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We study a functional linear regression model that deals with functional responses and allows for both functional covariates and high-dimensional vector covariates. The proposed model is flexible and nests several functional regression models in the literature as special cases. Based on the theory of reproducing kernel Hilbert spaces (RKHS), we propose a penalized least squares estimator that can accommodate functional variables observed on discrete sample points. Besides a conventional smoothness penalty, a group Lasso-type penalty is further imposed to induce sparsity in the high-dimensional vector predictors. We derive finite sample theoretical guarantees and show that the excess prediction risk of our estimator is minimax optimal. Furthermore, our analysis reveals an interesting phase transition phenomenon that the optimal excess risk is determined jointly by the smoothness and the sparsity of the functional regression coefficients. A novel efficient optimization algorithm based on iterative coordinate descent is devised to handle the smoothness and group penalties simultaneously. Simulation studies and real data applications illustrate the promising performance of the proposed approach compared to the state-of-the-art methods in the literature.

Causal inference using observational text data is becoming increasingly popular in many research areas. This paper presents the Bayesian Topic Regression (BTR) model that uses both text and numerical information to model an outcome variable. It allows estimation of both discrete and continuous treatment effects. Furthermore, it allows for the inclusion of additional numerical confounding factors next to text data. To this end, we combine a supervised Bayesian topic model with a Bayesian regression framework and perform supervised representation learning for the text features jointly with the regression parameter training, respecting the Frisch-Waugh-Lovell theorem. Our paper makes two main contributions. First, we provide a regression framework that allows causal inference in settings when both text and numerical confounders are of relevance. We show with synthetic and semi-synthetic datasets that our joint approach recovers ground truth with lower bias than any benchmark model, when text and numerical features are correlated. Second, experiments on two real-world datasets demonstrate that a joint and supervised learning strategy also yields superior prediction results compared to strategies that estimate regression weights for text and non-text features separately, being even competitive with more complex deep neural networks.

Partial orders and directed acyclic graphs are commonly recurring data structures that arise naturally in numerous domains and applications and are used to represent ordered relations between entities in the domains. Examples are task dependencies in a project plan, transaction order in distributed ledgers and execution sequences of tasks in computer programs, just to mention a few. We study the problem of order preserving hierarchical clustering of this kind of ordered data. That is, if we have $a < b$ in the original data and denote their respective clusters by $[a]$ and $[b]$, then we shall have $[a] < [b]$ in the produced clustering. The clustering is similarity based and uses standard linkage functions, such as single- and complete linkage, and is an extension of classical hierarchical clustering. To achieve this, we define the output from running classical hierarchical clustering on strictly ordered data to be partial dendrograms; sub-trees of classical dendrograms with several connected components. We then construct an embedding of partial dendrograms over a set into the family of ultrametrics over the same set. An optimal hierarchical clustering is defined as the partial dendrogram corresponding to the ultrametric closest to the original dissimilarity measure, measured in the p-norm. Thus, the method is a combination of classical hierarchical clustering and ultrametric fitting. A reference implementation is employed for experiments on both synthetic random data and real world data from a database of machine parts. When compared to existing methods, the experiments show that our method excels both in cluster quality and order preservation.

We present an objective function for similarity based hierarchical clustering of partially ordered data that preserves the partial order in the sense that if $x \leq y$, and if $[x]$ and $[y]$ are the respective clusters of $x$ and $y$, then there is an order relation $\leq'$ on the clusters for which $[x] \leq' |y]$. The model distinguishes itself from existing methods and models for clustering of ordered data in that the order relation and the similarity are combined to obtain an optimal hierarchical clustering seeking to satisfy both, and that the order relation is equipped with a pairwise level of comparability in the range $[0,1]$. In particular, if the similarity and the order relation are not aligned, then order preservation may have to yield in favor of clustering. Finding an optimal solution is NP-hard, so we provide a polynomial time approximation algorithm, with a relative performance guarantee of $O(\log^{3/2}n)$, based on successive applications of directed sparsest cut. The model is an extension of the Dasgupta cost function for divisive hierarchical clustering.

We consider a sparse deep ReLU network (SDRN) estimator obtained from empirical risk minimization with a Lipschitz loss function in the presence of a large number of features. Our framework can be applied to a variety of regression and classification problems. The unknown target function to estimate is assumed to be in a Korobov space. Functions in this space only need to satisfy a smoothness condition rather than having a compositional structure. We develop non-asymptotic excess risk bounds for our SDRN estimator. We further derive that the SDRN estimator can achieve the same minimax rate of estimation (up to logarithmic factors) as one-dimensional nonparametric regression when the dimension of the features is fixed, and the estimator has a suboptimal rate when the dimension grows with the sample size. We show that the depth and the total number of nodes and weights of the ReLU network need to grow as the sample size increases to ensure a good performance, and also investigate how fast they should increase with the sample size. These results provide an important theoretical guidance and basis for empirical studies by deep neural networks.

