Machine Learning is the study of methods for programming computers to learn. Computers are applied to a wide range of tasks, and for most of these it is relatively easy for programmers to design and implement the necessary software. However, there are many tasks for which this is difficult or impossible. These can be divided into four general categories. First, there are problems for which there exist no human experts. For example, in modern automated manufacturing facilities, there is a need to predict machine failures before they occur by analyzing sensor readings. Because the machines are new, there are no human experts who can be interviewed by a programmer to provide the knowledge necessary to build a computer system. A machine learning system can study recorded data and subsequent machine failures and learn prediction rules. Second, there are problems where human experts exist, but where they are unable to explain their expertise. This is the case in many perceptual tasks, such as speech recognition, hand-writing recognition, and natural language understanding. Virtually all humans exhibit expert-level abilities on these tasks, but none of them can describe the detailed steps that they follow as they perform them. Fortunately, humans can provide machines with examples of the inputs and correct outputs for these tasks, so machine learning algorithms can learn to map the inputs to the outputs. Third, there are problems where phenomena are changing rapidly. In finance, for example, people would like to predict the future behavior of the stock market, of consumer purchases, or of exchange rates. These behaviors change frequently, so that even if a programmer could construct a good predictive computer program, it would need to be rewritten frequently. A learning program can relieve the programmer of this burden by constantly modifying and tuning a set of learned prediction rules. Fourth, there are applications that need to be customized for each computer user separately. Consider, for example, a program to filter unwanted electronic mail messages. Different users will need different filters. It is unreasonable to expect each user to program his or her own rules, and it is infeasible to provide every user with a software engineer to keep the rules up-to-date. A machine learning system can learn which mail messages the user rejects and maintain the filtering rules automatically. Machine learning addresses many of the same research questions as the fields of statistics, data mining, and psychology, but with differences of emphasis. Statistics focuses on understanding the phenomena that have generated the data, often with the goal of testing different hypotheses about those phenomena. Data mining seeks to find patterns in the data that are understandable by people. Psychological studies of human learning aspire to understand the mechanisms underlying the various learning behaviors exhibited by people (concept learning, skill acquisition, strategy change, etc.).

Machine learning

In recent years, spectral clustering has become one of the most popular modern clustering algorithms. It is simple to implement, can be solved efficiently by standard linear algebra software, and very often outperforms traditional clustering algorithms such as the k-means algorithm. On the first glance spectral clustering appears slightly mysterious, and it is not obvious to see why it works at all and what it really does. The goal of this tutorial is to give some intuition on those questions. We describe different graph Laplacians and their basic properties, present the most common spectral clustering algorithms, and derive those algorithms from scratch by several different approaches. Advantages and disadvantages of the different spectral clustering algorithms are discussed.

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A tutorial on spectral clustering

The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i. e. the organization of vertices in clusters, with many edges joining vertices of the same cluster and comparatively few edges joining vertices of different clusters. Such clusters, or communities, can be considered as fairly independent compartments of a graph, playing a similar role like, e. g., the tissues or the organs in the human body. Detecting communities is of great importance in sociology, biology and computer science, disciplines where systems are often represented as graphs. This problem is very hard and not yet satisfactorily solved, despite the huge effort of a large interdisciplinary community of scientists working on it over the past few years. We will attempt a thorough exposition of the topic, from the definition of the main elements of the problem, to the presentation of most methods developed, with a special focus on techniques designed by statistical physicists, from the discussion of crucial issues like the significance of clustering and how methods should be tested and compared against each other, to the description of applications to real networks.

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Community detection in graphs

Today's Web-enabled deluge of electronic data calls for automated methods of data analysis. Machine learning provides these, developing methods that can automatically detect patterns in data and then use the uncovered patterns to predict future data. This textbook offers a comprehensive and self-contained introduction to the field of machine learning, based on a unified, probabilistic approach. The coverage combines breadth and depth, offering necessary background material on such topics as probability, optimization, and linear algebra as well as discussion of recent developments in the field, including conditional random fields, L1 regularization, and deep learning. The book is written in an informal, accessible style, complete with pseudo-code for the most important algorithms. All topics are copiously illustrated with color images and worked examples drawn from such application domains as biology, text processing, computer vision, and robotics. Rather than providing a cookbook of different heuristic methods, the book stresses a principled model-based approach, often using the language of graphical models to specify models in a concise and intuitive way. Almost all the models described have been implemented in a MATLAB software package--PMTK (probabilistic modeling toolkit)--that is freely available online. The book is suitable for upper-level undergraduates with an introductory-level college math background and beginning graduate students.

