RegScore: Interpretable Regression Scorecards

• RegScore is a regression scorecard framework that extends interpretable binary scorecards to continuous targets using k-sparse ridge regression.
• It applies binarization and discretization of tabular data to produce human-readable, additive score contributions, enhancing clinical transparency.
• Personalized extensions like PLR and PRS integrate multimodal data, maintaining interpretability while targeting accurate mPAP estimation.

RegScore is a scoring-system framework for regression that extends sparse, transparent scorecards from binary decision tasks to continuous targets. It was introduced for estimating mean Pulmonary Artery Pressure (mPAP) from tabular clinical and cardiac MRI-derived variables, with a downstream Pulmonary Hypertension (PH) diagnosis obtained by thresholding predicted mPAP. The method preserves the operational form of a clinical scorecard—human-readable rules with additive point contributions—but replaces the usual classification focus with a $k$-sparse ridge regression over binarized features, and relaxes the traditional restriction to integer-valued coefficients. The same paper also introduces multimodal personalized extensions, Personalized Linear Regression (PLR) and Personalized RegScore (PRS), built on the TIP (Tabular Image Pretraining) architecture to combine tabular data and cardiac MRI images while retaining an interpretable prediction layer (Grzeszczyk et al., 25 Jul 2025).

1. Motivation and problem setting

Scoring systems are widely used in medicine because they are easy to inspect, compute, and communicate. The paper situates RegScore in the tradition of scorecards such as CHADS$_2$ and NEWS$_2$, but argues that most existing data-driven scoring systems are designed for classification rather than regression (Grzeszczyk et al., 25 Jul 2025). That distinction is central in conditions where the clinically relevant diagnosis is defined by thresholding an underlying continuous physiological quantity.

Pulmonary Hypertension is the motivating example. Diagnosis depends on whether invasively measured mPAP exceeds a threshold, but the paper argues that directly estimating mPAP is more informative than predicting a binary class alone. In the experimental protocol, PH classification is recovered by thresholding predicted mPAP at $\text{mPAP} > 25 \text{ mmHg}$ (Grzeszczyk et al., 25 Jul 2025). This makes RegScore a regression model first and a classifier only secondarily.

The stated objective is therefore not merely to produce a sparse yes/no risk score, but to expose how tabular features contribute to an estimated continuous target. In the paper’s framing, this improves transparency because clinicians can inspect how each rule affects the predicted physiological quantity itself rather than only a class label.

2. Formal model and representation

RegScore operates on a binarized tabular representation. Given tabular features $$X^t = [x^t_1,\ldots,x^t_N] \in \mathbb{R}^{P \times N}$$, the paper constructs a binary matrix

$$\hat{X}^t = [\hat{x}^t_1,\ldots,\hat{x}^t_D] \in \{0,1\}^{P \times D}$$

by one-hot encoding $N_{\text{cat}}$ categorical variables and discretizing the remaining continuous variables with a discretization function $\xi$. The two discretizers considered are MDLP (Minimum Description Length Principle) and tertile binning (Grzeszczyk et al., 25 Jul 2025).

The core RegScore estimator is a sparse ridge regression: $$\min_{\beta} \left\| \mathbf{y} - \mathbf{\hat{X}}\boldsymbol{\beta} \right\|_2^2 + \lambda_2 \|\boldsymbol{\beta}\|_2^2 \quad \text{subject to} \quad \|\boldsymbol{\beta}\|_0 \le k.$$ Here $\boldsymbol{\beta}$ is the vector of feature weights or scores, $_2$0 is the ridge regularization coefficient, and $_2$1 is the sparsity budget. Operationally, prediction takes the additive scorecard form

$_2$2

with at most $_2$3 nonzero coefficients.

This formulation keeps the usual scorecard workflow—activate a small number of binary rules, attach coefficients, and sum them—but differs from classical systems such as RiskSLIM or FasterRisk in one specific respect: RegScore allows continuous coefficients rather than restricting weights to small integers (Grzeszczyk et al., 25 Jul 2025). The paper’s rationale is that, for regression targets such as mPAP, forcing integer points can reduce fidelity. A plausible implication is that RegScore treats interpretability as arising primarily from sparse, human-readable binary rules rather than from integer arithmetic alone.

3. Optimization and learning procedures

Because the constraint $_2$4 makes the optimization problem NP-hard, the paper introduces two training strategies (Grzeszczyk et al., 25 Jul 2025).