Clustering is one of the most fundamental and wide-spread techniques in exploratory data analysis. Yet, the basic approach to clustering has not really changed: a practitioner hand-picks a task-specific clustering loss to optimize and fit the given data to reveal the underlying cluster structure. Some types of losses---such as k-means, or its non-linear version: kernelized k-means (centroid based), and DBSCAN (density based)---are popular choices due to their good empirical performance on a range of applications. Although every so often the clustering output using these standard losses fails to reveal the underlying structure, and the practitioner has to custom-design their own variation. In this work we take an intrinsically different approach to clustering: rather than fitting a dataset to a specific clustering loss, we train a recurrent model that learns how to cluster. The model uses as training pairs examples of datasets (as input) and its corresponding cluster identities (as output). By providing multiple types of training datasets as inputs, our model has the ability to generalize well on unseen datasets (new clustering tasks). Our experiments reveal that by training on simple synthetically generated datasets or on existing real datasets, we can achieve better clustering performance on unseen real-world datasets when compared with standard benchmark clustering techniques. Our meta clustering model works well even for small datasets where the usual deep learning models tend to perform worse.

Meta learning is a promising solution to few-shot learning problems. However, existing meta learning methods are restricted to the scenarios where training and application tasks share the same out-put structure. To obtain a meta model applicable to the tasks with new structures, it is required to collect new training data and repeat the time-consuming meta training procedure. This makes them inefficient or even inapplicable in learning to solve heterogeneous few-shot learning tasks. We thus develop a novel and principled HierarchicalMeta Learning (HML) method. Different from existing methods that only focus on optimizing the adaptability of a meta model to similar tasks, HML also explicitly optimizes its generalizability across heterogeneous tasks. To this end, HML first factorizes a set of similar training tasks into heterogeneous ones and trains the meta model over them at two levels to maximize adaptation and generalization performance respectively. The resultant model can then directly generalize to new tasks. Extensive experiments on few-shot classification and regression problems clearly demonstrate the superiority of HML over fine-tuning and state-of-the-art meta learning approaches in terms of generalization across heterogeneous tasks.

In this paper, we present an accurate and scalable approach to the face clustering task. We aim at grouping a set of faces by their potential identities. We formulate this task as a link prediction problem: a link exists between two faces if they are of the same identity. The key idea is that we find the local context in the feature space around an instance (face) contains rich information about the linkage relationship between this instance and its neighbors. By constructing sub-graphs around each instance as input data, which depict the local context, we utilize the graph convolution network (GCN) to perform reasoning and infer the likelihood of linkage between pairs in the sub-graphs. Experiments show that our method is more robust to the complex distribution of faces than conventional methods, yielding favorably comparable results to state-of-the-art methods on standard face clustering benchmarks, and is scalable to large datasets. Furthermore, we show that the proposed method does not need the number of clusters as prior, is aware of noises and outliers, and can be extended to a multi-view version for more accurate clustering accuracy.

Because of continuous advances in mathematical programing, Mix Integer Optimization has become a competitive vis-a-vis popular regularization method for selecting features in regression problems. The approach exhibits unquestionable foundational appeal and versatility, but also poses important challenges. We tackle these challenges, reducing computational burden when tuning the sparsity bound (a parameter which is critical for effectiveness) and improving performance in the presence of feature collinearity and of signals that vary in nature and strength. Importantly, we render the approach efficient and effective in applications of realistic size and complexity - without resorting to relaxations or heuristics in the optimization, or abandoning rigorous cross-validation tuning. Computational viability and improved performance in subtler scenarios is achieved with a multi-pronged blueprint, leveraging characteristics of the Mixed Integer Programming framework and by means of whitening, a data pre-processing step.

Recently, graph neural networks (GNNs) have revolutionized the field of graph representation learning through effectively learned node embeddings, and achieved state-of-the-art results in tasks such as node classification and link prediction. However, current GNN methods are inherently flat and do not learn hierarchical representations of graphs---a limitation that is especially problematic for the task of graph classification, where the goal is to predict the label associated with an entire graph. Here we propose DiffPool, a differentiable graph pooling module that can generate hierarchical representations of graphs and can be combined with various graph neural network architectures in an end-to-end fashion. DiffPool learns a differentiable soft cluster assignment for nodes at each layer of a deep GNN, mapping nodes to a set of clusters, which then form the coarsened input for the next GNN layer. Our experimental results show that combining existing GNN methods with DiffPool yields an average improvement of 5-10% accuracy on graph classification benchmarks, compared to all existing pooling approaches, achieving a new state-of-the-art on four out of five benchmark data sets.