Linear regression studies the problem of estimating a model parameter $\beta^* \in \mathbb{R}^p$, from $n$ observations $\{(y_i,\mathbf{x}_i)\}_{i=1}^n$ from linear model $y_i = \langle \mathbf{x}_i,\beta^* \rangle + \epsilon_i$. We consider a significant generalization in which the relationship between $\langle \mathbf{x}_i,\beta^* \rangle$ and $y_i$ is noisy, quantized to a single bit, potentially nonlinear, noninvertible, as well as unknown. This model is known as the single-index model in statistics, and, among other things, it represents a significant generalization of one-bit compressed sensing. We propose a novel spectral-based estimation procedure and show that we can recover $\beta^*$ in settings (i.e., classes of link function $f$) where previous algorithms fail. In general, our algorithm requires only very mild restrictions on the (unknown) functional relationship between $y_i$ and $\langle \mathbf{x}_i,\beta^* \rangle$. We also consider the high dimensional setting where $\beta^*$ is sparse ,and introduce a two-stage nonconvex framework that addresses estimation challenges in high dimensional regimes where $p \gg n$. For a broad class of link functions between $\langle \mathbf{x}_i,\beta^* \rangle$ and $y_i$, we establish minimax lower bounds that demonstrate the optimality of our estimators in both the classical and high dimensional regimes.

A central goal of neuroscience is to understand how activity in the nervous system is related to features of the external world, or to features of the nervous system itself. A common approach is to model neural responses as a weighted combination of external features, or vice versa. The structure of the model weights can provide insight into neural representations. Often, neural input-output relationships are sparse, with only a few inputs contributing to the output. In part to account for such sparsity, structured regularizers are incorporated into model fitting optimization. However, by imposing priors, structured regularizers can make it difficult to interpret learned model parameters. Here, we investigate a simple, minimally structured model estimation method for accurate, unbiased estimation of sparse models based on Bootstrapped Adaptive Threshold Selection followed by ordinary least-squares refitting (BoATS). Through extensive numerical investigations, we show that this method often performs favorably compared to L1 and L2 regularizers. In particular, for a variety of model distributions and noise levels, BoATS more accurately recovers the parameters of sparse models, leading to more parsimonious explanations of outputs. Finally, we apply this method to the task of decoding human speech production from ECoG recordings.

We provide scientific foundations for athletic performance prediction on an individual level, exposing the phenomenology of individual athletic running performance in the form of a low-rank model dominated by an individual power law. We present, evaluate, and compare a selection of methods for prediction of individual running performance, including our own, \emph{local matrix completion} (LMC), which we show to perform best. We also show that many documented phenomena in quantitative sports science, such as the form of scoring tables, the success of existing prediction methods including Riegel's formula, the Purdy points scheme, the power law for world records performances and the broken power law for world record speeds may be explained on the basis of our findings in a unified way.

For analysis of a high-dimensional dataset, a common approach is to test a null hypothesis of statistical independence on all variable pairs using a non-parametric measure of dependence. However, because this approach attempts to identify any non-trivial relationship no matter how weak, it often identifies too many relationships to be useful. What is needed is a way of identifying a smaller set of relationships that merit detailed further analysis. Here we formally present and characterize equitability, a property of measures of dependence that aims to overcome this challenge. Notionally, an equitable statistic is a statistic that, given some measure of noise, assigns similar scores to equally noisy relationships of different types [Reshef et al. 2011]. We begin by formalizing this idea via a new object called the interpretable interval, which functions as an interval estimate of the amount of noise in a relationship of unknown type. We define an equitable statistic as one with small interpretable intervals. We then draw on the equivalence of interval estimation and hypothesis testing to show that under moderate assumptions an equitable statistic is one that yields well powered tests for distinguishing not only between trivial and non-trivial relationships of all kinds but also between non-trivial relationships of different strengths. This means that equitability allows us to specify a threshold relationship strength $x_0$ and to search for relationships of all kinds with strength greater than $x_0$. Thus, equitability can be thought of as a strengthening of power against independence that enables fruitful analysis of data sets with a small number of strong, interesting relationships and a large number of weaker ones. We conclude with a demonstration of how our two equivalent characterizations of equitability can be used to evaluate the equitability of a statistic in practice.

Exchangeable graphs arise via a sampling procedure from measurable functions known as graphons. A natural estimation problem is how well we can recover a graphon given a single graph sampled from it. One general framework for estimating a graphon uses step-functions obtained by partitioning the nodes of the graph according to some clustering algorithm. We propose an iterative step-function estimator (ISFE) that, given an initial partition, iteratively clusters nodes based on their edge densities with respect to the previous iteration's partition. We analyze ISFE and demonstrate its performance in comparison with other graphon estimation techniques.

