We propose a new model for dimensionality reduction, the PCA tree, which works like a regular autoencoder, having explicit projection and reconstruction mappings. The projection is effected by a sparse oblique tree, having hard, hyperplane splits using few features and linear leaves. The reconstruction mapping is a set of local linear mappings. Thus, rather than producing a global map as in t-SNE and other methods, which often leads to distortions, it produces a hierarchical set of local PCAs. The use of a sparse oblique tree and PCA makes the overall model interpretable and very fast to project or reconstruct new points. Joint optimization of all the parameters in the tree is a nonconvex nondifferentiable problem. We propose an algorithm that is guaranteed to decrease the error monotonically and which scales to large datasets without any approximation. In experiments, we show PCA trees are able to identify a wealth of low-dimensional and cluster structure in image and document datasets.

Learning a decision tree from data is a difficult optimization problem. The most widespread algorithm in practice, dating to the 1980s, is based on a greedy growth of the tree structure by recursively splitting nodes, and possibly pruning back the final tree. The parameters (decision function) of an internal node are approximately estimated by minimizing an impurity measure. We give an algorithm that, given an input tree (its structure and the parameter values at its nodes), produces a new tree with the same or smaller structure but new parameter values that provably lower or leave unchanged the misclassification error. This can be applied to both axis-aligned and oblique trees and our experiments show it consistently outperforms various other algorithms while being highly scalable to large datasets and trees. Further, the same algorithm can handle a sparsity penalty, so it can learn sparse oblique trees, having a structure that is a subset of the original tree and few nonzero parameters. This combines the best of axis-aligned and oblique trees: flexibility to model correlated data, low generalization error, fast inference and interpretable nodes that involve only a few features in their decision.

Neural Information Processing Systems http://nips.cc/

We propose a new model for dimensionality reduction, the PCA tree, which works like a regular autoencoder, having explicit projection and reconstruction mappings. The projection is effected by a sparse oblique tree, having hard, hyperplane splits using few features and linear leaves. The reconstruction mapping is a set of local linear mappings. Thus, rather than producing a global map as in t-SNE and other methods, which often leads to distortions, it produces a hierarchical set of local PCAs. The use of a sparse oblique tree and PCA makes the overall model interpretable and very fast to project or reconstruct new points. Joint optimization of all the parameters in the tree is a nonconvex nondifferentiable problem.

Their method requires an initial decision tree. The topology of this tree will be fixed, and only the decision rules at each node will be adjusted. The idea behind the proposed adjustment is based on the observation that, fixing all of the parameters of all the nodes except the parameters of node i, the likelihood function for the whole tree reduces to the likelihood function of a simple K-classes classifier. This simple classifier can be trained efficiently (using existing techniques) and doing so will always guarantee that the overall loss will decrease when compared to the loss for the initial decision tree.

The widespread deployment of deep nets in practical applications has lead to a growing desire to understand how and why such black-box methods perform prediction. Much work has focused on understanding what part of the input pattern (an image, say) is responsible for a particular class being predicted, and how the input may be manipulated to predict a different class. We focus instead on understanding which of the internal features computed by the neural net are responsible for a particular class. We achieve this by mimicking part of the neural net with an oblique decision tree having sparse weight vectors at the decision nodes. Using the recently proposed Tree Alternating Optimization (TAO) algorithm, we are able to learn trees that are both highly accurate and interpretable. Such trees can faithfully mimic the part of the neural net they replaced, and hence they can provide insights into the deep net black box. Further, we show we can easily manipulate the neural net features in order to make the net predict, or not predict, a given class, thus showing that it is possible to carry out adversarial attacks at the level of the features. These insights and manipulations apply globally to the entire training and test set, not just at a local (single-instance) level. We demonstrate this robustly in the MNIST and ImageNet datasets with LeNet5 and VGG networks.

Learning a decision tree from data is a difficult optimization problem. The most widespread algorithm in practice, dating to the 1980s, is based on a greedy growth of the tree structure by recursively splitting nodes, and possibly pruning back the final tree. The parameters (decision function) of an internal node are approximately estimated by minimizing an impurity measure. We give an algorithm that, given an input tree (its structure and the parameter values at its nodes), produces a new tree with the same or smaller structure but new parameter values that provably lower or leave unchanged the misclassification error. This can be applied to both axis-aligned and oblique trees and our experiments show it consistently outperforms various other algorithms while being highly scalable to large datasets and trees.

Learning a decision tree from data is a difficult optimization problem. The most widespread algorithm in practice, dating to the 1980s, is based on a greedy growth of the tree structure by recursively splitting nodes, and possibly pruning back the final tree. The parameters (decision function) of an internal node are approximately estimated by minimizing an impurity measure. We give an algorithm that, given an input tree (its structure and the parameter values at its nodes), produces a new tree with the same or smaller structure but new parameter values that provably lower or leave unchanged the misclassification error. This can be applied to both axis-aligned and oblique trees and our experiments show it consistently outperforms various other algorithms while being highly scalable to large datasets and trees. Further, the same algorithm can handle a sparsity penalty, so it can learn sparse oblique trees, having a structure that is a subset of the original tree and few nonzero parameters. This combines the best of axis-aligned and oblique trees: flexibility to model correlated data, low generalization error, fast inference and interpretable nodes that involve only a few features in their decision.

Learning a decision tree from data is a difficult optimization problem. The most widespread algorithm in practice, dating to the 1980s, is based on a greedy growth of the tree structure by recursively splitting nodes, and possibly pruning back the final tree. The parameters (decision function) of an internal node are approximately estimated by minimizing an impurity measure. We give an algorithm that, given an input tree (its structure and the parameter values at its nodes), produces a new tree with the same or smaller structure but new parameter values that provably lower or leave unchanged the misclassification error. This can be applied to both axis-aligned and oblique trees and our experiments show it consistently outperforms various other algorithms while being highly scalable to large datasets and trees. Further, the same algorithm can handle a sparsity penalty, so it can learn sparse oblique trees, having a structure that is a subset of the original tree and few nonzero parameters. This combines the best of axis-aligned and oblique trees: flexibility to model correlated data, low generalization error, fast inference and interpretable nodes that involve only a few features in their decision.