One minus correlation coefficient as a distance (RD)

The one minus correlation coefficient is a widely used distance metric in bioinformatics, particularly for measuring gene expression similarity [42]. Derived from the Pearson correlation coefficient, which quantifies the linear relationship between two variables, say X and Y, it transforms correlation into a dissimilarity measure as , ensuring larger values indicate greater feature dissimilarity.

A key advantage of one minus correlation coefficient is its scale invariance, making it ideal for biological datasets where gene expression levels vary. It remains robust to differences in magnitude, making it useful for clustering and classification by effectively capturing linear dependencies [43]. Additionally, it is computationally efficient, allowing fast calculations in large datasets. Beyond bioinformatics, one minus correlation coefficient is applied in time-series analysis, machine learning, and clustering, playing a critical role in pattern recognition and feature analysis.

Despite its benefits, one minus correlation coefficient has limitations. It only captures linear relationships, making it ineffective for nonlinear dependencies. It is sensitive to outliers, which can distort similarity measures and introduce bias. Moreover, it ignores absolute differences in magnitude, which is problematic in fields like medical diagnostics and financial risk analysis, where similar trends but vastly different values must be distinguished. Additionally, one minus correlation coefficient is not a true distance metric, as it fails the triangle inequality, limiting its use in certain distance-based algorithms and clustering methods.

Geodesic distance (GD)

Euclidean distance often misrepresents true proximity when data lies on a curved or nonlinear manifold, as it assumes a flat, linear space. In contrast, Geodesic distance accounts for the manifold’s intrinsic geometry by measuring the shortest path along its surface between two points. This makes it especially effective in capturing meaningful relationships within data embedded in complex, curved spaces. By respecting the underlying structure of the space, Geodesic distance offers a more accurate and informative measure of similarity than traditional linear metrics. However, a key limitation of Geodesic distance is that it does not satisfy the triangle inequality, making direct computation infeasible. To overcome this issue, approximation techniques such as the ISOMAP algorithm [44] are employed.

The ISOMAP algorithm combines the computational efficiency of Multidimensional Scaling (MDS) and Principal Component Analysis (PCA) with global optimization and asymptotic convergence to estimate Geodesic distances. It constructs a neighborhood graph by connecting nearby points in high-dimensional space, effectively capturing the local structure of the nonlinear space. The estimated Geodesic distances between feature variables in a tabular dataset X are computed as follows:

1. Compute the Euclidean distance d_X(i,j) between every pair of feature variables X_i and X_j, where .
2. Construct a graph G using the feature variables as nodes. Connect neighboring features based on their distances. If X_i and X_j are neighbors, set
   Otherwise, assignto indicate no direct connection between the two features.
3. Update d_G(i,j) iteratively using the equation:
   The resulting matrix represents the minimum distances and provides an estimate of the Geodesic distance among feature variables.

In this study, the neighborhood graph was constructed using a k-nearest neighbor method. To mitigate topological instability, often referred to as the “pinch problem” where the graph becomes disconnected, we adopted a single-linkage strategy where the distance between disconnected components was approximated by the minimum Euclidean distance between their closest points. By leveraging the ISOMAP algorithm, Geodesic distances can be effectively estimated, allowing for a more accurate representation of relationships within nonlinear feature spaces. This method enhances the ability of CNNs to capture complex feature structures when transforming tabular data into image representations.

Jensen-Shannon distance (JD)

The Jensen-Shannon (JS) distance [45] is a statistical measure used to quantify the similarity between two probability distributions. It is based on the Jensen-Shannon divergence, which, unlike the Kullback-Leibler (KL) divergence, is both symmetric and satisfies the triangle inequality, qualifying it as a true metric. The JS distance between two distributions X and Y is defined as:

where and KL denotes the Kullback-Leibler divergence.

This distance ranges from 0 to 1, where 0 indicates identical distributions and 1 indicates completely different distributions. JS distance is sensitive to both the shape and magnitude of the input distributions. Unlike Euclidean distance, which assumes feature homogeneity and fails to account for probabilistic structure, JS distance excels at comparing normalized distributions, such as frequency profiles or categorical probabilities. For high-dimensional feature spaces where variables may not follow a common distribution, JS distance offers a robust alternative to traditional metrics. It effectively captures the local structure and relationships among features by comparing the underlying probability distributions, making it especially useful when the data exhibits non-uniform or categorical characteristics.

