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This paper studies low-rank matrix completion in the presence of heavy-tailed and possibly asymmetric noise, where we aim to estimate an underlying low-rank matrix given a set of highly incomplete noisy entries. Though the matrix completion problem has attracted much attention in the past decade, there is still lack of theoretical understanding when the observations are contaminated by heavy-tailed noises. Prior theory falls short of explaining the empirical results and is unable to capture the optimal dependence of the estimation error on the noise level. In this paper, we adopt an adaptive Huber loss to accommodate heavy-tailed noise, which is robust against large and possibly asymmetric errors when the parameter in the loss function is carefully designed to balance the Huberization biases and robustness to outliers. Then, we propose an efficient nonconvex algorithm via a balanced low-rank Burer-Monteiro matrix factorization and gradient decent with robust spectral initialization. We prove that under merely bounded second moment condition on the error distributions, rather than the sub-Gaussian assumption, the Euclidean error of the iterates generated by the proposed algorithm decrease geometrically fast until achieving a minimax-optimal statistical estimation error, which has the same order as that in the sub-Gaussian case. The key technique behind this significant advancement is a powerful leave-one-out analysis framework. The theoretical results are corroborated by our simulation studies.

This paper studies inference in randomized controlled trials with multiple treatments, where treatment status is determined according to a "matched tuples" design. Here, by a matched tuples design, we mean an experimental design where units are sampled i.i.d. from the population of interest, grouped into "homogeneous" blocks with cardinality equal to the number of treatments, and finally, within each block, each treatment is assigned exactly once uniformly at random. We first study estimation and inference for matched tuples designs in the general setting where the parameter of interest is a vector of linear contrasts over the collection of average potential outcomes for each treatment. Parameters of this form include but are not limited to standard average treatment effects used to compare one treatment relative to another. We first establish conditions under which a sample analogue estimator is asymptotically normal and construct a consistent estimator of its corresponding asymptotic variance. Combining these results establishes the asymptotic validity of tests based on these estimators. In contrast, we show that a common testing procedure based on a linear regression with block fixed effects and the usual heteroskedasticity-robust variance estimator is invalid in the sense that the resulting test may have a limiting rejection probability under the null hypothesis strictly greater than the nominal level. We then apply our results to study the asymptotic properties of what we call "fully-blocked" $2^K$ factorial designs, which are simply matched tuples designs applied to a full factorial experiment. Leveraging our previous results, we establish that our estimator achieves a lower asymptotic variance under the fully-blocked design than that under any stratified factorial design. A simulation study and empirical application illustrate the practical relevance of our results.

A matchstick graph is a crossing-free unit-distance graph in the plane. Harborth (1981) proposed the problem of determining whether there exists a matchstick graph in which every vertex has degree exactly $5$. In 1982, Blokhuis gave a proof of non-existence. A shorter proof was found by Kurz and Pinchasi (2011) using a charging method. We combine their method with the isoperimetric inequality to show that there are $\Omega(\sqrt{n})$ vertices in a matchstick graph on $n$ vertices that are of degree at most $4$, which is asymptotically tight.

In linear regression we wish to estimate the optimum linear least squares predictor for a distribution over $d$-dimensional input points and real-valued responses, based on a small sample. Under standard random design analysis, where the sample is drawn i.i.d. from the input distribution, the least squares solution for that sample can be viewed as the natural estimator of the optimum. Unfortunately, this estimator almost always incurs an undesirable bias coming from the randomness of the input points, which is a significant bottleneck in model averaging. In this paper we show that it is possible to draw a non-i.i.d. sample of input points such that, regardless of the response model, the least squares solution is an unbiased estimator of the optimum. Moreover, this sample can be produced efficiently by augmenting a previously drawn i.i.d. sample with an additional set of $d$ points, drawn jointly according to a certain determinantal point process constructed from the input distribution rescaled by the squared volume spanned by the points. Motivated by this, we develop a theoretical framework for studying volume-rescaled sampling, and in the process prove a number of new matrix expectation identities. We use them to show that for any input distribution and $\epsilon>0$ there is a random design consisting of $O(d\log d+ d/\epsilon)$ points from which an unbiased estimator can be constructed whose expected square loss over the entire distribution is bounded by $1+\epsilon$ times the loss of the optimum. We provide efficient algorithms for generating such unbiased estimators in a number of practical settings and support our claims experimentally.

The study of representations is of fundamental importance to any form of communication, and our ability to exploit them effectively is paramount. This article presents a novel theory -- Representational Systems Theory -- that is designed to abstractly encode a wide variety of representations from three core perspectives: syntax, entailment, and their properties. By introducing the concept of a construction space, we are able to encode each of these core components under a single, unifying paradigm. Using our Representational Systems Theory, it becomes possible to structurally transform representations in one system into representations in another. An intrinsic facet of our structural transformation technique is representation selection based on properties that representations possess, such as their relative cognitive effectiveness or structural complexity. A major theoretical barrier to providing general structural transformation techniques is a lack of terminating algorithms. Representational Systems Theory permits the derivation of partial transformations when no terminating algorithm can produce a full transformation. Since Representational Systems Theory provides a universal approach to encoding representational systems, a further key barrier is eliminated: the need to devise system-specific structural transformation algorithms, that are necessary when different systems adopt different formalisation approaches. Consequently, Representational Systems Theory is the first general framework that provides a unified approach to encoding representations, supports representation selection via structural transformations, and has the potential for widespread practical application.

