In these notes, we wish to provide an introduction to the signature method, focusing on its basic theoretical properties and recent numerical applications. The notes are split into two parts. The first part focuses on the definition and fundamental properties of the signature of a path, or the path signature. We have aimed for a minimalistic approach, assuming only familiarity with classical real analysis and integration theory, and supplementing theory with straightforward examples. We have chosen to focus in detail on the principle properties of the signature which we believe are fundamental to understanding its role in applications. We also present an informal discussion on some of its deeper properties and briefly mention the role of the signature in rough paths theory, which we hope could serve as a light introduction to rough paths for the interested reader. The second part of these notes discusses practical applications of the path signature to the area of machine learning. The signature approach represents a non-parametric way for extraction of characteristic features from data. The data are converted into a multi-dimensional path by means of various embedding algorithms and then processed for computation of individual terms of the signature which summarise certain information contained in the data. The signature thus transforms raw data into a set of features which are used in machine learning tasks. We will review current progress in applications of signatures to machine learning problems.

Distance metric learning (DML) approaches learn a transformation to a representation space where distance is in correspondence with a predefined notion of similarity. While such models offer a number of compelling benefits, it has been difficult for these to compete with modern classification algorithms in performance and even in feature extraction. In this work, we propose a novel approach explicitly designed to address a number of subtle yet important issues which have stymied earlier DML algorithms. It maintains an explicit model of the distributions of the different classes in representation space. It then employs this knowledge to adaptively assess similarity, and achieve local discrimination by penalizing class distribution overlap. We demonstrate the effectiveness of this idea on several tasks. Our approach achieves state-of-the-art classification results on a number of fine-grained visual recognition datasets, surpassing the standard softmax classifier and outperforming triplet loss by a relative margin of 30-40%. In terms of computational performance, it alleviates training inefficiencies in the traditional triplet loss, reaching the same error in 5-30 times fewer iterations. Beyond classification, we further validate the saliency of the learnt representations via their attribute concentration and hierarchy recovery properties, achieving 10-25% relative gains on the softmax classifier and 25-50% on triplet loss in these tasks.

Decentralized partially observable Markov decision processes (Dec-POMDPs) provide a general model for decision-making under uncertainty in decentralized settings, but are difficult to solve optimally (NEXP-Complete). As a new way of solving these problems, we introduce the idea of transforming a Dec-POMDP into a continuous-state deterministic MDP with a piecewise-linear and convex value function. This approach makes use of the fact that planning can be accomplished in a centralized offline manner, while execution can still be decentralized. This new Dec-POMDP formulation, which we call an occupancy MDP, allows powerful POMDP and continuous-state MDP methods to be used for the first time. To provide scalability, we refine this approach by combining heuristic search and compact representations that exploit the structure present in multi-agent domains, without losing the ability to converge to an optimal solution. In particular, we introduce a feature-based heuristic search value iteration (FB-HSVI) algorithm that relies on feature-based compact representations, point-based updates and efficient action selection. A theoretical analysis demonstrates that FB-HSVI terminates in finite time with an optimal solution. We include an extensive empirical analysis using well-known benchmarks, thereby demonstrating that our approach provides significant scalability improvements compared to the state of the art.

Automatic description generation from natural images is a challenging problem that has recently received a large amount of interest from the computer vision and natural language processing communities. In this survey, we classify the existing approaches based on how they conceptualize this problem, viz., models that cast description as either generation problem or as a retrieval problem over a visual or multimodal representational space. We provide a detailed review of existing models, highlighting their advantages and disadvantages. Moreover, we give an overview of the benchmark image datasets and the evaluation measures that have been developed to assess the quality of machine-generated image descriptions.

Performing exact posterior inference in complex generative models is often difficult or impossible due to an expensive to evaluate or intractable likelihood function. Approximate Bayesian computation (ABC) is an inference framework that constructs an approximation to the true likelihood based on the similarity between the observed and simulated data as measured by a predefined set of summary statistics. Although the choice of appropriate problem-specific summary statistics crucially influences the quality of the likelihood approximation and hence also the quality of the posterior sample in ABC, there are only few principled general-purpose approaches to the selection or construction of such summary statistics. In this paper, we develop a novel framework for this task using kernel-based distribution regression. We model the functional relationship between data distributions and the optimal choice (with respect to a loss function) of summary statistics using kernel-based distribution regression. We show that our approach can be implemented in a computationally and statistically efficient way using the random Fourier features framework for large-scale kernel learning. In addition to that, our framework shows superior performance when compared to related methods on toy and real-world problems.

