Decision trees with binary splits are popularly constructed using Classification and Regression Trees (CART) methodology. For regression models, this approach recursively divides the data into two near-homogenous daughter nodes according to a split point that maximizes the reduction in sum of squares error (the impurity) along a particular variable. This paper aims to study the statistical properties of regression trees constructed with CART methodology. In doing so, we find that the training error is governed by the Pearson correlation between the optimal decision stump and response data in each node, which we bound by constructing a prior distribution on the split points and solving a nonlinear optimization problem. We leverage this connection between the training error and Pearson correlation to show that CART with cost-complexity pruning achieves an optimal complexity/goodnessof-fit tradeoff when the depth scales with the logarithm of the sample size. Data dependent quantities, which adapt to the dimensionality and latent structure of the regression model, are seen to govern the rates of convergence of the prediction error.

Link prediction in graphs is an important task in the fields of network science and machine learning. We propose a flexible means of regularization for link prediction based on an approximation of the Kolmogorov complexity of graphs. Informally, the Kolmogorov complexity of an object is the length of the shortest computer program that produces the object. Complex networks are often generated, in part, by simple mechanisms; for example, many citation networks and social networks are approximately scale-free and can be explained by preferential attachment. A preference for predicting graphs with simpler generating mechanisms motivates our choice of Kolmogorov complexity as a regularization term. Our method is differentiable, fast and compatible with recent advances in link prediction algorithms based on graph neural networks. We demonstrate the effectiveness of our regularization technique on a set of diverse real-world networks.

What do data tell us about physics-and what don't they tell us? There has been a surge of interest in using machine learning models to discover governing physical laws such as differential equations from data, but current methods lack uncertainty quantification to communicate their credibility. This work addresses this shortcoming from a Bayesian perspective. We introduce a novel model comprising "leaf" modules that learn to represent distinct experiments' spatiotemporal functional data as neural networks and a single "root" module that expresses a nonparametric distribution over their governing nonlinear differential operator as a Gaussian process. Automatic differentiation is used to compute the required partial derivatives from the leaf functions as inputs to the root. Our approach quantifies the reliability of the learned physics in terms of a posterior distribution over operators and propagates this uncertainty to solutions of novel initial-boundary value problem instances. Numerical experiments demonstrate the method on several nonlinear PDEs.

Deep neural networks are often ignorant about what they do not know and overconfident when they make uninformed predictions. Some recent approaches quantify classification uncertainty directly by training the model to output high uncertainty for the data samples close to class boundaries or from the outside of the training distribution. These approaches use an auxiliary data set during training to represent out-of-distribution samples. However, selection or creation of such an auxiliary data set is non-trivial, especially for high dimensional data such as images. In this work we develop a novel neural network model that is able to express both aleatoric and epistemic uncertainty to distinguish decision boundary and out-of-distribution regions of the feature space. To this end, variational autoencoders and generative adversarial networks are incorporated to automatically generate out-of-distribution exemplars for training. Through extensive analysis, we demonstrate that the proposed approach provides better estimates of uncertainty for in- and out-of-distribution samples, and adversarial examples on well-known data sets against state-of-the-art approaches including recent Bayesian approaches for neural networks and anomaly detection methods.

Mining and learning the shape variability of underlying population has benefited the applications including parametric shape modeling, 3D animation, and image segmentation. The current statistical shape modeling method works well on learning unstructured shape variations without obvious pose changes (relative rotations of the body parts). Studying the pose variations within a shape population involves segmenting the shapes into different articulated parts and learning the transformations of the segmented parts. This paper formulates the pose learning problem as mixtures of factor analyzers. The segmentation is obtained by components posterior probabilities and the rotations in pose variations are learned by the factor loading matrices. To guarantee that the factor loading matrices are composed by rotation matrices, constraints are imposed and the corresponding closed form optimal solution is derived. Based on the proposed method, the pose variations are automatically learned from the given shape populations. The method is applied in motion animation where new poses are generated by interpolating the existing poses in the training set. The obtained results are smooth and realistic.

