Proof of Lemma 2. Let U be the data set associated to ν. Proof of Lemma 3. First, we prove that the property holds for the root node. We wish to prove the property for some unexplored leaf after the iteration. This is trivial if the leaf ν is not expanded in that iteration. Suppose the leaf ν is expanded. Proof of Lemma 5. From Lemma 2, we note that Q Consider any path from the root to a leaf whose length is mK for some integer K > 0. We note that for each node ν and any of its children ν (Lemma 5).

's leaf nodes to form Given the definition of our attention in Eq. 9 in the main text, the highest polynomial order is Before providing the proof of Theorem 4, we establish Lemma 1 as its foundation. We follow the principle of Y an et al's work [ Figure 1, we consider two kinds of value functions, i.e., In P AM-Net, we set the number of levels to 2. A grid search is performed over different configurations We conduct grid searches on the dropout rate over {0, 0.1, 0.2} and the initial

Learning feature interactions can be the key for multivariate predictive modeling. ReLU-activated neural networks create piecewise linear prediction models.

As a summary, this paper makes the following contributions.

Self-supervised learning (SSL) has recently advanced through non-contrastive methods that couple an invariance term with variance, covariance, or redundancy-reduction penalties. While such objectives shape first- and second-order statistics of the representation, they largely ignore the local geometry of the underlying data manifold. In this paper, we introduce CurvSSL, a curvature-regularized self-supervised learning framework, and its RKHS extension, kernel CurvSSL. Our approach retains a standard two-view encoder-projector architecture with a Barlow Twins-style redundancy-reduction loss on projected features, but augments it with a curvature-based regularizer. Each embedding is treated as a vertex whose $k$ nearest neighbors define a discrete curvature score via cosine interactions on the unit hypersphere; in the kernel variant, curvature is computed from a normalized local Gram matrix in an RKHS. These scores are aligned and decorrelated across augmentations by a Barlow-style loss on a curvature-derived matrix, encouraging both view invariance and consistency of local manifold bending. Experiments on MNIST and CIFAR-10 datasets with a ResNet-18 backbone show that curvature-regularized SSL yields competitive or improved linear evaluation performance compared to Barlow Twins and VICReg. Our results indicate that explicitly shaping local geometry is a simple and effective complement to purely statistical SSL regularizers.

As a summary, this paper makes the following contributions.