We consider finding a counterfactual explanation for a classification or regression forest, such as a random forest. This requires solving an optimization problem to find the closest input instance to a given instance for which the forest outputs a desired value. Finding an exact solution has a cost that is exponential on the number of leaves in the forest. We propose a simple but very effective approach: we constrain the optimization to input space regions populated by actual data points. The problem reduces to a form of nearest-neighbor search using a certain distance on a certain dataset. This has two advantages: first, the solution can be found very quickly, scaling to large forests and high-dimensional data, and enabling interactive use. Second, the solution found is more likely to be realistic in that it is guided towards high-density areas of input space.

Interpreting the image datasets is a difficult task, as each image contains a lot of irrelevant data. This paper presents a simple yet effective method to interpret the image datasets. We achieve this by using sparse oblique trees as a tool to select features from the dataset. These trees are not only accurate but also very interpretable. The hierarchical structure of the tree helps to visualize the underlying patterns in the dataset. By studying the weights of the nodes, we can determine what set of features differentiate between classes or groups of classes. We effectively demonstrate our results in multiple image datasets.

Natural language processing datasets, for example for document classification or sentiment analysis, are characterized by sparse, high-dimensional feature vectors, often based on bag-of-words approaches. Such datasets contain a wealth of information not just about the predictive task in question, but also about the language itself, and it is of interest to do data mining on such data. While one way to do this is to use standard exploratory data analysis techniques such as clustering or dimensionality reduction, here we propose a different approach, which can be used if we have access to a labeled dataset. The idea is to use sparse oblique decision trees, a type of interpretable model having the structure of a decision tree but where the decision nodes use hyperplanes involving few input features. Such trees can be trained using the Tree Alternating Optimization (TAO) algorithm. Our approach is to train a sparse oblique tree that is as small and sparse as possible while achieving a good enough predictive accuracy, and then to inspect the weights in the tree decision nodes in order to establish a relationship between input features and classes. This reveals interesting patterns about the classifier and about the data itself. For example, we determine how small, specific subsets of features are used for specific classes (say, certain words for certain document topics), both globally or for a single input instance. The hierarchical structure of the tree also explains the common theme among a group of instances. We demonstrate this using the AG news dataset.

We consider counterfactual explanations, the problem of minimally adjusting features in a source input instance so that it is classified as a target class under a given classifier. This has become a topic of recent interest as a way to query a trained model and suggest possible actions to overturn its decision. Mathematically, the problem is formally equivalent to that of finding adversarial examples, which also has attracted significant attention recently. Most work on either counterfactual explanations or adversarial examples has focused on differentiable classifiers, such as neural nets. We focus on classification trees, both axis-aligned and oblique (having hyperplane splits). Although here the counterfactual optimization problem is nonconvex and nondifferentiable, we show that an exact solution can be computed very efficiently, even with high-dimensional feature vectors and with both continuous and categorical features, and demonstrate it in different datasets and settings. The results are particularly relevant for finance, medicine or legal applications, where interpretability and counterfactual explanations are particularly important.

With the recent success of deep neural networks in computer vision, it is important to understand the internal working of these networks. What does a given neuron represent? The concepts captured by a neuron may be hard to understand or express in simple terms. The approach we propose in this paper is to characterize the region of input space that excites a given neuron to a certain level; we call this the inverse set. This inverse set is a complicated high dimensional object that we explore by an optimization-based sampling approach. Inspection of samples of this set by a human can reveal regularities that help to understand the neuron. This goes beyond approaches which were limited to finding an image which maximally activates the neuron [1] or using Markov chain Monte Carlo to sample images [2], but this is very slow, generates samples with little diversity and lacks control over the activation value of the generated samples. Our approach also allows us to explore the intersection of inverse sets of several neurons and other variations.

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 what internal features computed by the neural net are responsible for a particular class. We achieve this by mimicking part of the net with a decision tree having sparse weight vectors at the nodes. We are able to learn trees that are both highly accurate and interpretable, so 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. We demonstrate this robustly in MNIST and ImageNet with LeNet5 and VGG networks.

Learning decision trees is a difficult optimization problem: nonconvex, nondifferentiable and over a huge number of tree structures. The dominant paradigm in practice, established in the 1980s, are axis-aligned trees trained with a greedy recursive partitioning algorithm such as CART or C5.0. Several non-greedy optimization algorithms have been proposed recently, which optimize all the nodes' parameters jointly, and we compare experimentally some of them in a range of classification and regression datasets, in terms of accuracy, training time and tree size. The non-greedy algorithms do not improve over CART significantly with one exception, tree alternating optimization (TAO). TAO scales to large datasets and produces axis-aligned and especially oblique trees that consistently outperform all other algorithms, often by a large margin. TAO makes oblique trees preferable to axis-aligned ones in many cases, since they are much more accurate while remaining small and interpretable. This suggests a change in paradigm in practical applications of decision trees.

Arbitrary style transfer is an important problem in computer vision that aims to transfer style patterns from an arbitrary style image to a given content image. However, current methods either rely on slow iterative optimization or fast pre-determined feature transformation, but at the cost of compromised visual quality of the styled image; especially, distorted content structure. In this work, we present an effective and efficient approach for arbitrary style transfer that seamlessly transfers style patterns as well as keep content structure intact in the styled image. We achieve this by aligning style features to content features using rigid alignment; thus modifying style features, unlike the existing methods that do the opposite. We demonstrate the effectiveness of the proposed approach by generating high-quality stylized images and compare the results with the current state-of-the-art techniques for arbitrary style transfer.