Markov chain Monte Carlo methods are often deemed too computationally intensive to be of any practical use for big data applications, and in particular for inference on datasets containing a large number $n$ of individual data points, also known as tall datasets. In scenarios where data are assumed independent, various approaches to scale up the Metropolis-Hastings algorithm in a Bayesian inference context have been recently proposed in machine learning and computational statistics. These approaches can be grouped into two categories: divide-and-conquer approaches and, subsampling-based algorithms. The aims of this article are as follows. First, we present a comprehensive review of the existing literature, commenting on the underlying assumptions and theoretical guarantees of each method. Second, by leveraging our understanding of these limitations, we propose an original subsampling-based approach which samples from a distribution provably close to the posterior distribution of interest, yet can require less than $O(n)$ data point likelihood evaluations at each iteration for certain statistical models in favourable scenarios. Finally, we have only been able so far to propose subsampling-based methods which display good performance in scenarios where the Bernstein-von Mises approximation of the target posterior distribution is excellent. It remains an open challenge to develop such methods in scenarios where the Bernstein-von Mises approximation is poor.

We describe a new instance-based learning algorithm called the Boundary Forest (BF) algorithm, that can be used for supervised and unsupervised learning. The algorithm builds a forest of trees whose nodes store previously seen examples. It can be shown data points one at a time and updates itself incrementally, hence it is naturally online. Few instance-based algorithms have this property while being simultaneously fast, which the BF is. This is crucial for applications where one needs to respond to input data in real time. The number of children of each node is not set beforehand but obtained from the training procedure, which makes the algorithm very flexible with regards to what data manifolds it can learn. We test its generalization performance and speed on a range of benchmark datasets and detail in which settings it outperforms the state of the art. Empirically we find that training time scales as O(DNlog(N)) and testing as O(Dlog(N)), where D is the dimensionality and N the amount of data,

Community detection is one of the fundamental problems of network analysis, for which a number of methods have been proposed. Most model-based or criteria-based methods have to solve an optimization problem over a discrete set of labels to find communities, which is computationally infeasible. Some fast spectral algorithms have been proposed for specific methods or models, but only on a case-by-case basis. Here we propose a general approach for maximizing a function of a network adjacency matrix over discrete labels by projecting the set of labels onto a subspace approximating the leading eigenvectors of the expected adjacency matrix. This projection onto a low-dimensional space makes the feasible set of labels much smaller and the optimization problem much easier. We prove a general result about this method and show how to apply it to several previously proposed community detection criteria, establishing its consistency for label estimation in each case and demonstrating the fundamental connection between spectral properties of the network and various model-based approaches to community detection. Simulations and applications to real-world data are included to demonstrate our method performs well for multiple problems over a wide range of parameters.

Tucker decomposition is the cornerstone of modern machine learning on tensorial data analysis, which have attracted considerable attention for multiway feature extraction, compressive sensing, and tensor completion. The most challenging problem is related to determination of model complexity (i.e., multilinear rank), especially when noise and missing data are present. In addition, existing methods cannot take into account uncertainty information of latent factors, resulting in low generalization performance. To address these issues, we present a class of probabilistic generative Tucker models for tensor decomposition and completion with structural sparsity over multilinear latent space. To exploit structural sparse modeling, we introduce two group sparsity inducing priors by hierarchial representation of Laplace and Student-t distributions, which facilitates fully posterior inference. For model learning, we derived variational Bayesian inferences over all model (hyper)parameters, and developed efficient and scalable algorithms based on multilinear operations. Our methods can automatically adapt model complexity and infer an optimal multilinear rank by the principle of maximum lower bound of model evidence. Experimental results and comparisons on synthetic, chemometrics and neuroimaging data demonstrate remarkable performance of our models for recovering ground-truth of multilinear rank and missing entries.

The Gaussian process latent variable model (GP-LVM) is a popular approach to non-linear probabilistic dimensionality reduction. One design choice for the model is the number of latent variables. We present a spike and slab prior for the GP-LVM and propose an efficient variational inference procedure that gives a lower bound of the log marginal likelihood. The new model provides a more principled approach for selecting latent dimensions than the standard way of thresholding the length-scale parameters. The effectiveness of our approach is demonstrated through experiments on real and simulated data. Further, we extend multi-view Gaussian processes that rely on sharing latent dimensions (known as manifold relevance determination) with spike and slab priors. This allows a more principled approach for selecting a subset of the latent space for each view of data. The extended model outperforms the previous state-of-the-art when applied to a cross-modal multimedia retrieval task.