In practice, JS distance has been widely used in fields like natural language processing, particularly for topic modeling and document classification [46], and in single-cell RNA sequencing, where it aids in comparing cell-type-specific gene expression distributions [47]. Its robustness, boundedness, and interpretability make it a valuable tool for analyzing structured, probability-based data.

To apply the Jensen-Shannon distance to continuous tabular data, such as gene expression profiles, we treat each feature vector as a discrete Probability Mass Function (PMF) defined over the sample space. Specifically, let represent the vector of values for the j-th feature across n samples. We normalize this vector to sum to unity:

where P_ij represents the probability mass associated with sample i for feature j. If the data had contained negative values, such as z-scores, a Softmax normalization would be applied instead. Unlike a Kernel Density Estimation (KDE) which estimates the distribution of values, this PMF approach preserves the sample-wise structure, allowing the JS distance to quantify the dissimilarity between feature profiles.

Wasserstein distance (WD)

The Wasserstein distance [48], also referred to as the Earth Mover’s Distance (EMD) or Mallows distance, quantifies the minimal cost required to transform one probability distribution into another by optimally transporting probability mass. Unlike traditional pointwise metrics, it considers both the amount of mass moved and the distance it is moved, making it sensitive to the geometry of distributions. To compute the Wasserstein distance between two continuous distributions F_X and F_Y on a two-dimensional plane, the following steps are taken:

1. For quantile functions and , the Wasserstein distance is computed as:
2. Closed-form formulae are only available when X and Y follow a Gaussian distribution. If and , then the squared Wasserstein distance is represented as:where is the Pearson correlation coefficient.

This distance is particularly advantageous for comparing complex, non-uniform distributions in high-dimensional spaces. It captures both shape and location differences between distributions, offering a more nuanced measure than divergence-based metrics like KL or JS divergence. Furthermore, Wasserstein distance is inherently robust to sample variability and outliers, making it suitable for analyzing empirical distributions such as histograms or quantiles.

Due to these properties, Wasserstein distance has become a foundational tool in various machine learning applications. It is central to generative modeling, most notably in Wasserstein GANs [49], where it addresses instability issues in adversarial training. In spatial transcriptomics, it facilitates the comparison of gene expression distributions across spatial regions [50]. Additionally, it has proven effective in tasks like histogram-based image classification, where it accurately reflects the underlying structure of the data.

Tropical distance (TD)

Recent research has highlighted the growing importance of non-Euclidean distance measures in machine learning and data analysis. Among these, the tropical distance, also known as the generalized Hilbert projective metric, is capable of capturing more complex data geometries and hierarchical relationships. The tropical distance originates from the field of tropical geometry [51], in which standard arithmetic operations are replaced by the “min-plus” or “max-plus” algebra. This framework provides a piecewise-linear and combinatorial alternative to classical Euclidean geometry, enabling the modeling of high-dimensional and structured data.

In this study, we define the tropical distance between two feature variables, X_i and X_j, as

The resulting matrix represents the tropical distance among feature variables. The tropical distance measures the range of coordinate-wise differences and reflects the relative structural variations between vectors rather than their absolute magnitudes. Owing to its scale-invariant and polyhedral nature, tropical geometry provides a robust foundation for representing complex relationships among multivariate data.

This approach has demonstrated significant potential in advancing neural network capabilities. For instance, Yoshida, Aliatimis, and Miura (2024) [52] proposed tropical neural networks, which have been successfully applied to the classification of complex and structured data such as phylogenetic trees, illustrating their effectiveness in capturing hierarchical patterns. Furthermore, Pasque et al. (2024) [53] introduced tropical decision boundaries that have been shown to provide notable robustness against adversarial attacks, revealing an inherent stability often lacking in Euclidean-based models. These advances collectively highlight the promise of tropical and other non-Euclidean metrics in enhancing data representation, model robustness, and classification performance within modern computational frameworks. Motivated by these findings, we integrate the tropical distance into our comparative analysis of non-Euclidean metrics within the IGTD framework for converting tabular data into images, aiming to evaluate its potential for improving both the performance and resilience of subsequent CNN-based classification models.