An estimated 17.5 million people died from a cardiovascular disease in 2012, representing 31% of all global deaths. Most acute coronary events result from rupture of the protective fibrous cap overlying an atherosclerotic plaque. The task of early identification of plaque types that can potentially rupture is, therefore, of great importance. The state-of-the-art approach to imaging blood vessels is intravascular optical coherence tomography (IVOCT). However, currently, this is an offline approach where the images are first collected and then manually analyzed a frame at a time to identify regions at risk of thrombosis. This process is extremely laborious, time consuming and prone to human error. We are building a system that, when complete, will provide interactive 3D visualization of a blood vessel as an IVOCT is in progress. The visualization will highlight different plaque types and enable quick identification of regions at risk for thrombosis. In this paper, we describe our approach, focusing on machine learning methods that are a key enabling technology. Our empirical results using real OCT data show that our approach can identify different plaque types efficiently with high accuracy across multiple patients.

Capturing domain knowledge can be a time-consuming process that typically requires the collaboration of a Subject Matter Expert and a modeling expert to encode the knowledge. In a number of domains and applications, this situation is further exacerbated by the fact that the Subject Matter Expert may find it difficult to articulate the domain knowledge as a procedure or rules, but instead may find it easier to classify instance data. To facilitate this type of knowledge elicitation from Subject Matter Experts, we have developed a system that automatically generates formal and executable rules from provided labeled instance data. We do this by leveraging the techniques of Inductive Logic Programming (ILP) to generate Horn clause based rules to separate out positive and negative instance data. We illustrate our approach on a Design For Manufacturability (DFM) platform where the goal is to design products that are easy to manufacture by providing early manufacturability feedback. Specifically we show how our approach can be used to generate feature recognition rules from positive and negative instance data supplied by Subject Matter Experts. Our platform is interactive, provides visual feedback and is iterative. The feature identification rules generated can be inspected, manually refined and vetted.

Poaching is a serious threat to the conservation of key species and whole ecosystems. While conducting foot patrols is the most commonly used approach in many countries to prevent poaching, such patrols often do not make the best use of limited patrolling resources. To remedy this situation, prior work introduced a novel emerging application called PAWS (Protection Assistant for Wildlife Security); PAWS was proposed as a game-theoretic (``security games'') decision aid to optimize the use of patrolling resources. This paper reports on PAWS's significant evolution from a proposed decision aid to a regularly deployed application, reporting on the lessons from the first tests in Africa in Spring 2014, through its continued evolution since then, to current regular use in Southeast Asia and plans for future worldwide deployment. In this process, we have worked closely with two NGOs (Panthera and Rimba) and incorporated extensive feedback from professional patrolling teams. We outline key technical advances that lead to PAWS's regular deployment: (i) incorporating complex topographic features, e.g., ridgelines, in generating patrol routes; (ii) handling uncertainties in species distribution (game theoretic payoffs); (iii) ensuring scalability for patrolling large-scale conservation areas with fine-grained guidance; and (iv) handling complex patrol scheduling constraints.

The stochastic blockmodel (SBM) is a generative model for network data introduced in Holland et al. (1983). The SBM is a member of the general class of latent position random graph models introduced in Hoff et al. (2002). These models have been used in various application domains as diverse as social networks (vertices may represent people with edges indicating social interaction), citation networks (who cites whom), connectomics (brain connectivity networks; vertices may represent neurons with edges indicating axon-synapse-dendrite connections, or vertices may represent brain regions with edges indicating connectivity between regions), and many others. For comprehensive reviews of statistical models and applications, see Fienberg (2010), Goldenberg et al. (2010), Fienberg (2012). In general, statistical inference on graphs is becoming essential in many areas of science, engineering, and business. The SBM supposes that each of n vertices is assigned to one of K blocks. The probability of an 1 edge between two vertices depends only on their respective block memberships, and the presence of edges are conditionally independent given block memberships.

In this paper, we extend the Maximum Satisfiability (MaxSAT) problem to {\L}ukasiewicz logic. The MaxSAT problem for a set of formulae {\Phi} is the problem of finding an assignment to the variables in {\Phi} that satisfies the maximum number of formulae. Three possible solutions (encodings) are proposed to the new problem: (1) Disjunctive Linear Relations (DLRs), (2) Mixed Integer Linear Programming (MILP) and (3) Weighted Constraint Satisfaction Problem (WCSP). Like its Boolean counterpart, the extended fuzzy MaxSAT will have numerous applications in optimization problems that involve vagueness.