We study --both in theory and practice-- the use of momentum motions in classic iterative hard thresholding (IHT) methods. By simply modifying plain IHT, we investigate its convergence behavior on convex optimization criteria with non-convex constraints, under standard assumptions. In diverse scenaria, we observe that acceleration in IHT leads to significant improvements, compared to state of the art projected gradient descent and Frank-Wolfe variants. As a byproduct of our inspection, we study the impact of selecting the momentum parameter: similar to convex settings, two modes of behavior are observed --"rippling" and linear-- depending on the level of momentum.

Deep neural networks are demonstrating excellent performance on several classical vision problems. However, these networks are vulnerable to adversarial examples, minutely modified images that induce arbitrary attacker-chosen output from the network. We propose a mechanism to protect against these adversarial inputs based on a generative model of the data. We introduce a pre-processing step that projects on the range of a generative model using gradient descent before feeding an input into a classifier. We show that this step provides the classifier with robustness against first-order, substitute model, and combined adversarial attacks. Using a min-max formulation, we show that there may exist adversarial examples even in the range of the generator, natural-looking images extremely close to the decision boundary for which the classifier has unjustifiedly high confidence. We show that adversarial training on the generative manifold can be used to make a classifier that is robust to these attacks. Finally, we show how our method can be applied even without a pre-trained generative model using a recent method called the deep image prior. We evaluate our method on MNIST, CelebA and Imagenet and show robustness against the current state of the art attacks.

Unsupervised clustering is one of the most fundamental challenges in machine learning. A popular hypothesis is that data are generated from a union of low-dimensional nonlinear manifolds; thus an approach to clustering is identifying and separating these manifolds. In this paper, we present a novel approach to solve this problem by using a mixture of autoencoders. Our model consists of two parts: 1) a collection of autoencoders where each autoencoder learns the underlying manifold of a group of similar objects, and 2) a mixture assignment neural network, which takes the concatenated latent vectors from the autoencoders as input and infers the distribution over clusters. By jointly optimizing the two parts, we simultaneously assign data to clusters and learn the underlying manifolds of each cluster.

A central question in statistical learning is to design algorithms that not only perform well on training data, but also generalize to new and unseen data. In this paper, we tackle this question by formulating a distributionally robust stochastic optimization (DRSO) problem, which seeks a solution that minimizes the worst-case expected loss over a family of distributions that are close to the empirical distribution in Wasserstein distances. We establish a connection between such Wasserstein DRSO and regularization. More precisely, we identify a broad class of loss functions, for which the Wasserstein DRSO is asymptotically equivalent to a regularization problem with a gradient-norm penalty. Such relation provides new interpretations for problems involving regularization, including a great number of statistical learning problems and discrete choice models (e.g. multinomial logit). The connection suggests a principled way to regularize high-dimensional, non-convex problems. This is demonstrated through the training of Wasserstein generative adversarial networks in deep learning.

Kernel PCA is a widely used nonlinear dimension reduction technique in machine learning, but storing the kernel matrix is notoriously challenging when the sample size is large. Inspired by Yi et al. [2016], where the idea of partial matrix sampling followed by nonconvex optimization is proposed for matrix completion and robust PCA, we apply a similar approach to memory-efficient Kernel PCA. In theory, with no assumptions on the kernel matrix in terms of eigenvalues or eigenvectors, we established a model-free theory for the low-rank approximation based on any local minimum of the proposed objective function. As interesting byproducts, when the underlying positive semidefinite matrix is assumed to be low-rank and highly structured, corollaries of our main theorem improve the state-of-the-art results of Ge et al. [2016, 2017] for nonconvex matrix completion with no spurious local minima. Numerical experiments also show that our approach is competitive in terms of approximation accuracy compared to the well-known Nystr\"{o}m algorithm for Kernel PCA.

Kernel-based methods enjoy powerful generalization capabilities in handling a variety of learning tasks. When such methods are provided with sufficient training data, broadly-applicable classes of nonlinear functions can be approximated with desired accuracy. Nevertheless, inherent to the nonparametric nature of kernel-based estimators are computational and memory requirements that become prohibitive with large-scale datasets. In response to this formidable challenge, the present work puts forward a low-rank, kernel-based, feature extraction approach that is particularly tailored for online operation, where data streams need not be stored in memory. A novel generative model is introduced to approximate high-dimensional (possibly infinite) features via a low-rank nonlinear subspace, the learning of which leads to a direct kernel function approximation. Offline and online solvers are developed for the subspace learning task, along with affordable versions, in which the number of stored data vectors is confined to a predefined budget. Analytical results provide performance bounds on how well the kernel matrix as well as kernel-based classification and regression tasks can be approximated by leveraging budgeted online subspace learning and feature extraction schemes. Tests on synthetic and real datasets demonstrate and benchmark the efficiency of the proposed method when linear classification and regression is applied to the extracted features.

We provide a novel notion of what it means to be interpretable, looking past the usual association with human understanding. Our key insight is that interpretability is not an absolute concept and so we define it relative to a target model, which may or may not be a human. We define a framework that allows for comparing interpretable procedures by linking it to important practical aspects such as accuracy and robustness. We characterize many of the current state-of-the-art interpretable methods in our framework portraying its general applicability. Finally, principled interpretable strategies are proposed and empirically evaluated on synthetic data, as well as on the largest public olfaction dataset that was made recently available \cite{olfs}. We also experiment on MNIST with a simple target model and different oracle models of varying complexity. This leads to the insight that the improvement in the target model is not only a function of the oracle models performance, but also its relative complexity with respect to the target model.

Python is a dynamic programming language used in a wide range of domains by programmers who find it simple yet powerful. In today's world, everyone wants to gain insights from the deluge of data coming their way. Data mining provides a way of finding these insights, and Python is one of the most popular languages for data mining, providing both power and flexibility in analysis. Python has become the language of choice for data scientists for data analysis, visualization, and machine learning. In this course, you will discover the key concepts of data mining and learn how to apply different data mining techniques to find the valuable insights hidden in real-world data.

Comprehensive Linear Modeling with R provides a wide overview of numerous contemporary linear and non-linear modeling approaches for the analysis of research data. These include basic, conditional and simultaneous inference techniques; analysis of variance (ANOVA); linear regression; survival analysis; generalized linear models (GLMs); parametric and non-parametric smoothers and generalized additive models (GAMs); longitudinal and mixed-effects, split-plot and other nested model designs. R Commander is a popular GUI-based "front-end" to the broad range of embedded statistical functionality in R software. R Commander is an'SPSS-like' GUI that enables the implementation of a large variety of statistical and graphical techniques using both menus and scripts. Please note that the R Commander GUI is written in the RGtk2 R-specific visual language (based on GTK) which is known to have problems running on a Mac computer.

As the amount of data continues to grow at an almost incomprehensible rate, being able to understand and process data is becoming a key differentiator for competitive organizations. Machine learning applications are everywhere, from self-driving cars, spam detection, document search, and trading strategies, to speech recognition. This makes machine learning well-suited to the present-day era of Big Data and Data Science. The main challenge is how to transform data into actionable knowledge. In this book you will learn all the important Machine Learning algorithms that are commonly used in the field of data science.