In this task, you are given a part of an article. Your task is to generate headline (title) for this text. Preferred headlines are under fifteen words.
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Question: In classical machine learning, regression is treated as a black box process of identifying a suitable function from a hypothesis set without attempting to gain insight into the mechanism connecting inputs and outputs. In the natural sciences, however, finding an interpretable function for a phenomenon is the prime goal as it allows to understand and generalize results. This paper proposes a novel type of function learning network, called equation learner (EQL), that can learn analytical expressions and is able to extrapolate to unseen domains. It is implemented as an end-to-end differentiable feed-forward network and allows for efficient gradient based training. Due to sparsity regularization concise interpretable expressions can be obtained. Often the true underlying source expression is identified. INTRODUCTION The quality of a model is typically measured by its ability to generalize from a training set to previously unseen data from the same distribution. In regression tasks generalization essentially boils down to interpolation if the training data is sufficiently dense. As long as models are selected correctly, i. e. in a way to not overfit the data, the regression problem is well understood and can – at least conceptually – be considered solved. However, when working with data from real-world devices, e. g. controlling a robotic arm, interpolation might not be sufficient. It could happen that future data lies outside of the training domain, e. g. when the arm is temporarily operated outside of its specifications. For the sake of robustness and safety it is desirable in such a case to have a regression model that continues to make good predictions, or at least does not fail catastrophically. This setting, which we call extrapolation generalization, is the topic of the present paper. We are particularly interested in regression tasks for systems that can be described by real-valued analytic expression, e. g. mechanical systems such as a pendulum or a robotic arm. These are typically governed by a highly nonlinear function but it is nevertheless possible, in principle, to infer their behavior on an extrapolation domain from their behavior elsewhere. We make two main contributions: 1) a new type of network that can learn analytical expressions and is able to extrapolate to unseen domains and 2) a model selection strategy tailored to the extrapolation setting. The following section describes the setting of regression and extrapolation. Afterwards we introduce our method and discuss the architecture, its training, and its relation to prior art. We present our results in the Section Experimental evaluation and close with conclusions. REGRESSION AND EXTRAPOLATION We consider a multivariate regression problem with a training set {(x1, y1), . . . , (xN , yN )} with x ∈ R, y ∈ R. Because our main interest lies on extrapolation in the context of learning the dynamics of physical systems we assume the data originates from an unknown analytical function (or system of functions), φ : R → R with additive zero-mean noise, ξ, i. e. y = φ(x) + ξ and Eξ = 0. The function φ may, for instance, reflect a system of ordinary differential equations that govern the movements of a robot arm or the like. The general task is to learn a function ψ : R → R that approximates the true functional relation as well as possible in the squared loss sense, i. e. achieves minimal expected error E‖ψ(x) − φ(x)‖2. In practice, we only have particular examples of the function values available and measure the quality of predicting in terms of the empirical error on

Answer: EXTRAPOLATION AND LEARNING EQUATIONS


Question: An RDF data shape is a description of the expected contents of an RDF document (aka graph) or dataset. A major part of this description is the set of constraints that the document or dataset is required to satisfy. W3C recently (2014) chartered the RDF Data Shapes Working Group to define SHACL, a standard RDF data shape language. We refer to the ability to name and reference shape language elements as recursion. This article provides a precise definition of the meaning of recursion as used in Resource Shape 2.0. The definition of recursion presented in this article is largely independent of language-specific details. We speculate that it also applies to ShEx and to all three of the current proposals for SHACL. In particular, recursion is not permitted in the SHACL-SPARQL proposal, but we conjecture that recursion could be added by using the definition proposed here as a top-level control structure.

Answer: Recursion in RDF Data Shape Languages


Question: A new algorithm named EXPected Similarity Estimation (EXPoSE) was recently proposed to solve the problem of large-scale anomaly detection. It is a non-parametric and distribution free kernel method based on the Hilbert space embedding of probability measures. Given a dataset of n samples, EXPoSE needs only O(n) (linear time) to build a model and O(1) (constant time) to make a prediction. In this work we improve the linear computational complexity and show that an -accurate model can be estimated in constant time, which has significant implications for large-scale learning problems. To achieve this goal, we cast the original EXPoSE formulation into a stochastic optimization problem. It is crucial that this approach allows us to determine the number of iteration based on a desired accuracy , independent of the dataset size n. We will show that the proposed stochastic gradient descent algorithm works in general (possible infinite-dimensional) Hilbert spaces, is easy to implement and requires no additional step-size parameters.

Answer:
CONSTANT TIME EXPECTED SIMILARITY ESTIMATION USING STOCHASTIC OPTIMIZATION