Euclidean distance-based model

In this setting, the first 100 predictors are simulated from a multivariate normal distribution with independent and identically distributed components, capturing a linear structure well-suited to Euclidean geometry. The binary response variable is generated using a logistic regression model, where the linear predictor is formed from these signal features. Euclidean distance is particularly effective when data resides in a flat, linear space with uncorrelated and equally scaled features. Under this model, the predictors follow a multivariate normal distribution with identity covariance, naturally inducing a Euclidean geometry.

Let the signal predictors be denoted by , where each component is independently drawn from a standard normal distribution. The model is specified as follows:

The binary response Y_i is generated from a Bernoulli distribution with probability P(Y_i = 1). This formulation assumes that the similarity between observations is best captured by their Euclidean proximity in the high-dimensional predictor space. We purposely maintained a high noise-to-signal ratio (2400 noise features versus 100 signal features) in this setting. This parameter choice was intended to serve as a stress test for the curse of dimensionality, evaluating whether the distance metrics could recover a linear signal buried in substantial noise without prior feature selection.

One minus correlation-based model

To construct a dataset where the similarity among observations is best captured by one minus correlation distance, the first 100 predictors are generated from a multivariate normal distribution with a strong Toeplitz correlation structure: , where . This structure ensures high linear dependencies among features, which enhances the sensitivity of correlation-based metrics. Unlike random coefficients, we design the coefficient vector to follow a smooth monotonic trend, increasing gradually across dimensions, which helps align the discriminative signal with the correlation structure.

Let represent the signal features drawn from:

We define a structured coefficient vector:

which distributes signal smoothly across features in a correlated fashion.

The linear predictor is scaled to amplify the signal:

Finally, the binary response is generated from a Bernoulli distribution using the logistic probabilities. The remaining 2400 predictors are added as standard Gaussian noise, resulting in a high-dimensional dataset with 2500 total features. This formulation ensures that class-discriminative information is embedded in the linear correlation patterns, favoring the one minus correlation distance (IGTD(RD)) as the most appropriate similarity measure.

Geodesic distance-based model

The signal features are sampled from a nonlinear manifold, such as a Swiss roll, embedded in high-dimensional space. The binary response Y_i is determined based on the Geodesic distance along the manifold, reflecting intrinsic, non-Euclidean feature relationships. Let lie on a low-dimensional nonlinear manifold , e.g., a Swiss roll:

Let be a nonlinear function defined over the Geodesic distance from a reference point, the binary response Y_i is obtained by

where is the median of .

Jensen-Shannon distance-based model

In this model, the first 100 predictors represent compositional or probability-valued features, drawn from class-specific Dirichlet distributions. This setup is designed to favor the Jensen-Shannon (JS) distance, which effectively captures differences between discrete probability distributions. For each observation i, the binary response variable determines the distribution from which the signal features are generated. Let denote a probability vector lying on the 99-simplex. Then, the class-conditional generation is given by:

Here, and are chosen to generate distinct yet plausible class-specific distributions: produces a uniform distribution, while concentrates the probability mass more evenly, resulting in lower entropy. This construction induces differences in the distributional structure of the features, such that the JS distance between observations from different classes is maximized in expectation. The remaining 2400 predictors are standard Gaussian noise, completing the high-dimensional structure. This simulation design ensures that class separation is primarily encoded in distributional divergence, favoring the Jensen-Shannon distance as an appropriate similarity measure.

Wasserstein distance-based model

To construct a setting where the Wasserstein distance effectively captures class differences, the signal features are simulated as empirical quantile vectors derived from class-specific distributions. Specifically, for each observation i, the signal vector is generated by sorting a sample of 100 values drawn from an underlying distribution depending on the class label . The class-conditional distributions are defined as follows:

Let be a fixed template distribution, defined as the average quantile vector computed from all class 0 observations. The Wasserstein distance is computed as:

The binary class label is then determined by:

where is the empirical median of the Wasserstein distances across all observations. This construction ensures that class separation is encoded via both location and shape differences in the empirical distributions, making Wasserstein distance the most suitable similarity measure for classification in this setting.