Machine Learning : A Probabilistic Perspective

Consider a neighborhood graph, for example a k-nearest neighbor graph, that is constructed on sample points drawn according to some density p. Our goal is to re-weight the graph's edges such that all cuts and volumes behave as if the graph was built on a different sample drawn from an alternative density p. We introduce the f -adjusted graph and prove that it provides the correct cuts and volumes as the sample size tends to infinity. From an algebraic perspective, we show that its normalized Laplacian, denoted as the f -adjusted Laplacian, represents a natural family of diagonal perturbations of the original normalized Laplacian. Our technique allows to apply any cut and volume based algorithm to the f -adjusted graph, for example spectral clustering, in order to study the given graph as if it were built on an unaccessible sample from a different density. We point out applications in sample bias correction, data uniformization, and multi-scale analysis of graphs.

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The f-Adjusted Graph Laplacian: a Diagonal Modification with a Geometric Interpretation

We consider machine learning in a comparison-based setting where we are given a set of points in a metric space, but we have no access to the actual distances between the points. Instead, we can only ask an oracle whether the distance between two points $i$ and $j$ is smaller than the distance between the points $i$ and $k$. We are concerned with data structures and algorithms to find nearest neighbors based on such comparisons. We focus on a simple yet effective algorithm that recursively splits the space by first selecting two random pivot points and then assigning all other points to the closer of the two (comparison tree). We prove that if the metric space satisfies certain expansion conditions, then with high probability the height of the comparison tree is logarithmic in the number of points, leading to efficient search performance. We also provide an upper bound for the failure probability to return the true nearest neighbor. Experiments show that the comparison tree is competitive with algorithms that have access to the actual distance values, and needs less triplet comparisons than other competitors.

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Comparison-Based Nearest Neighbor Search

The study of networks leads to a wide range of high dimensional inference problems. In many practical applications, one needs to draw inference from one or few large sparse networks. The present paper studies hypothesis testing of graphs in this high-dimensional regime, where the goal is to test between two populations of inhomogeneous random graphs defined on the same set of $n$ vertices. The size of each population $m$ is much smaller than $n$, and can even be a constant as small as 1. The critical question in this context is whether the problem is solvable for small $m$. 
We answer this question from a minimax testing perspective. Let $P,Q$ be the population adjacencies of two sparse inhomogeneous random graph models, and $d$ be a suitably defined distance function. Given a population of $m$ graphs from each model, we derive minimax separation rates for the problem of testing $P=Q$ against $d(P,Q)>\rho$. We observe that if $m$ is small, then the minimax separation is too large for some popular choices of $d$, including total variation distance between corresponding distributions. This implies that some models that are widely separated in $d$ cannot be distinguished for small $m$, and hence, the testing problem is generally not solvable in these cases. 
We also show that if $m>1$, then the minimax separation is relatively small if $d$ is the Frobenius norm or operator norm distance between $P$ and $Q$. For $m=1$, only the latter distance provides small minimax separation. Thus, for these distances, the problem is solvable for small $m$. We also present near-optimal two-sample tests in both cases, where tests are adaptive with respect to sparsity level of the graphs.

Two-sample Hypothesis Testing for Inhomogeneous Random Graphs

Interpretability of machine learning algorithm is a pressing need. Numerous methods appeared in recent years, but do they make sense in simple cases? In this paper, we present a thorough theoretical analysis of Tabular LIME. In particular, we show that the explanations provided by Tabular LIME are close to an explicit expression in the large sample limit. We leverage this knowledge when the function to explain has some nice algebraic structure (linear, multiplicative, or depending on a subset of the coordinates) and provide some interesting insights on the explanations provided in these cases. In particular, we show that Tabular LIME provides explanations that are proportional to the coefficients of the function to explain in the linear case, and provably discards coordinates unused by the function to explain in the general case.

Looking deeper into LIME

To investigate objects without a describable notion of distance, one can gather ordinal information by asking triplet comparisons of the form "Is object $x$ closer to $y$ or is $x$ closer to $z$?" In order to learn from such data, the objects are typically embedded in a Euclidean space while satisfying as many triplet comparisons as possible. In this paper, we introduce empirical uncertainty estimates for standard embedding algorithms when few noisy triplets are available, using a bootstrap and a Bayesian approach. In particular, simulations show that these estimates are well calibrated and can serve to select embedding parameters or to quantify uncertainty in scientific applications.

Ulrike von Luxburg

Papers

The f-Adjusted Graph Laplacian: a Diagonal Modification with a Geometric Interpretation

Comparison-Based Nearest Neighbor Search

Two-sample Hypothesis Testing for Inhomogeneous Random Graphs

Looking deeper into LIME

Uncertainty Estimates for Ordinal Embeddings.