In this work, we revisit fast dimension reduction approaches, as with random projections and random sampling. Our goal is to summarize the data to decrease computational costs and memory footprint of subsequent analysis. Such dimension reduction can be very efficient when the signals of interest have a strong structure, such as with images. We focus on this setting and investigate feature clustering schemes for data reductions that capture this structure. An impediment to fast dimension reduction is that good clustering comes with large algorithmic costs. We address it by contributing a linear-time agglomerative clustering scheme, Recursive Nearest Agglomeration (ReNA). Unlike existing fast agglomerative schemes, it avoids the creation of giant clusters. We empirically validate that it approximates the data as well as traditional variance-minimizing clustering schemes that have a quadratic complexity. In addition, we analyze signal approximation with feature clustering and show that it can remove noise, improving subsequent analysis steps. As a consequence, data reduction by clustering features with ReNA yields very fast and accurate models, enabling to process large datasets on budget. Our theoretical analysis is backed by extensive experiments on publicly-available data that illustrate the computation efficiency and the denoising properties of the resulting dimension reduction scheme.

Low-rank signal modeling has been widely leveraged to capture non-local correlation in image processing applications. We propose a new method that employs low-rank tensor factor analysis for tensors generated by grouped image patches. The low-rank tensors are fed into the alternative direction multiplier method (ADMM) to further improve image reconstruction. The motivating application is compressive sensing (CS), and a deep convolutional architecture is adopted to approximate the expensive matrix inversion in CS applications. An iterative algorithm based on this low-rank tensor factorization strategy, called NLR-TFA, is presented in detail. Experimental results on noiseless and noisy CS measurements demonstrate the superiority of the proposed approach, especially at low CS sampling rates.

We consider a high dimensional linear regression problem where the goal is to efficiently recover an unknown vector $\beta^*$ from $n$ noisy linear observations $Y=X\beta^*+W \in \mathbb{R}^n$, for known $X \in \mathbb{R}^{n \times p}$ and unknown $W \in \mathbb{R}^n$. Unlike most of the literature on this model we make no sparsity assumption on $\beta^*$. Instead we adopt a regularization based on assuming that the underlying vectors $\beta^*$ have rational entries with the same denominator $Q \in \mathbb{Z}_{>0}$. We call this $Q$-rationality assumption. We propose a new polynomial-time algorithm for this task which is based on the seminal Lenstra-Lenstra-Lovasz (LLL) lattice basis reduction algorithm. We establish that under the $Q$-rationality assumption, our algorithm recovers exactly the vector $\beta^*$ for a large class of distributions for the iid entries of $X$ and non-zero noise $W$. We prove that it is successful under small noise, even when the learner has access to only one observation ($n=1$). Furthermore, we prove that in the case of the Gaussian white noise for $W$, $n=o\left(p/\log p\right)$ and $Q$ sufficiently large, our algorithm tolerates a nearly optimal information-theoretic level of the noise.

It is common in modern prediction problems for many predictor variables to be counts of rarely occurring events. This leads to design matrices in which many columns are highly sparse. The challenge posed by such "rare features" has received little attention despite its prevalence in diverse areas, ranging from natural language processing (e.g., rare words) to biology (e.g., rare species). We show, both theoretically and empirically, that not explicitly accounting for the rareness of features can greatly reduce the effectiveness of an analysis. We next propose a framework for aggregating rare features into denser features in a flexible manner that creates better predictors of the response. Our strategy leverages side information in the form of a tree that encodes feature similarity. We apply our method to data from TripAdvisor, in which we predict the numerical rating of a hotel based on the text of the associated review. Our method achieves high accuracy by making effective use of rare words; by contrast, the lasso is unable to identify highly predictive words if they are too rare. A companion R package, called rare, implements our new estimator, using the alternating direction method of multipliers.