Although the Wasserstein distance is theoretically capable of capturing differences in both distribution shape (e.g., variance, skewness) and location, this simulation focuses exclusively on a location shift between class-conditional distributions. This design choice was made to isolate the location effect and provide a controlled evaluation of the metric’s sensitivity to mean differences, ensuring that the source of the discriminative signal is unambiguous.

Tropical distance-based model

To construct a simulation setting in which the tropical distance is uniquely aligned with the intrinsic discriminative structure, we design a feature space whose geometry is governed by coordinate-wise dominance patterns rather than Euclidean magnitude, linear correlation, or smooth manifold structure. The tropical distance between two vectors is defined as

which measures the range of coordinate-wise differences and is invariant to additive shifts. This metric is therefore particularly sensitive to max-min dominance relations.

In this model, the first 100 predictors encode the tropical signal, while the remaining p–100 predictors act as independent noise. Let denote the signal features for the ith observation and the corresponding binary class label, with independently for . The index set is randomly partitioned into S disjoint segments of approximately equal size, each representing a plateau of nearly constant feature values. For each observation i and segment , we first draw an independent baseline level

To introduce global class-dependent structure, two distinct segments s⁺ and s⁻ are selected once per dataset, uniformly from with . These segments are shifted upward and downward, respectively, for all observations in class 1, while class 0 observations retain only the baseline levels:

where controls the strength of the tropical signal. To avoid degeneracy and introduce mild within-segment variability, we add a small Gaussian perturbation and set

Unless otherwise stated, we set S = 5, choose , and use .

After simulating the 100 signal features, we optionally apply column-wise centering across samples,

which removes mean differences between feature vectors. This operation alters Euclidean and correlation-based distances between features but leaves the tropical distance d_T unchanged, because subtracting a constant from both vectors does not affect the range of their coordinate-wise differences. Thus, column centering suppresses some of the geometric cues preferred by Euclidean and correlation metrics while preserving the dominance structure emphasized by the tropical distance. The remaining p–100 predictors are generated as independent Gaussian noise:

and the full predictor vector is defined as

From a geometric perspective, this construction induces a clear block structure in the tropical distance matrix between features. For any two feature vectors belonging to the same segment I_s, the coordinate-wise differences across samples remain small and nearly constant, yielding a small tropical distance d_T. In contrast, pairs of features from different segments, and especially those between the globally shifted segments and , exhibit large and systematic coordinate-wise dominance gaps, resulting in substantially larger tropical distances. Consequently, when IGTD is built using the tropical distance, the resulting feature distance rank matrix closely approximates a block-structured ideal, and the optimized pixel arrangement places segment features into coherent spatial clusters on the image grid.

By comparison, Euclidean and one minus correlation distances primarily depend on differences in global means, variances, or linear dependencies, which are moderated by the symmetric baseline and further reduced by column-wise centering. Geodesic, Jensen-Shannon, and Wasserstein distances rely on smooth manifold geometry or fine-grained distributional form, both of which are weakened by the piecewise-constant plateau construction and the addition of isotropic Gaussian noise. As a result, these non-tropical metrics fail to reveal the strong segment-wise geometry that determines the class boundary, whereas the tropical distance, being explicitly sensitive to coordinate-wise dominance and invariant to translation, remains uniquely aligned with the true discriminative structure. Within the IGTD framework, this alignment enables IGTD(TD) to generate pseudo-images with distinct, localized intensity patterns that are strongly associated with the class label, allowing CNN-based classifiers to achieve substantially higher accuracy than when IGTD is built on Euclidean, correlation-based, Geodesic, Jensen-Shannon, or Wasserstein distances.

Results

The generated datasets were processed using the IGTD algorithm to produce image representations based on six different distance metrics. Note that the computation of pixel distances in the IGTD algorithm is always based on Euclidean distances. These images were then input into a standardized, pre-trained convolutional neural network (CNN). The CNN architecture comprised two convolutional layers with 3 3 kernels, followed by max-pooling layers and batch normalization. We used a stride of 1 and same padding (zero padding to preserve spatial dimensions) for the convolutional layers. Binary cross-entropy was used as the loss function. Model training and evaluation were performed using ten-fold cross-validation repeated across ten iterations. A learning rate reduction strategy and early stopping mechanism were employed to optimize training efficiency and prevent overfitting. The full architecture of the CNN is illustrated in Fig 1.

