Multilayer neural networks trained with the back-propagation algorithm constitute the best example of a successful gradient based learning technique. Given an appropriate network architecture, gradient-based learning algorithms can be used to synthesize a complex decision surface that can classify high-dimensional patterns, such as handwritten characters, with minimal preprocessing. This paper reviews various methods applied to handwritten character recognition and compares them on a standard handwritten digit recognition task. Convolutional neural networks, which are specifically designed to deal with the variability of 2D shapes, are shown to outperform all other techniques. Real-life document recognition systems are composed of multiple modules including field extraction, segmentation recognition, and language modeling. A new learning paradigm, called graph transformer networks (GTN), allows such multimodule systems to be trained globally using gradient-based methods so as to minimize an overall performance measure. Two systems for online handwriting recognition are described. Experiments demonstrate the advantage of global training, and the flexibility of graph transformer networks. A graph transformer network for reading a bank cheque is also described. It uses convolutional neural network character recognizers combined with global training techniques to provide record accuracy on business and personal cheques. It is deployed commercially and reads several million cheques per day.

Gradient-based learning applied to document recognition

LIBSVM is a library for Support Vector Machines (SVMs). We have been actively developing this package since the year 2000. The goal is to help users to easily apply SVM to their applications. LIBSVM has gained wide popularity in machine learning and many other areas. In this article, we present all implementation details of LIBSVM. Issues such as solving SVM optimization problems theoretical convergence multiclass classification probability estimates and parameter selection are discussed in detail.

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LIBSVM: A library for support vector machines

The increasing volume of data in modern business and science calls for more complex and sophisticated tools. Although advances in data mining technology have made extensive data collection much easier, it's still always evolving and there is a constant need for new techniques and tools that can help us transform this data into useful information and knowledge. Since the previous edition's publication, great advances have been made in the field of data mining. Not only does the third of edition of Data Mining: Concepts and Techniques continue the tradition of equipping you with an understanding and application of the theory and practice of discovering patterns hidden in large data sets, it also focuses on new, important topics in the field: data warehouses and data cube technology, mining stream, mining social networks, and mining spatial, multimedia and other complex data. Each chapter is a stand-alone guide to a critical topic, presenting proven algorithms and sound implementations ready to be used directly or with strategic modification against live data. This is the resource you need if you want to apply today's most powerful data mining techniques to meet real business challenges. * Presents dozens of algorithms and implementation examples, all in pseudo-code and suitable for use in real-world, large-scale data mining projects. * Addresses advanced topics such as mining object-relational databases, spatial databases, multimedia databases, time-series databases, text databases, the World Wide Web, and applications in several fields. *Provides a comprehensive, practical look at the concepts and techniques you need to get the most out of real business data

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Data Mining: Concepts and Techniques

This paper describes a machine learning approach for visual object detection which is capable of processing images extremely rapidly and achieving high detection rates. This work is distinguished by three key contributions. The first is the introduction of a new image representation called the "integral image" which allows the features used by our detector to be computed very quickly. The second is a learning algorithm, based on AdaBoost, which selects a small number of critical visual features from a larger set and yields extremely efficient classifiers. The third contribution is a method for combining increasingly more complex classifiers in a "cascade" which allows background regions of the image to be quickly discarded while spending more computation on promising object-like regions. The cascade can be viewed as an object specific focus-of-attention mechanism which unlike previous approaches provides statistical guarantees that discarded regions are unlikely to contain the object of interest. In the domain of face detection the system yields detection rates comparable to the best previous systems. Used in real-time applications, the detector runs at 15 frames per second without resorting to image differencing or skin color detection.

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Rapid object detection using a boosted cascade of simple features

The tutorial starts with an overview of the concepts of VC dimension and structural risk minimization. We then describe linear Support Vector Machines (SVMs) for separable and non-separable data, working through a non-trivial example in detail. We describe a mechanical analogy, and discuss when SVM solutions are unique and when they are global. We describe how support vector training can be practically implemented, and discuss in detail the kernel mapping technique which is used to construct SVM solutions which are nonlinear in the data. We show how Support Vector machines can have very large (even infinite) VC dimension by computing the VC dimension for homogeneous polynomial and Gaussian radial basis function kernels. While very high VC dimension would normally bode ill for generalization performance, and while at present there exists no theory which shows that good generalization performance is guaranteed for SVMs, there are several arguments which support the observed high accuracy of SVMs, which we review. Results of some experiments which were inspired by these arguments are also presented. We give numerous examples and proofs of most of the key theorems. There is new material, and I hope that the reader will find that even old material is cast in a fresh light.

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A Tutorial on Support Vector Machines for Pattern Recognition

Logistic regression is one of the most popular methods in binary classification, wherein estimation of model parameters is carried out by solving the maximum likelihood (ML) optimization problem, and the ML estimator is defined to be the optimal solution of this problem. It is well known that the ML estimator exists when the data is non-separable, but fails to exist when the data is separable. First-order methods are the algorithms of choice for solving large-scale instances of the logistic regression problem. In this paper, we introduce a pair of condition numbers that measure the degree of non-separability or separability of a given dataset in the setting of binary classification, and we study how these condition numbers relate to and inform the properties and the convergence guarantees of first-order methods. When the training data is non-separable, we show that the degree of non-separability naturally enters the analysis and informs the properties and convergence guarantees of two standard first-order methods: steepest descent (for any given norm) and stochastic gradient descent. Expanding on the work of Bach, we also show how the degree of non-separability enters into the analysis of linear convergence of steepest descent (without needing strong convexity), as well as the adaptive convergence of stochastic gradient descent. When the training data is separable, first-order methods rather curiously have good empirical success, which is not well understood in theory. In the case of separable data, we demonstrate how the degree of separability enters into the analysis of $\ell_2$ steepest descent and stochastic gradient descent for delivering approximate-maximum-margin solutions with associated computational guarantees as well. This suggests that first-order methods can lead to statistically meaningful solutions in the separable case, even though the ML solution does not exist.