The positive-unlabeled (PU) classification is a common scenario in real-world applications such as healthcare, text classification, and bioinformatics, in which we only observe a few samples labeled as "positive" together with a large volume of "unlabeled" samples that may contain both positive and negative samples. Building robust classifier for the PU problem is very challenging, especially for complex data where the negative samples overwhelm and mislabeled samples or corrupted features exist. To address these three issues, we propose a robust learning framework that unifies AUC maximization (a robust metric for biased labels), outlier detection (for excluding wrong labels), and feature selection (for excluding corrupted features). The generalization error bounds are provided for the proposed model that give valuable insight into the theoretical performance of the method and lead to useful practical guidance, e.g., to train a model, we find that the included unlabeled samples are sufficient as long as the sample size is comparable to the number of positive samples in the training process. Empirical comparisons and two real-world applications on surgical site infection (SSI) and EEG seizure detection are also conducted to show the effectiveness of the proposed model.

Lactate threshold is considered an essential parameter when assessing performance of elite and recreational runners and prescribing training intensities in endurance sports. However, the measurement of blood lactate concentration requires expensive equipment and the extraction of blood samples, which are inconvenient for frequent monitoring. Furthermore, most recreational runners do not have access to routine assessment of their physical fitness by the aforementioned equipment so they are not able to calculate the lactate threshold without resorting to an expensive and specialized centre. Therefore, the main objective of this study is to create an intelligent system capable of estimating the lactate threshold of recreational athletes participating in endurance running sports. The solution here proposed is based on a machine learning system which models the lactate evolution using recurrent neural networks and includes the proposal of standardization of the temporal axis as well as a modification of the stratified sampling method. The results show that the proposed system accurately estimates the lactate threshold of 89.52% of the athletes and its correlation with the experimentally measured lactate threshold is very high (R=0,89). Moreover, its behaviour with the test dataset is as good as with the training set, meaning that the generalization power of the model is high. Therefore, in this study a machine learning based system is proposed as alternative to the traditional invasive lactate threshold measurement tests for recreational runners.

We provide adaptive inference methods for linear functionals of sparse linear approximations to the conditional expectation function. Examples of such functionals include average derivatives, policy effects, average treatment effects, and many others. The construction relies on building Neyman-orthogonal equations that are approximately invariant to perturbations of the nuisance parameters, including the Riesz representer for the linear functionals. We use L1-regularized methods to learn approximations to the regression function and the Riesz representer, and construct the estimator for the linear functionals as the solution to the orthogonal estimating equations. We establish that under weak assumptions the estimator concentrates in a 1/root n neighborhood of the target with deviations controlled by the normal laws, and the estimator attains the semi-parametric efficiency bound in many cases. In particular, either the approximation to the regression function or the approximation to the Riesz representer can be "dense" as long as one of them is sufficiently "sparse". Our main results are non-asymptotic and imply asymptotic uniform validity over large classes of models.

A Gaussian classifier or nearest mean classifier just uses the mean of each class to make classifications. It therefore assumes classes are spherical Gaussians. The softmax on a CNN by contrast is more like multinomial logistic regression, which learns linear boundaries like a nearest mean classifier, but not the same boundary. I want to force the CNN to learn features in the last layer such that it has a prototype (mean) for each category, and uses a nearest mean rule to make classifications.

Community detection is a fundamental unsupervised learning problem for unlabeled networks which has a broad range of applications. Many community detection algorithms assume that the number of clusters $r$ is known apriori. In this paper, we propose an approach based on semi-definite relaxations, which does not require prior knowledge of model parameters like many existing convex relaxation methods and recovers the number of clusters and the clustering matrix exactly under a broad parameter regime, with probability tending to one. On a variety of simulated and real data experiments, we show that the proposed method often outperforms state-of-the-art techniques for estimating the number of clusters.

I've been swamped with a bit of a travel binge and am hopelessly behind on blogging. After my last post on nominal control, I received an email from Pavel Christof pointing out that if we switch from stochastic gradient descent to Adam, policy gradient works much better. Indeed, I implemented this myself, and he's totally right. Let's revisit the last post with a revised Jupyter notebook. First, I coded up Adam in pure python to avoid introducing any deep learning package dependencies (it's only 4 lines of python, after all).