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Fig 1. Schematic of the CNN architecture used for classification.

The network comprises two sequential blocks, each containing a convolutional layer with 3 3 kernels (stride 1, same padding), batch normalization, and a max-pooling layer. The final layers are tailored for binary classification, utilizing binary cross-entropy loss. Training was optimized with a learning rate reduction strategy and early stopping.

https://doi.org/10.1371/journal.pone.0340005.g001

To assess classification performance, we used standard validations indices (VIs) including accuracy, precision, recall, and F1-score. The performance of the IGTD algorithm using non-Euclidean distance metrics was compared against a baseline model that employed the Euclidean distance. Specifically, we denote IGTD(ED) for the Euclidean distance, IGTD(RD) for the one minus correlation distance, IGTD(GD) for the Geodesic distance, IGTD(JS) for the Jensen-Shannon distance, IGTD(WD) for the Wasserstein distance, and IGTD(TD) for the Tropical distance. Ten-fold cross-validation classification results (mean and standard deviation) across six simulated model datasets, using six different distance metrics within the IGTD framework are shown in Table 1. The classification results across the six simulated models demonstrate that each distance metric within the IGTD framework performs best under specific data structures that align with its mathematical properties.

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Table 1. Ten-fold cross-validation classification results (mean and standard deviation) across six simulated model datasets, using six different distance metrics within the IGTD framework.

https://doi.org/10.1371/journal.pone.0340005.t001

In the Euclidean distance-based model, IGTD(ED) yielded the highest mean accuracy (0.52) relative to the non-Euclidean alternatives, although the absolute performance was low across all methods. This is attributed to two factors: the high-dimensional noise (2400 noise features vs. 100 signal features) and the use of a standardized CNN architecture. To ensure a fair comparison across all simulation scenarios, we did not fine-tune the CNN hyperparameters specifically for the Euclidean model. Consequently, these results serve as a baseline validation: they demonstrate that under strict “ceteris paribus” conditions, utilizing more complex non-Euclidean metrics offers no predictive benefit when the underlying data structure is strictly linear and i.i.d. Thus, while the challenging experimental conditions dampened the absolute classification power, the relative performance confirms that Euclidean distance remains the most appropriate and parsimonious metric when the data lacks intrinsic non-linear manifolds.

For the correlation-based model, IGTD(RD) emerges as the strongest performer with an accuracy of 0.78, precision of 0.79, and F1-score of 0.78. This reflects the model’s deliberate design: the use of Toeplitz covariance and a gradually increasing coefficient vector emphasizes linear dependency across features, which is effectively captured by the one minus correlation distance. Other metrics, particularly IGTD(JD) and IGTD(WD), show lower and more variable performance, indicating they are not well-suited for such correlated structures.

In the Geodesic distance-based model, IGTD(GD) significantly outperforms all other metrics, with an accuracy of 0.87 and F1-score of 0.88. This aligns well with the underlying data, which are sampled from a nonlinear manifold. The strength of IGTD(GD) lies in its ability to preserve local geometry, making it particularly effective for data residing on curved or non-Euclidean spaces. Metrics like IGTD(ED) and IGTD(RD) are less capable of capturing this structure and thus perform moderately.

Under the Jensen-Shannon model, IGTD(JD) shows excellent performance with an accuracy of 0.99 and F1-score of 0.99, though IGTD(ED) and IGTD(WD) still achieve perfect accuracy in this setting. While this suggests that several metrics can handle compositional features well, IGTD(JD) remains the most interpretable and theoretically appropriate choice for data represented by probability vectors. The fact that all metrics perform strongly here might be due to the pronounced class differences introduced via the Dirichlet parameters.

In the Wasserstein distance-based model, IGTD(WD) performs best overall, achieving the highest accuracy (0.99) and F1-score (0.99), with perfect recall. This is consistent with the model’s design, which encodes class differences through quantile-based distributional shifts. The Wasserstein distance’s sensitivity to both location and shape differences in distributions makes it particularly powerful in this context. Other metrics also perform well, but not as consistently or robustly.