The first is beam-search RegScore, denoted $_2$5. It adapts the beam-search phase of FasterRisk from logistic classification to sparse ridge regression. The method assumes that a good model of size $_2$6 likely contains a good model of size $_2$7. It therefore grows candidate sparse supports incrementally, optimizes the nonzero coefficients by ridge regression for each candidate, keeps only the top $_2$8 candidates at each stage, and then fine-tunes the nonzero weights among the retained supports. The paper sets the beam width to

$_2$9

The second is OKRidge RegScore, denoted $_2$0, which uses branch-and-bound for optimal $_2$1-sparse ridge regression. The search-tree logic is described in terms of computing a lower bound for a node, checking whether that bound is promising relative to the current incumbent, updating the best solution when appropriate, and pruning unpromising regions.

The paper also emphasizes computational practicality. The beam-search variant can be generated in minutes, approximately $_2$2 seconds, which is contrasted with deep learning models that require hours (Grzeszczyk et al., 25 Jul 2025). This is presented as enabling clinicians or domain experts to inspect multiple near-optimal sparse models rather than a single opaque solution.

4. Personalized bimodal extensions

The multimodal extensions use TIP, a bimodal framework combining tabular data and cardiac MRI images. TIP consists of a convolutional image encoder $_2$3, a transformer-based tabular encoder $_2$4, and a multimodal interaction module $_2$5. Given image representation

$_2$6

and tabular representation

$_2$7

the multimodal block outputs

$_2$8

TIP is pretrained with contrastive learning between image and tabular representations, image-tabular matching, and tabular reconstruction from $_2$9 (Grzeszczyk et al., 25 Jul 2025).

PLR replaces a standard scalar regression head with a personalized linear regression over tabular features. The CLS token is mapped to $\text{mPAP} > 25 \text{ mmHg}$0 parameters: $\text{mPAP} > 25 \text{ mmHg}$1 personalized coefficients $\text{mPAP} > 25 \text{ mmHg}$2 and one personalized bias $\text{mPAP} > 25 \text{ mmHg}$3. Prediction is

$\text{mPAP} > 25 \text{ mmHg}$4

The coefficients are patient-specific because they are generated from the multimodal representation rather than shared globally.

PRS applies the same idea to the binary scorecard representation. It scores the binary rules and keeps only the top $\text{mPAP} > 25 \text{ mmHg}$5 features for each patient. The gating equations are

$\text{mPAP} > 25 \text{ mmHg}$6

$\text{mPAP} > 25 \text{ mmHg}$7

with

$\text{mPAP} > 25 \text{ mmHg}$8

Here $\text{mPAP} > 25 \text{ mmHg}$9 is the hard top-$$X^t = [x^t_1,\ldots,x^t_N] \in \mathbb{R}^{P \times N}$$0 selector used at inference and $$X^t = [x^t_1,\ldots,x^t_N] \in \mathbb{R}^{P \times N}$$1 is the differentiable approximation used in training (Grzeszczyk et al., 25 Jul 2025).

These extensions preserve interpretability at the final layer. In PLR, the output remains a linear expression in clinically meaningful tabular variables. In PRS, the active rules can differ across patients, but the decision still depends on a small subset of human-readable binary conditions. This suggests a shift from a single global scorecard to patient-specific sparse scorecards without abandoning additive structure.

5. Data, preprocessing, and evaluation design

The experimental data come from the ASPIRE Registry and include 2051 invasively measured mPAP values matched with cardiac MRI videos from one cardiac cycle, drawn from 1918 patients. The cohort contains 1171 female and 747 male patients, with age $$X^t = [x^t_1,\ldots,x^t_N] \in \mathbb{R}^{P \times N}$$2 years. Some patients had repeated procedures over time (Grzeszczyk et al., 25 Jul 2025).

For tabular data, the authors retain demographic and MRI-derived variables with fewer than 500 missing values. Numerical features are standardized, and only statistically significant ones are retained using f-regression. For imaging, cardiac MRIs from Siemens, Philips, and GE are reduced to systolic, diastolic, and intermediate frames as a 3-channel image, resampled to $$X^t = [x^t_1,\ldots,x^t_N] \in \mathbb{R}^{P \times N}$$3 and resized to $$X^t = [x^t_1,\ldots,x^t_N] \in \mathbb{R}^{P \times N}$$4 (Grzeszczyk et al., 25 Jul 2025).

