In this paper, we study the problem of semi-supervised structured output prediction, which aims to learn predictors for structured outputs, such as sequences, tree nodes, vectors, etc., from a set of data points of both inputoutput pairs and single inputs without outputs. The traditional methods to solve this problem usually learns one single predictor for all the data points, and ignores the variety of the different data points. Different parts of the data set may have different local distributions, and requires different optimal local predictors. To overcome this disadvantage of existing methods, we propose to learn different local predictors for neighborhoods of different data points, and the missing structured outputs simultaneously. In the neighborhood of each data point, we proposed to learn a linear predictor by minimizing both the complexity of the predictor and the upper bound of the structured prediction loss. The minimization is conducted by gradient descent algorithms. Experiments over four benchmark data sets, including DDSM mammography medical images, SUN natural image data set, Cora research paper data set, and Spanish news wire article sentence data set, show the advantages of the proposed method.
@cite_14 proposed to regularize the structured outputs by the manifold constructed from the input space directly. This method constructs a nearest neighbor graph from the input features, and use it to represent the manifold. Then the manifold is used to regularize the learning of the missing outputs of the unlabeled data points. The outputs and the predictor are learned simultaneously, and they regularize each other in the learning process.
@cite_14
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Manifold regularization (, 2006) is a geometrically motivated framework for machine learning within which several semi-supervised algorithms have been constructed. Here we try to provide some theoretical understanding of this approach. Our main result is to expose the natural structure of a class of problems on which manifold regularization methods are helpful. We show that for such problems, no supervised learner can learn effectively. On the other hand, a manifold based learner (that knows the manifold or "learns" it from unlabeled examples) can learn with relatively few labeled examples. Our analysis follows a minimax style with an emphasis on finite sample results (in terms of n: the number of labeled examples). These results allow us to properly interpret manifold regularization and related spectral and geometric algorithms in terms of their potential use in semi-supervised learning. Structured output prediction aims to learn a predictor to predict a structured output from a input data vector . The structured outputs include vector, tree, sequence, etc. We usually assume that we have a training set of input---output pairs to train the predictor. However, in many real-world applications, it is difficult to obtain the output for a input, and thus for many training input data points, the structured outputs are missing. In this paper, we discuss how to learn from a training set composed of some input---output pairs and some input data points without outputs. This problem is called semi-supervised structured output prediction. We propose a novel method for this problem by constructing a nearest neighbor graph from the input space to present the manifold structure and use it to regularize the structured output space directly. We define a slack structured output for each training data point and propose to predict it by learning a structured output predictor. The learning of both slack structured outputs and the predictor are unified within one single minimization problem. In this problem, we propose to minimize the structured loss between the slack structured outputs of neighboring data points and the prediction error measured by the structured loss. The problem is optimized by an iterative algorithm. Experiment results over three benchmark data sets show its advantage. Iterative methods that calculate their steps from approximate subgradient directions have proved to be useful for stochastic learning problems over large and streaming data sets. When the objective consists of a loss function plus a nonsmooth regularization term, the solution often lies on a low-dimensional manifold of parameter space along which the regularizer is smooth. (When an l1 regularizer is used to induce sparsity in the solution, for example, this manifold is defined by the set of nonzero components of the parameter vector.) This paper shows that a regularized dual averaging algorithm can identify this manifold, with high probability, before reaching the solution. This observation motivates an algorithmic strategy in which, once an iterate is suspected of lying on an optimal or near-optimal manifold, we switch to a "local phase" that searches in this manifold, thus converging rapidly to a near-optimal point. Computational results are presented to verify the identification property and to illustrate the effectiveness of this approach. Recently, manifold learning has been widely exploited in pattern recognition, data analysis, and machine learning. This paper presents a novel framework, called Riemannian manifold learning (RML), based on the assumption that the input high-dimensional data lie on an intrinsically low-dimensional Riemannian manifold. The main idea is to formulate the dimensionality reduction problem as a classical problem in Riemannian geometry, that is, how to construct coordinate charts for a given Riemannian manifold? We implement the Riemannian normal coordinate chart, which has been the most widely used in Riemannian geometry, for a set of unorganized data points. First, two input parameters (the neighborhood size k and the intrinsic dimension d) are estimated based on an efficient simplicial reconstruction of the underlying manifold. Then, the normal coordinates are computed to map the input high-dimensional data into a low- dimensional space. Experiments on synthetic data, as well as real-world images, demonstrate that our algorithm can learn intrinsic geometric structures of the data, preserve radial geodesic distances, and yield regular embeddings.