Finally, in the Tropical distance-based model, IGTD(TD) demonstrated superior performance metrics, achieving the highest precision (0.9457) and F1-score (0.9041), alongside a high accuracy (0.9150) comparable to the Euclidean baseline. This suggests that while Euclidean distance can capture some signal in max-plus algebraic structures due to scaling effects, the Tropical distance provides a more precise characterization of the coordinate-wise dominance relationships inherent in the data generation process. The distinct advantage in precision highlights the specific alignment of IGTD(TD) with the tropical geometry of the features.

In summary, these results affirm that each non-Euclidean distance metric excels under the conditions it was designed to model. While the performance gaps between the proposed non-Euclidean metrics and the Euclidean baseline are not always large in these controlled simulations, the results validate a crucial principle of structural alignment. In every scenario, the distance metric that mathematically corresponds to the underlying generative process (e.g., Geodesic distance for manifold data, Wasserstein for distributional shifts, Tropical distance for max-plus structures) consistently achieved the top-ranking performance. This confirms that the IGTD framework functions as intended: it effectively translates specific geometric and distributional relationships into image representations. The simulation study thus serves as a proof of concept, demonstrating that choosing the correct metric provides a consistent, albeit sometimes marginal, advantage in controlled settings, a benefit that becomes more pronounced in complex, real-world data where multiple structural characteristics may coexist.

Datasets

Building on the successful validation through statistical simulations, our study progresses to applying the proposed method to real-world datasets. We utilize the same CNN models employed in the simulation study to maintain consistency in evaluation. Performance assessment is conducted using key evaluation metrics, including accuracy, precision, recall, and F1-score, ensuring a comprehensive comparison of model effectiveness. Below, we provide a summarized description of the genetic datasets used in this study (Table 2).

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Table 2. The summary of the datasets used in this study.

https://doi.org/10.1371/journal.pone.0340005.t002

ARCENE dataset. [54] ARCENE was constructed by merging three mass spectrometry datasets: NCI ovarian cancer data, NCI prostate cancer data, and EVMS prostate cancer data. The dataset includes 100 training samples and 100 validation samples, each containing 10,000 features, of which 7000 are predictive and 3000 are spurious. Since the testing set does not provide binary labels, we combined the training and validation sets, resulting in 112 control samples labeled as class 0 and 88 cancer samples labeled as class 1.

Colon cancer dataset. [55] This dataset consists of the 2000 genes with the highest minimal intensity across samples and contains 62 samples, including 22 normal and 40 tumor tissues. The binary target variable y is set to class 0 for normal tissues and class 1 for tumor tissues.

Leukemia dataset. [56] This dataset differentiates between acute myeloid leukemia (AML) and acute lymphoblastic leukemia (ALL). It comprises 72 samples, with 25 AML cases and 47 ALL cases, and includes 3571 gene expression features. The binary target variable y is defined as class 0 for AML and class 1 for ALL.

Ovarian cancer dataset. [57] This dataset consists of 253 samples, including 91 normal ovarian tissue samples and 162 ovarian cancer samples, with 15,154 molecular mass/charge (M/Z) features. The binary target variable y is labeled as class 0 for normal samples and class 1 for cancer samples.

Prostate cancer dataset. [58] This dataset contains prostate tissue samples from 102 patients, comprising 52 tumor samples and 50 non-tumor samples, with 6033 gene expression features. The binary target variable y is defined as class 0 for non-tumor samples and class 1 for tumor samples.

Results of the feature distance rank matrices

To evaluate the effectiveness of different distance metrics in capturing spatial and structural relationships among features, we compare feature distance rank matrices derived from five real-world tabular datasets against a theoretical “true model.” This reference model represents the rank matrix of Euclidean distances between all pixel pairs in a p p image grid, arranged row-wise. Its rank values, visualized through grayscale intensities, exhibit a smooth, symmetric structure, with lighter tones near the diagonal and darker shades radiating outward, reflecting spatial continuity and locality in a well-ordered 2D layout. Fig 2 presents the rank matrices obtained using six distance metrics, Euclidean, one minus correlation, Geodesic, Jensen-Shannon, Wasserstein, and Tropical after feature reordering and optimization to approximate this ideal structure.