The split is 1790 training samples and 261 test samples, with 5-fold cross-validation on the training set. Patient-level separation is enforced so that a patient appears in only one split, and stratification is based on mPAP divided into four bins. Regression is evaluated using MAE and Pearson correlation $$X^t = [x^t_1,\ldots,x^t_N] \in \mathbb{R}^{P \times N}$$5; downstream PH classification is evaluated using Accuracy and F1 after thresholding predicted mPAP at $$X^t = [x^t_1,\ldots,x^t_N] \in \mathbb{R}^{P \times N}$$6 (Grzeszczyk et al., 25 Jul 2025).

The principal hyperparameters reported are: $$X^t = [x^t_1,\ldots,x^t_N] \in \mathbb{R}^{P \times N}$$7 RegScore uses MDLP discretization, whereas PRS uses tertiles. Deep models are trained in PyTorch on an NVIDIA A100 80GB GPU for up to 500 epochs of self-supervised learning and fine-tuning using Adam (Grzeszczyk et al., 25 Jul 2025).

6. Empirical results and interpretability

On tabular scoring-system experiments, RegScore improves over classification-oriented scoring methods even when assessed on downstream PH diagnosis. For $$X^t = [x^t_1,\ldots,x^t_N] \in \mathbb{R}^{P \times N}$$8, the reported results are: $$X^t = [x^t_1,\ldots,x^t_N] \in \mathbb{R}^{P \times N}$$9 and

$$\hat{X}^t = [\hat{x}^t_1,\ldots,\hat{x}^t_D] \in \{0,1\}^{P \times D}$$0

The paper contrasts these with MISS $$\hat{X}^t = [\hat{x}^t_1,\ldots,\hat{x}^t_D] \in \{0,1\}^{P \times D}$$1 at Accuracy $$\hat{X}^t = [\hat{x}^t_1,\ldots,\hat{x}^t_D] \in \{0,1\}^{P \times D}$$2, F1 $$\hat{X}^t = [\hat{x}^t_1,\ldots,\hat{x}^t_D] \in \{0,1\}^{P \times D}$$3; RiskSLIM $$\hat{X}^t = [\hat{x}^t_1,\ldots,\hat{x}^t_D] \in \{0,1\}^{P \times D}$$4 at Accuracy $$\hat{X}^t = [\hat{x}^t_1,\ldots,\hat{x}^t_D] \in \{0,1\}^{P \times D}$$5, F1 $$\hat{X}^t = [\hat{x}^t_1,\ldots,\hat{x}^t_D] \in \{0,1\}^{P \times D}$$6; and FasterRisk $$\hat{X}^t = [\hat{x}^t_1,\ldots,\hat{x}^t_D] \in \{0,1\}^{P \times D}$$7 at Accuracy $$\hat{X}^t = [\hat{x}^t_1,\ldots,\hat{x}^t_D] \in \{0,1\}^{P \times D}$$8, F1 $$\hat{X}^t = [\hat{x}^t_1,\ldots,\hat{x}^t_D] \in \{0,1\}^{P \times D}$$9 (Grzeszczyk et al., 25 Jul 2025).

At $N_{\text{cat}}$0, performance further improves: $N_{\text{cat}}$1

$N_{\text{cat}}$2

The paper identifies these as the best results in that table (Grzeszczyk et al., 25 Jul 2025).

In the multimodal setting, the strongest black-box baseline is TIP: $N_{\text{cat}}$3 The interpretable personalized variants remain close: $N_{\text{cat}}$4

$N_{\text{cat}}$5

Thus TIP has the best pure regression performance, while PRS has the best classification metrics in that table (Grzeszczyk et al., 25 Jul 2025).

The ablation results also identify specific dependencies. For PLR, removing images changes performance from MAE $N_{\text{cat}}$6, $N_{\text{cat}}$7 to MAE $N_{\text{cat}}$8, $N_{\text{cat}}$9, while removing self-supervised learning yields MAE $\xi$0, $\xi$1. For PRS, the full model gives MAE $\xi$2, $\xi$3, compared with MAE $\xi$4, $\xi$5 without self-supervised learning and MAE $\xi$6, $\xi$7 without image input (Grzeszczyk et al., 25 Jul 2025). The paper therefore attributes part of the multimodal performance to both image information and self-supervised pretraining.