We revisit the classic regular expression matching problem, that is, given a regular expression $R$ and a string $Q$, decide if $Q$ matches any of the strings specified by $R$. A standard textbook solution [Thompson, CACM 1968] solves this problem in $O(nm)$ time, where $n$ is the length of $Q$ and $m$ is the number of characters in $R$. More recently, several results that improve this bound by polylogarithmic factor have appeared. All of these solutions are essentially based on constructing and simulation a non-deterministic finite automaton. On the other hand, assuming the strong exponential time hypotheses we cannot solve regular expression $O((nm)^{1-\epsilon})$ [Backurs and Indyk, FOCS 2016]. Hence, a natural question is if we can design algorithms that can take advantage of other parameters of the problem to obtain more fine-grained bounds. We present the first algorithm for regular expression matching that can take advantage of sparsity of the automaton simulation. More precisely, we define the \emph{density}, $\Delta$, of the instance to be the total number of states in a simulation of a natural automaton for $R$. The density is always at most $nm+1$ but may be significantly smaller for many typical scenarios, e.g., when a string only matches a small part of the regular expression. Our main result is a new algorithm that solves the problem in $$O\left(\Delta \log \log \frac{nm}{\Delta} + n + m\right)$$ time. This result essentially replaces $nm$ with $\Delta$ in the complexity of regular expression matching. Prior to this work no non-trivial bound in terms of $\Delta$ was known. The key technical contribution is a new linear space representation of the classic position automaton that supports fast state-set transition computation in near-linear time in the size of the input and output state sets.

Graph Neural Networks (GNNs) have been shown to be effective models for different predictive tasks on graph-structured data. Recent work on their expressive power has focused on isomorphism tasks and countable feature spaces. We extend this theoretical framework to include continuous features - which occur regularly in real-world input domains and within the hidden layers of GNNs - and we demonstrate the requirement for multiple aggregation functions in this context. Accordingly, we propose Principal Neighbourhood Aggregation (PNA), a novel architecture combining multiple aggregators with degree-scalers (which generalize the sum aggregator). Finally, we compare the capacity of different models to capture and exploit the graph structure via a novel benchmark containing multiple tasks taken from classical graph theory, alongside existing benchmarks from real-world domains, all of which demonstrate the strength of our model. With this work, we hope to steer some of the GNN research towards new aggregation methods which we believe are essential in the search for powerful and robust models.

Deep learning has become the most widely used approach for cardiac image segmentation in recent years. In this paper, we provide a review of over 100 cardiac image segmentation papers using deep learning, which covers common imaging modalities including magnetic resonance imaging (MRI), computed tomography (CT), and ultrasound (US) and major anatomical structures of interest (ventricles, atria and vessels). In addition, a summary of publicly available cardiac image datasets and code repositories are included to provide a base for encouraging reproducible research. Finally, we discuss the challenges and limitations with current deep learning-based approaches (scarcity of labels, model generalizability across different domains, interpretability) and suggest potential directions for future research.

Medical image segmentation requires consensus ground truth segmentations to be derived from multiple expert annotations. A novel approach is proposed that obtains consensus segmentations from experts using graph cuts (GC) and semi supervised learning (SSL). Popular approaches use iterative Expectation Maximization (EM) to estimate the final annotation and quantify annotator's performance. Such techniques pose the risk of getting trapped in local minima. We propose a self consistency (SC) score to quantify annotator consistency using low level image features. SSL is used to predict missing annotations by considering global features and local image consistency. The SC score also serves as the penalty cost in a second order Markov random field (MRF) cost function optimized using graph cuts to derive the final consensus label. Graph cut obtains a global maximum without an iterative procedure. Experimental results on synthetic images, real data of Crohn's disease patients and retinal images show our final segmentation to be accurate and more consistent than competing methods.

Learning from a few examples remains a key challenge in machine learning. Despite recent advances in important domains such as vision and language, the standard supervised deep learning paradigm does not offer a satisfactory solution for learning new concepts rapidly from little data. In this work, we employ ideas from metric learning based on deep neural features and from recent advances that augment neural networks with external memories. Our framework learns a network that maps a small labelled support set and an unlabelled example to its label, obviating the need for fine-tuning to adapt to new class types. We then define one-shot learning problems on vision (using Omniglot, ImageNet) and language tasks. Our algorithm improves one-shot accuracy on ImageNet from 87.6% to 93.2% and from 88.0% to 93.8% on Omniglot compared to competing approaches. We also demonstrate the usefulness of the same model on language modeling by introducing a one-shot task on the Penn Treebank.