Machine learning is the science of discovering statistical dependencies in data, and the use of those dependencies to perform predictions. During the last decade, machine learning has made spectacular progress, surpassing human performance in complex tasks such as object recognition, car driving, and computer gaming. However, the central role of prediction in machine learning avoids progress towards general-purpose artificial intelligence. As one way forward, we argue that causal inference is a fundamental component of human intelligence, yet ignored by learning algorithms. Causal inference is the problem of uncovering the cause-effect relationships between the variables of a data generating system. Causal structures provide understanding about how these systems behave under changing, unseen environments. In turn, knowledge about these causal dynamics allows to answer "what if" questions, describing the potential responses of the system under hypothetical manipulations and interventions. Thus, understanding cause and effect is one step from machine learning towards machine reasoning and machine intelligence. But, currently available causal inference algorithms operate in specific regimes, and rely on assumptions that are difficult to verify in practice. This thesis advances the art of causal inference in three different ways. First, we develop a framework for the study of statistical dependence based on copulas and random features. Second, we build on this framework to interpret the problem of causal inference as the task of distribution classification, yielding a family of novel causal inference algorithms. Third, we discover causal structures in convolutional neural network features using our algorithms. The algorithms presented in this thesis are scalable, exhibit strong theoretical guarantees, and achieve state-of-the-art performance in a variety of real-world benchmarks.

The GMM (generalized min-max) kernel was recently proposed (Li, 2016) as a measure of data similarity and was demonstrated effective in machine learning tasks. In order to use the GMM kernel for large-scale datasets, the prior work resorted to the (generalized) consistent weighted sampling (GCWS) to convert the GMM kernel to linear kernel. We call this approach as ``GMM-GCWS''. In the machine learning literature, there is a popular algorithm which we call ``RBF-RFF''. That is, one can use the ``random Fourier features'' (RFF) to convert the ``radial basis function'' (RBF) kernel to linear kernel. It was empirically shown in (Li, 2016) that RBF-RFF typically requires substantially more samples than GMM-GCWS in order to achieve comparable accuracies. The Nystrom method is a general tool for computing nonlinear kernels, which again converts nonlinear kernels into linear kernels. We apply the Nystrom method for approximating the GMM kernel, a strategy which we name as ``GMM-NYS''. In this study, our extensive experiments on a set of fairly large datasets confirm that GMM-NYS is also a strong competitor of RBF-RFF.

High-dimensional big data appears in many research fields such as image recognition, biology and collaborative filtering. Often, the exploration of such data by classic algorithms is encountered with difficulties due to `curse of dimensionality' phenomenon. Therefore, dimensionality reduction methods are applied to the data prior to its analysis. Many of these methods are based on principal components analysis, which is statistically driven, namely they map the data into a low-dimension subspace that preserves significant statistical properties of the high-dimensional data. As a consequence, such methods do not directly address the geometry of the data, reflected by the mutual distances between multidimensional data point. Thus, operations such as classification, anomaly detection or other machine learning tasks may be affected. This work provides a dictionary-based framework for geometrically driven data analysis that includes dimensionality reduction, out-of-sample extension and anomaly detection. It embeds high-dimensional data in a low-dimensional subspace. This embedding preserves the original high-dimensional geometry of the data up to a user-defined distortion rate. In addition, it identifies a subset of landmark data points that constitute a dictionary for the analyzed dataset. The dictionary enables to have a natural extension of the low-dimensional embedding to out-of-sample data points, which gives rise to a distortion-based criterion for anomaly detection. The suggested method is demonstrated on synthetic and real-world datasets and achieves good results for classification, anomaly detection and out-of-sample tasks.

The well known maximum-entropy principle due to Jaynes, which states that given mean parameters, the maximum entropy distribution matching them is in an exponential family, has been very popular in machine learning due to its "Occam's razor" interpretation. Unfortunately, calculating the potentials in the maximum-entropy distribution is intractable \cite{bresler2014hardness}. We provide computationally efficient versions of this principle when the mean parameters are pairwise moments: we design distributions that approximately match given pairwise moments, while having entropy which is comparable to the maximum entropy distribution matching those moments. We additionally provide surprising applications of the approximate maximum entropy principle to designing provable variational methods for partition function calculations for Ising models without any assumptions on the potentials of the model. More precisely, we show that in every temperature, we can get approximation guarantees for the log-partition function comparable to those in the low-temperature limit, which is the setting of optimization of quadratic forms over the hypercube. \cite{alon2006approximating}