Interpretability is illustrated by an explicit RegScore example. The scorecard contains a bias term $\xi$8 and the following rules: $\xi$9 contributes $$\min_{\beta} \left\| \mathbf{y} - \mathbf{\hat{X}}\boldsymbol{\beta} \right\|_2^2 + \lambda_2 \|\boldsymbol{\beta}\|_2^2 \quad \text{subject to} \quad \|\boldsymbol{\beta}\|_0 \le k.$$0; $$\min_{\beta} \left\| \mathbf{y} - \mathbf{\hat{X}}\boldsymbol{\beta} \right\|_2^2 + \lambda_2 \|\boldsymbol{\beta}\|_2^2 \quad \text{subject to} \quad \|\boldsymbol{\beta}\|_0 \le k.$$1 contributes $$\min_{\beta} \left\| \mathbf{y} - \mathbf{\hat{X}}\boldsymbol{\beta} \right\|_2^2 + \lambda_2 \|\boldsymbol{\beta}\|_2^2 \quad \text{subject to} \quad \|\boldsymbol{\beta}\|_0 \le k.$$2; $$\min_{\beta} \left\| \mathbf{y} - \mathbf{\hat{X}}\boldsymbol{\beta} \right\|_2^2 + \lambda_2 \|\boldsymbol{\beta}\|_2^2 \quad \text{subject to} \quad \|\boldsymbol{\beta}\|_0 \le k.$$3 contributes $$\min_{\beta} \left\| \mathbf{y} - \mathbf{\hat{X}}\boldsymbol{\beta} \right\|_2^2 + \lambda_2 \|\boldsymbol{\beta}\|_2^2 \quad \text{subject to} \quad \|\boldsymbol{\beta}\|_0 \le k.$$4; $$\min_{\beta} \left\| \mathbf{y} - \mathbf{\hat{X}}\boldsymbol{\beta} \right\|_2^2 + \lambda_2 \|\boldsymbol{\beta}\|_2^2 \quad \text{subject to} \quad \|\boldsymbol{\beta}\|_0 \le k.$$5 contributes $$\min_{\beta} \left\| \mathbf{y} - \mathbf{\hat{X}}\boldsymbol{\beta} \right\|_2^2 + \lambda_2 \|\boldsymbol{\beta}\|_2^2 \quad \text{subject to} \quad \|\boldsymbol{\beta}\|_0 \le k.$$6; and $$\min_{\beta} \left\| \mathbf{y} - \mathbf{\hat{X}}\boldsymbol{\beta} \right\|_2^2 + \lambda_2 \|\boldsymbol{\beta}\|_2^2 \quad \text{subject to} \quad \|\boldsymbol{\beta}\|_0 \le k.$$7 contributes $$\min_{\beta} \left\| \mathbf{y} - \mathbf{\hat{X}}\boldsymbol{\beta} \right\|_2^2 + \lambda_2 \|\boldsymbol{\beta}\|_2^2 \quad \text{subject to} \quad \|\boldsymbol{\beta}\|_0 \le k.$$8. For a patient satisfying all five conditions,

$$\min_{\beta} \left\| \mathbf{y} - \mathbf{\hat{X}}\boldsymbol{\beta} \right\|_2^2 + \lambda_2 \|\boldsymbol{\beta}\|_2^2 \quad \text{subject to} \quad \|\boldsymbol{\beta}\|_0 \le k.$$9

This is exactly the operational interpretation the paper emphasizes: a small table of rules whose contributions can be inspected and summed manually (Grzeszczyk et al., 25 Jul 2025).

7. Limitations and relation to adjacent scoring-system research

The paper explicitly notes a performance–interpretability trade-off. Moving from the strongest black-box multimodal model, TIP, to RegScore, PLR, or PRS improves transparency but can reduce raw regression accuracy because the interpretable variants constrain the prediction to sparse or additive expressions in tabular variables (Grzeszczyk et al., 25 Jul 2025).

Other stated limitations are dependence on discretization and binning, dependence on feature engineering and feature availability, the fact that PLR and PRS still rely on a deep multimodal encoder even though the final layer is interpretable, and the concentration of empirical validation in a single clinical domain. The authors identify incorporating binning into the optimization process as a direction for future work (Grzeszczyk et al., 25 Jul 2025).

Within the broader scoring-system literature, RegScore occupies a specific position. “Score Engineered Logistic Regression” formulates constrained logistic scorecards for binary outcomes using sequential quadratic programming and linear equality and inequality score-engineering constraints (Hoadley, 2020). “Gradient Boosted Risk Scores” learns compact additive point systems from boosted decision stumps and supports regression, classification, and time-to-event tasks (Georgantas et al., 4 May 2026). This suggests that RegScore belongs to a broader research program that extends interpretable additive scoring systems beyond classical classification scorecards, but does so in a distinct way: by combining binary rule representations, $\boldsymbol{\beta}$0-sparse ridge regression, and multimodal personalized extensions targeted at continuous clinical prediction.