An Enhanced Projection Pursuit Tree Classifier with Visual Methods for Assessing Algorithmic Improvements

Predictive models in clinical and regulated settings must be accurate and fully auditable. Tree ensembles deliver strong accuracy on tabular data, but their sequential boosting couples structure discovery with coefficient estimation, making compact per-prediction auditing difficult. Interpretable alternatives impose structural constraints that limit expressiveness: generalized additive models typically restrict interactions to pairwise terms and post-hoc rule extractors produce overlapping rules that hinder compact interpretation. We introduce Residual Pattern Tree Ensemble (RPTE), a three-stage learning approach, that is built on three key principles: bounded feature budget, source disjointness, and separate coefficient estimation. Stage1 builds a supervised symbolic feature vocabulary. Stage2 grows shallow trees under a source-disjointness constraint, where each raw variable is allocated to at most one tree, and retains only the discovered tree structures. Stage~3 solves a single $\ell_1$-regularized logistic regression over leaf-region indicators, yielding jointly optimal sparse coefficients. This learning approach ensures that every prediction decomposes into an algebraic sum of named, non-overlapping rule contributions, enabling full auditability by design. Empirical evaluation on twelve clinical-domain binary classification benchmarks using repeated stratified 5-fold cross-validation shows that RPTE performs competitively against tuned opaque ensembles and interpretable baselines. RPTE reduces model inspection units by 9$\times$ to 87$\times$ relative to XGBoost and maintains lower audit complexity than EBM on all 12 datasets. RuleFit requires comparable or fewer inspection units on three datasets where its rule count is small, but without source-disjointness guarantees. The source code is available at this https URL.

Optimal decision trees have attracted increasing attention in recent years, yet existing algorithms face two closely related challenges. First, the trees they construct have limited expressivity because their splitting rules are typically restricted to axis-parallel splits or binary features. Second, these algorithms generally scale poorly to large datasets. These challenges compound one another: more expressive splitting rules increase the combinatorial complexity of the search space, making the resulting optimal decision tree (ODT) problem even more difficult to solve. Building on the proper decision tree framework of \citet{he2025odt}, we propose, to the best of our knowledge, the first algorithm for constructing optimal decision trees with hypersurface splits. The algorithm has worst-case work complexity $O\left(K!K N^{KG+1}\right)$ when $N\geq K$ and fixed-dimensional feasibility tests are treated as constant-cost operations, where $G=\binom{D+M}{M}-1$ depends on the hypersurface polynomial degree $M$ and the data dimension $D$. Its structure is naturally amenable to vectorization and parallelization, enabling efficient execution on GPUs, and the generic algorithm design can be used to accelerate other ODT variants, such as axis-parallel decision (ADT) trees. We further develop a safe pruning strategy that substantially reduces the number of candidate configurations and an incremental generation procedure that reduces the associated feasibility-checking cost from $O(K^2)$ to $O(K)$.

Tree ensembles are machine learning models with strong predictive performance and interpretability, and remain widely used for tabular data. Standard pruning methods for tree ensembles typically optimize an accuracy-compression trade-off and may change a subset of predictions, potentially compromising decision consistency. Faithful pruning methods address this issue by preserving prediction equivalence over the entire input space, but this requirement leads to lower compression ratios. We propose PINE, a pruning method that provides strong guarantees within an in-distribution region. PINE preserves prediction equivalence within this region and controls the region size using a single parameter $α$ via conformal calibration. Experiments on 12 public tabular datasets show that PINE improves the compression ratio by up to 30% while preserving predictions at a comparable level to existing faithful pruning methods.