Condition Number Analysis of Logistic Regression, and its Implications for Standard First-Order Solution Methods.

Partial differential equations (PDEs) are commonly used to model a wide variety of physical phenomena. A PDE model of a physical problem is typically described by conservation laws, constitutive laws, material properties, boundary conditions, boundary data, and geometry. In most practical applications, however, the PDE model is only an approximation to the real physical problem due to both (i) the deliberate mathematical simplification of the model to keep it tractable and (ii) the inherent uncertainty of the physical parameters. In such cases, the PDE model may not produce a good prediction of the true state of the underlying physical problem. In this paper, we introduce a functional regression method that incorporates observations into a deterministic linear PDE model to improve its prediction of the true state. Our method is devised as follows. First, we augment the PDE model with a random Gaussian functional which serves to represent various sources of uncertainty in the model. We next derive a linear...

Functional Regression for State Prediction Using Linear PDE Models and Observations

We present a new analysis of the Frank-Wolfe method proposed in Dvurechensky et al. applied to constrained minimization problems where the objective function $f$ is a $\theta$-logarithmically-homogeneous self-concordant barrier. We show that the Frank-Wolfe method requires $O((\delta_0 + \theta)\ln(\delta_0) + \theta^2/\varepsilon)$ iterations to produce an $\varepsilon$-approximate solution, where $\delta_0$ denotes the initial optimality gap. This result establishes certain intrinsic connections between $\theta$-logarithmically homogeneous barriers and the Frank-Wolfe method. When specialized to the $D$-optimal design problem, we essentially recover the complexity obtained by Kachiyan using the Frank-Wolfe method with exact line-search. We present numerical experiments on simulated PET problem instances that demonstrate the potential for practical usefulness of the Frank-Wolfe method on this class of problems. In addition, we present and analyze an adaptive step-size Frank-Wolfe method when $f$ is Holder smooth.

Analysis of the Frank-Wolfe Method for Logarithmically-Homogeneous Barriers, with an Extension

The design of materials is currently a fertile research domain. However, most of the material designs described in the literature arise from physical intuition, and often assume infinite periodicity. There is a need for a design methodology capable of computing patterns and designs involving two different materials where the underlying design variables correspond to a finite set of pixels in a 2-dimensional mesh, and where the goal is a design with prescribed material properties. This naturally leads to the consideration of binary optimization models in contrast to classical (continuous) gradient-based methods, which generically provide continuous solutions that then need to be “rounded” to binary values. While the potential drawback of binary optimization is that its computational complexity is usually NP-hard, and hence theoretically unattractive, we show herein that binary optimization combined with a reduced basis approach can relatively efficiently produce good solutions to material design problems of interest.

Binary Optimization Techniques for Linear PDE-governed Material Design

We present and analyze a new generalized Frank-Wolfe method for the composite optimization problem $(P):{\min}_{x\in\mathbb{R}^n}\; f(\mathsf{A} x) + h(x)$, where $f$ is a $\theta$-logarithmically-homogeneous self-concordant barrier, $\mathsf{A}$ is a linear operator and the function $h$ has bounded domain but is possibly non-smooth. We show that our generalized Frank-Wolfe method requires $O((\delta_0 + \theta + R_h)\ln(\delta_0) + (\theta + R_h)^2/\varepsilon)$ iterations to produce an $\varepsilon$-approximate solution, where $\delta_0$ denotes the initial optimality gap and $R_h$ is the variation of $h$ on its domain. This result establishes certain intrinsic connections between $\theta$-logarithmically homogeneous barriers and the Frank-Wolfe method. When specialized to the $D$-optimal design problem, we essentially recover the complexity obtained by Khachiyan using the Frank-Wolfe method with exact line-search. We also study the (Fenchel) dual problem of $(P)$, and we show that our new method is equivalent to an adaptive-step-size mirror descent method applied to the dual problem. This enables us to provide iteration complexity bounds for the mirror descent method despite even though the dual objective function is non-Lipschitz and has unbounded domain. In addition, we present computational experiments that point to the potential usefulness of our generalized Frank-Wolfe method on Poisson image de-blurring problems with TV regularization, and on simulated PET problem instances.

Robert M. Freund

Papers

Condition Number Analysis of Logistic Regression, and its Implications for Standard First-Order Solution Methods.

Functional Regression for State Prediction Using Linear PDE Models and Observations

Analysis of the Frank-Wolfe Method for Logarithmically-Homogeneous Barriers, with an Extension

Binary Optimization Techniques for Linear PDE-governed Material Design

Analysis of the Frank-Wolfe Method for Convex Composite Optimization involving a Logarithmically-Homogeneous Barrier