Graph Convolutional Networks (GCNs) and their variants have received significant attention and achieved start-of-the-art performances on various recommendation tasks. However, many existing GCN models tend to perform recursive aggregations among all related nodes, which arises severe computational burden. Moreover, they favor multi-layer architectures in conjunction with complicated modeling techniques. Though effective, the excessive amount of model parameters largely hinder their applications in real-world recommender systems. To this end, in this paper, we propose the single-layer GCN model which is able to achieve superior performance along with remarkably less complexity compared with existing models. Our main contribution is three-fold. First, we propose a principled similarity metric named distribution-aware similarity (DA similarity), which can guide the neighbor sampling process and evaluate the quality of the input graph explicitly. We also prove that DA similarity has a positive correlation with the final performance, through both theoretical analysis and empirical simulations. Second, we propose a simplified GCN architecture which employs a single GCN layer to aggregate information from the neighbors filtered by DA similarity and then generates the node representations. Moreover, the aggregation step is a parameter-free operation, such that it can be done in a pre-processing manner to further reduce red the training and inference costs. Third, we conduct extensive experiments on four datasets. The results verify that the proposed model outperforms existing GCN models considerably and yields up to a few orders of magnitude speedup in training, in terms of the recommendation performance.

We establish a bridge between spectral clustering and Gromov-Wasserstein Learning (GWL), a recent optimal transport-based approach to graph partitioning. This connection both explains and improves upon the state-of-the-art performance of GWL. The Gromov-Wasserstein framework provides probabilistic correspondences between nodes of source and target graphs via a quadratic programming relaxation of the node matching problem. Our results utilize and connect the observations that the GW geometric structure remains valid for any rank-2 tensor, in particular the adjacency, distance, and various kernel matrices on graphs, and that the heat kernel outperforms the adjacency matrix in producing stable and informative node correspondences. Using the heat kernel in the GWL framework provides new multiscale graph comparisons without compromising theoretical guarantees, while immediately yielding improved empirical results. A key insight of the GWL framework toward graph partitioning was to compute GW correspondences from a source graph to a template graph with isolated, self-connected nodes. We show that when comparing against a two-node template graph using the heat kernel at the infinite time limit, the resulting partition agrees with the partition produced by the Fiedler vector. This in turn yields a new insight into the $k$-cut graph partitioning problem through the lens of optimal transport. Our experiments on a range of real-world networks achieve comparable results to, and in many cases outperform, the state-of-the-art achieved by GWL.

Learning distributions of graphs can be used for automatic drug discovery, molecular design, complex network analysis, and much more. We present an improved framework for learning generative models of graphs based on the idea of deep state machines. To learn state transition decisions we use a set of graph and node embedding techniques as memory of the state machine. Our analysis is based on learning the distribution of random graph generators for which we provide statistical tests to determine which properties can be learned and how well the original distribution of graphs is represented. We show that the design of the state machine favors specific distributions. Models of graphs of size up to 150 vertices are learned. Code and parameters are publicly available to reproduce our results.

Balancing utility and differential privacy by shuffling or \textit{BUDS} is an approach towards crowd-sourced, statistical databases, with strong privacy and utility balance using differential privacy theory. Here, a novel algorithm is proposed using one-hot encoding and iterative shuffling with the loss estimation and risk minimization techniques, to balance both the utility and privacy. In this work, after collecting one-hot encoded data from different sources and clients, a step of novel attribute shuffling technique using iterative shuffling (based on the query asked by the analyst) and loss estimation with an updation function and risk minimization produces a utility and privacy balanced differential private report. During empirical test of balanced utility and privacy, BUDS produces $\epsilon = 0.02$ which is a very promising result. Our algorithm maintains a privacy bound of $\epsilon = ln [t/((n_1 - 1)^S)]$ and loss bound of $c' \bigg|e^{ln[t/((n_1 - 1)^S)]} - 1\bigg|$.

We analyze a sequential decision making model in which decision makers (or, players) take their decisions based on their own private information as well as the actions of previous decision makers. Such decision making processes often lead to what is known as the \emph{information cascade} or \emph{herding} phenomenon. Specifically, a cascade develops when it seems rational for some players to abandon their own private information and imitate the actions of earlier players. The risk, however, is that if the initial decisions were wrong, then the whole cascade will be wrong. Nonetheless, information cascade are known to be fragile: there exists a sequence of \emph{revealing} probabilities $\{p_{\ell}\}_{\ell\geq1}$, such that if with probability $p_{\ell}$ player $\ell$ ignores the decisions of previous players, and rely on his private information only, then wrong cascades can be avoided. Previous related papers which study the fragility of information cascades always assume that the revealing probabilities are known to all players perfectly, which might be unrealistic in practice. Accordingly, in this paper we study a mismatch model where players believe that the revealing probabilities are $\{q_\ell\}_{\ell\in\mathbb{N}}$ when they truly are $\{p_\ell\}_{\ell\in\mathbb{N}}$, and study the effect of this mismatch on information cascades. We consider both adversarial and probabilistic sequential decision making models, and derive closed-form expressions for the optimal learning rates at which the error probability associated with a certain decision maker goes to zero. We prove several novel phase transitions in the behaviour of the asymptotic learning rate.