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Fig 2. Feature distance rank matrices across different metrics.

Feature distance rank matrices based on various distance metrics (from left to right: Euclidean distance, one minus correlation, Geodesic distance, Jensen-Shannon distance, Wasserstein distance, and Tropical distance) computed between all pairs of variables across five datasets, following optimization and feature reordering. The grey level indicates the rank value.

https://doi.org/10.1371/journal.pone.0340005.g002

Across datasets, Euclidean-based matrices show moderate spatial coherence. For instance, in Prostate, they retain some diagonal structure and smooth transitions. However, in smaller or noisier datasets like Colon and Leukemia, Euclidean matrices appear disordered, failing to reveal consistent feature relationships. One minus correlation distance yields mixed outcomes; while it reveals some block structures in datasets like Ovarian and Prostate, it generally captures linear dependencies rather than continuous spatial gradients, deviating from the true model. In contrast, Geodesic distance produces smoother transitions and clearer diagonal formations in Ovarian and Prostate, indicating its ability to uncover intrinsic geometric structures. Jensen-Shannon distance demonstrates variability: it shows some spatial coherence in Prostate but generates fragmented and noisy patterns in Colon, Leukemia, and Ovarian, likely due to unstable distribution estimates in sparse data. Wasserstein distance shows strong performance in datasets like Colon, Leukemia, and Prostate, maintaining smooth, block-like rank structures. However, in ARCENE, the matrices become less coherent, likely due to sample noise and the difficulty of estimating reliable empirical distributions. The Tropical distance generates rank matrices with distinct structural characteristics; for the Prostate dataset, it produces a highly coherent diagonal pattern similar to the Euclidean and Geodesic metrics, suggesting it successfully captures the hierarchical dominance relationships in this data. However, for the Colon dataset, the Tropical distance matrix appears more fragmented, indicating that the max-plus metric structure may be less suitable for this specific gene expression profile.

Overall, Wasserstein distance most closely approximates the spatial layout of the true model across well-behaved datasets, offering more faithful and interpretable representations of feature relationships. While Euclidean and correlation-based metrics are more limited in capturing nonlinear structure. The performance between Geodesic distance and Jensen-Shannon distance are similar. Tropical distance shows promise in datasets with specific hierarchical structures but exhibits variability in others. These findings emphasize the value of integrating non-Euclidean distances, particularly Geodesic, Wasserstein, and Tropical into the IGTD framework, enhancing its ability to transform tabular data into meaningful spatial representations for CNN-based learning, while also revealing its boundaries in challenging scenarios.

Pseudo-image visualization

Fig 3 examines the quality and structure of pseudo-images generated from tabular data for binary classification tasks using IGTD framework. For each of the five datasets, representative samples from class 0 and class 1 are visualized as grayscale images. These images were generated using six different distance metrics, and arranged from left to right as follows: Euclidean distance, one minus correlation distance, Geodesic distance, Jensen-Shannon distance, Wasserstein distance, and Tropical distance.

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Fig 3. Representative pseudo-images for two classes under varied distance metrics.

The first row displays pseudo-images from class 0, while the second row shows pseudo-images from class 1, for representative samples from each of the five datasets. Images are generated using six distance metrics, Euclidean distance, one minus correlation, Geodesic distance, Jensen-Shannon distance, Wasserstein distance, and Tropical distance, arranged from left to right.

https://doi.org/10.1371/journal.pone.0340005.g003

In ARCENE, pseudo-images generated using one minus correlation, Geodesic, Jensen-Shannon, Wasserstein distances, and Tropical distance revealed special patterns with clear class separability, in contrast to the diffuse, less structured patterns under Euclidean. This highlights the superiority of non-Euclidean metrics in capturing underlying feature geometry. In the Colon dataset, where the sample size is small and dimensionality high, all images appeared noisy, but Geodesic and Jensen-Shannon distances still produced marginally class separability, suggesting a slight advantage even under data constraints. The Leukemia dataset further emphasized the strength of Geodesic, Wasserstein, and Tropical metrics, which yielded more centered and structured intensity distributions, revealing clearer distinctions between classes. Euclidean distance again failed to preserve meaningful structure. In the Ovarian dataset, which benefits from a larger sample size and variable numbers, all distance metrics produced the most visually distinct class patterns. Finally, for Prostate, class differences were more distinct for Euclidean, one minus correlation, Geodesic and Jensen-Shannon distance metrics.