In the problem of modeling and predicting statistical processes, (wide-sense) stationarity is a helpful assumption, that allows us to learn the spectral characteristics of a process using very few samples. Especially for time-series prediction, learning from few samples is crucial, as one needs to estimate future values after only partially observing a single realization of the statistical process. This is the main reason why classical models for estimation and prediction of univariate processes, such as Wiener filters and auto-regressive moving average models (ARMA), rely on stationarity to produce predictions. For multivariate statistical processes, following the same methodology is often problematic, as the number of parameters to be estimated increases quadratically with the number variables, often rendering the problem intractable. A common way to deal with this dimensionality issue is to assume that there is an inherent structure to the process that can be captured by a graph. The graph assumption appears frequently in the machine learning and signal processing literature, and has been shown invaluable for tasks such as clustering [1], [18], low-rank extraction [14], spectral estimation [12], [10] and semi-supervised learning [2], [17]. Nevertheless, despite their promise, so far state-of-the-art graph-based methods predominantly ignore the time-dimension of data. The objective of this paper is to identify multivariate models that exploit the graph structure inherent to the data so as to facilitate the task of prediction.

Finding relevant information from large document collections such as the World Wide Web is a common task in our daily lives. Estimation of a user's interest or search intention is necessary to recommend and retrieve relevant information from these collections. We introduce a brain-information interface used for recommending information by relevance inferred directly from brain signals. In experiments, participants were asked to read Wikipedia documents about a selection of topics while their EEG was recorded. Based on the prediction of word relevance, the individual's search intent was modeled and successfully used for retrieving new, relevant documents from the whole English Wikipedia corpus. The results show that the users' interests towards digital content can be modeled from the brain signals evoked by reading. The introduced brain-relevance paradigm enables the recommendation of information without any explicit user interaction, and may be applied across diverse information-intensive applications.

According to a recent study, machine learning algorithms are expected to replace 25% of the jobs across the world, in the next 10 years. With the rapid growth of big data and availability of programming tools like Python and R –machine learning is gaining mainstream presence for data scientists. Machine learning applications are highly automated and self-modifying which continue to improve over time with minimal human intervention as they learn with more data. For instance, Netflix's recommendation algorithm learns more about the likes and dislikes of a viewer based on the shows every viewer watches. To address the complex nature of various real world data problems, specialized machine learning algorithms have been developed that solve these problems perfectly.

In this post, we are going to introduce you to the Support Vector Machine (SVM) machine learning algorithm. We will follow a similar process to our recent post Naive Bayes for Dummies; A Simple Explanation by keeping it short and not overly-technical. The aim is to give those of you who are new to machine learning a basic understanding of the key concepts of this algorithm. A Support Vector Machine (SVM) is a supervised machine learning algorithm that can be employed for both classification and regression purposes. SVMs are more commonly used in classification problems and as such, this is what we will focus on in this post.

I did a webcast earlier today about Bayesian statistics. Some time in the next week, the video should be available from O'Reilly. In the meantime, you can see my slides here: And here's a transcript of what I said: Thanks everyone for joining me for this webcast. At the bottom of this slide you can see the URL for my slides, so you can follow along at home. I'm Allen Downey and I'm a professor at Olin College, which is a new engineering college right outside Boston. Our mission is to fix engineering education, and one of the ways I'm working on that is by teaching Bayesian statistics. Bayesian methods have been the victim of a 200 year smear campaign. If you are interested in the history and the people involved, I recommend this book, The Theory That Would Not Die.

Many datasets can be viewed as a noisy sampling of an underlying space, and tools from topological data analysis can characterize this structure for the purpose of knowledge discovery. One such tool is persistent homology, which provides a multiscale description of the homological features within a dataset. A useful representation of this homological information is a persistence diagram (PD). Efforts have been made to map PDs into spaces with additional structure valuable to machine learning tasks. We convert a PD to a finite-dimensional vector representation which we call a persistence image (PI), and prove the stability of this transformation with respect to small perturbations in the inputs. The discriminatory power of PIs is compared against existing methods, showing significant performance gains. We explore the use of PIs with vector-based machine learning tools, such as linear sparse support vector machines, which identify features containing discriminating topological information. Finally, high accuracy inference of parameter values from the dynamic output of a discrete dynamical system (the linked twist map) and a partial differential equation (the anisotropic Kuramoto-Sivashinsky equation) provide a novel application of the discriminatory power of PIs.