
rand-once	39.6	Moderate	Limited spread, static parameters
rand-mfreq/LHS	52	Moderate	Broader coverage, periodic reseeding
NMCS (various)	44–46	Fast	Simulation-limited by default coverage
Hill climbing	52.7	236	Highest coverage, statistical control

3. Performance Metrics and Experimental Framework

Evaluation is formalized via Feature Space Hypercube Coverage (FSHC), measuring the number of unique “cells” (combinations of discretized feature values) populated in the preferred hypercube. Normalized FSHC compares filled versus total possible cells and thus furnishes a direct, interpretable measure of how effectively the space has been “illuminated”.

The GödelTest-based experimental system utilizes two main stochastic choice models for arithmetic expression generation to exercise these algorithms, thereby exposing the interplay between structured data generation mechanisms and coverage capacity (Feldt et al., 2017).

4. Trade-offs and Challenges in Diversity Preservation

Several key trade-offs arise in the pursuit of diversity-preserving feature maps:

• Directed search (hill climbing) offers rapid, high-coverage fill but can bias the exploration, missing atypical or anomalous cases.
• Pure random or uniformly stratified search avoids such bias but becomes inefficient in higher-dimensional or sparse feasible spaces.
• Model expressivity: The inherent coverage capacity of the stochastic choice model—particularly when extending to recursion-depth-dependent models (e.g., RecDepth5)—controls the granularity at which the feature space can be filled and the feasibility of random or systematic sampling.
• Infeasibility risk: Some parameter configurations produce invalid examples (e.g., excessive recursion depth); methods must include mechanisms for error recovery, parameter reinitialization, and adaptive modeling.
• Statistical model feedback: Effective control requires integrating statistical comparisons of sampled density (e.g., via Mann–Whitney U test) and points towards future use of Gaussian Processes or similar to directly link feature and parameter spaces, improving “recurrent” (e.g., regression testing) diversity acquisition.

5. Implications for Broader Testing and Feature Mapping Contexts

The study’s conclusions carry broad implications for the design and use of feature maps:

• Domain-specific mapping enables interpretable, actionable control over the diversity properties of generated data, in contrast to universal metrics grounded in information theory.
• Statistically-guided search can outperform naive sampling, especially when feature spaces are highly structured or when test input feasibility is nontrivial to maintain.
• Illumination algorithms—mapping test generation to a feature-targeting process analogous to the “illumination” principle in the MAP-Elites algorithm—can be generalized to related tasks in software testing, evolutionary search, and other diversity-critical domains.
• Hybrid strategies combining search, random, and stratified sampling merit further exploration—for instance, using NMCS guided by enhanced underlying models or dynamic mixture strategies based on observed density and feasibility statistics.
• Pointwise and statistical feedback during test generation supports adaptive targeting, potentially resulting in more robust regression testing and faster filling of novel or critical regions of the feature space.

6. Mathematical Formulations and Optimization Strategies

The paper provides concrete mathematical formulations to undergird diversity preservation strategies:

FSHC:

If hypercube bins are defined for feature values $(f_1, f_2)$ within the preferred region $[a_1, b_1] \times [a_2, b_2]$, with discretization into $k$ bins per feature, then

$$FSHC = \frac{\text{Number of unique (binned) feature pairs observed}}{k^2}$$

Hill climbing candidate acceptance (for parameter vector $P'$):

1. Generate a sample set with $P'$.
2. If infeasibility rate exceeds threshold, reject.
3. Compare the candidate's density profile against the current using the Mann–Whitney U test. If $p < 0.2$, accept $P'$.

This search operates in parameter space but is evaluated in the induced feature map, linking realized diversity to model configuration.

7. Applications, Limitations, and Prospective Developments

• Methods targeting feature diversity are critical in software testing to ensure wide behavioral coverage and in domains such as search-based SE or robust fuzzing.
• While random and search methods can be tuned for high domain-specific coverage, maintaining generalization to unforeseen domains or scaling to high-dimensional features remains an open problem.
• Integrating statistical surrogates (e.g., Gaussian Processes) that directly model the mapping between parameter samples and resulting feature-space densities has the potential to accelerate diverse input acquisition in repetitive or evolving test environments.
• Ultimately, targeted diversity preservation via explicit feature mapping offers practitioners concrete levers for coverage control, statistical evaluation, and iterative refinement in a variety of data generation and algorithmic search settings.

The cumulative evidence (Feldt et al., 2017) demonstrates that domain-specific feature maps, guided search and sampling, and statistically motivated feedback are central to effective diversity preservation. Such mechanisms underpin robust test data generation, support broader exploration in search-based systems, and provide a model for feature map-driven diversity control in applied AI and software engineering.