These visual comparisons demonstrate that non-Euclidean metrics such as Geodesic, Wasserstein, and Tropical distances consistently outperformed others in generating coherent and class-discriminative pseudo-images, regardless of sample size or feature complexity. These visual findings align with quantitative classification results, reinforcing the utility of non-Euclidean metrics in enhancing CNN-based learning from high-dimensional tabular data.

Classification performance evaluation

Ten-fold cross-validation classification results (mean validation indices with one standard deviation) across five real-world datasets using six different distance metrics are given in Table 3. Overall, the results demonstrate that non-Euclidean metrics often outperform the baseline Euclidean distance in terms of accuracy and predictive performance. In the ARCENE dataset, IGTD(GD) achieved the highest accuracy (0.8650), outperforming IGTD(ED) (0.8150), with IGTD(WD) also showing strong performance (0.8500). The Colon dataset exhibited a notable increase in accuracy with IGTD(GD) (0.8357), compared to only 0.7452 for IGTD(ED). For Leukemia, IGTD(RD) and IGTD(JD) yielded higher accuracy scores (0.9464 and 0.9429, respectively) than the Euclidean-based method. In the Ovarian dataset, all methods performed exceptionally well, but IGTD(GD) and IGTD(RD) slightly surpassed IGTD(ED), with accuracy values approaching 0.99. For the Prostate dataset, IGTD(TD) utilizing Tropical distance achieved the highest accuracy (0.8718) and precision (0.9133) among all metrics, surpassing IGTD(RD) (0.8609) and IGTD(ED) (0.8227). This significant result underscores the effectiveness of tropical geometry in capturing the specific feature relationships inherent in prostate cancer genomic data. Conversely, IGTD(TD) showed variable performance across other datasets, such as Colon (0.5833), suggesting its utility may be domain-specific.

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Table 3. Ten-fold cross-validation classification results (mean with one standard deviation) across five real-world datasets using six different distance metrics within the IGTD framework.

https://doi.org/10.1371/journal.pone.0340005.t003

In terms of stability, measured through standard deviations, IGTD(GD) consistently maintained a good balance between high accuracy and moderate variability, indicating robustness in modeling nonlinear structures. While IGTD(RD) and IGTD(JD) sometimes exhibited higher variance, particularly in smaller or more imbalanced datasets like Colon and Leukemia, IGTD(WD) generally combined strong performance with lower variability. These observations suggest that Wasserstein and Geodesic distances are both effective and reliable for capturing intrinsic relationships in high-dimensional feature spaces.

The influence of dataset characteristics, such as sample size and number of features, is also apparent in the results. Smaller datasets with high dimensionality, such as Colon (n = 62, p = 2000) and Leukemia (n = 72, p = 3571), showed the most pronounced performance gains when non-Euclidean metrics were used, particularly IGTD(GD). These improvements likely stem from the ability of non-Euclidean metrics to capture complex, nonlinear dependencies that traditional Euclidean distance may fail to detect.

These empirical findings underscore the critical importance of spatial coherence in the transformed data. The relevance of aligning with the theoretical true model extends beyond visual interpretability; it is fundamental to the mechanics of CNN-based classification. CNNs rely on the principle of locality, utilizing convolutional kernels to extract features from spatially adjacent pixels within a receptive field. A feature distance rank matrix that approximates the smooth, continuous structure of the true model indicates that the chosen distance metric has successfully mapped intrinsically similar features to neighboring coordinates on the image grid. This spatial coherence ensures that the CNN’s filters can effectively capture local correlations and structural patterns. Conversely, a fragmented or disordered rank matrix implies that related features remain scattered across the grid, negating the advantages of the convolutional architecture and hindering the model’s ability to extract robust representations.