Optimizing High-Dimensional Oblique Splits

• URL: http://arxiv.org/abs/2503.14381v1
• Date: Tue, 18 Mar 2025 16:14:38 GMT
• Title: Optimizing High-Dimensional Oblique Splits
• Authors: Chien-Ming Chi,
• Abstract summary: This paper explores optimizing high-dimensional $s$-sparse oblique splits from $(vecw, vecwtopboldsymbolX_i) : iin 1,dots, n, vecw in mathbbRp, | vecw |_0 leq s $ for growing oblique trees.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Orthogonal-split trees perform well, but evidence suggests oblique splits can enhance their performance. This paper explores optimizing high-dimensional $s$-sparse oblique splits from $\{(\vec{w}, \vec{w}^{\top}\boldsymbol{X}_{i}) : i\in \{1,\dots, n\}, \vec{w} \in \mathbb{R}^p, \| \vec{w} \|_{2} = 1, \| \vec{w} \|_{0} \leq s \}$ for growing oblique trees, where $ s $ is a user-defined sparsity parameter. We establish a connection between SID convergence and $s_0$-sparse oblique splits with $s_0\ge 1$, showing that the SID function class expands as $s_0$ increases, enabling the capture of more complex data-generating functions such as the $s_0$-dimensional XOR function. Thus, $s_0$ represents the unknown potential complexity of the underlying data-generating function. Learning these complex functions requires an $s$-sparse oblique tree with $s \geq s_0$ and greater computational resources. This highlights a trade-off between statistical accuracy, governed by the SID function class size depending on $s_0$, and computational cost. In contrast, previous studies have explored the problem of SID convergence using orthogonal splits with $ s_0 = s = 1 $, where runtime was less critical. Additionally, we introduce a practical framework for oblique trees that integrates optimized oblique splits alongside orthogonal splits into random forests. The proposed approach is assessed through simulations and real-data experiments, comparing its performance against various oblique tree models.

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