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. Manifold regularized sparse coding shows promising performance for various applications. The key issue that must be considered in the application is how to adaptively select the suitable graph hyper-parameters in manifold learning for the sparse coding task. Usually, cross validation is applied, but it does not necessarily scale up and easily leads to overfitting. In this article, multiple graph sparse coding (MGrSc) and multiple Hypergraph sparse coding (MHGrSc) for image representation are proposed. Inspired by the Ensemble Manifold Regularizer, we formulate multiple graph and multiple Hypergraph regularizers to guarantee the smoothness of sparse codes along the geodesics of a data manifold, which is characterized by fusing the multiple previously given graph Laplacians or Hypergraph Laplacians. Then, the proposed regularziers, respectively, are incorporated into the traditional sparse coding framework, which results in two unified objective functions of sparse coding. Alternating optimization is used to optimize the objective functions, and two, novel manifold regularized sparse coding algorithms are presented. The proposed two sparse coding methods learn both the composite manifold and the sparse coding jointly, and it is fully automatic for learning the graph hyper-parameters in the manifold learning. Image clustering tests on real world datasets demonstrated that the proposed sparse coding methods are superior to the state-of-the-art methods. The ever-increasing size of modern datasets combined with the difficulty of obtaining label information has made semi-supervised learning of significant practical importance in modern machine learning applications. Compared with supervised learning, the key difficulty in semi-supervised learning is how to make full use of the unlabeled data. In order to utilize manifold information provided by unlabeled data, we propose a novel regularization called the tangent-normal adversarial regularization, which is composed by two parts. The two terms complement with each other and jointly enforce the smoothness along two different directions that are crucial for semi-supervised learning. One is applied along the tangent space of the data manifold, aiming to enforce local invariance of the classifier on the manifold, while the other is performed on the normal space orthogonal to the tangent space, intending to impose robustness on the classifier against the noise causing the observed data deviating from the underlying data manifold. Both of the two regularizers are achieved by the strategy of virtual adversarial training. Our method has achieved state-of-the-art performance on semi-supervised learning tasks on both artificial dataset and FashionMNIST dataset. In this paper, we study a family of semisupervised learning algorithms for “aligning” different data sets that are characterized by the same underlying manifold. The optimizations of these algorithms are based on graphs that provide a discretized approximation to the manifold. Partial alignments of the data sets—obtained from prior knowledge of their manifold structure or from pairwise correspondences of subsets of labeled examples— are completed by integrating supervised signals with unsupervised frameworks for manifold learning. As an illustration of this semisupervised setting, we show how to learn mappings between different data sets of images that are parameterized by the same underlying modes of variability (e.g., pose and viewing angle). The curse of dimensionality in these problems is overcome by exploiting the low dimensional structure of image manifolds. In this paper, we address the problem of 3D object instance recognition and pose estimation of localized objects in cluttered environments using convolutional neural networks. Inspired by the descriptor learning approach of , we propose a method that introduces the dynamic margin in the manifold learning triplet loss function. Such a loss function is designed to map images of different objects under different poses to a lower-dimensional, similarity-preserving descriptor space on which efficient nearest neighbor search algorithms can be applied. Introducing the dynamic margin allows for faster training times and better accuracy of the resulting low-dimensional manifolds. Furthermore, we contribute the following: adding in-plane rotations (ignored by the baseline method) to the training, proposing new background noise types that help to better mimic realistic scenarios and improve accuracy with respect to clutter, adding surface normals as another powerful image modality representing an object surface leading to better performance than merely depth, and finally implementing an efficient online batch generation that allows for better variability during the training phase. We perform an exhaustive evaluation to demonstrate the effects of our contributions. Additionally, we assess the performance of the algorithm on the large BigBIRD dataset to demonstrate good scalability properties of the pipeline with respect to the number of models. In this work, we propose the use of large set of unlabeled images as a source of regularization data for learning robust visual representation. Given a visual model trained by a labeled dataset in a supervised fashion, we augment our training samples by incorporating large number of unlabeled data and train a semi-supervised model. We demonstrate that our proposed learning approach leverages an abundance of unlabeled images and boosts the visual recognition performance which alleviates the need to rely on large labeled datasets for learning robust representation. To increment the number of image instances needed to learn robust visual models in our approach, each labeled image propagates its label to its nearest unlabeled image instances. These retrieved unlabeled images serve as local perturbations of each labeled image to perform Visual Self-Regularization (VISER). To retrieve such visual self regularizers, we compute the cosine similarity in a semantic space defined by the penultimate layer in a fully convolutional neural network. We use the publicly available Yahoo Flickr Creative Commons 100M dataset as the source of our unlabeled image set and propose a distributed approximate nearest neighbor algorithm to make retrieval practical at that scale. Using the labeled instances and their regularizer samples we show that we significantly improve object categorization and localization performance on the MS COCO and Visual Genome datasets where objects appear in context. We propose an automatic approximation of the intrinsic manifold for general semi-supervised learning problems. Unfortunately, it is not trivial to define an optimization function to obtain optimal hyperparameters. Usually, pure cross-validation is considered but it does not necessarily scale up. A second problem derives from the suboptimality incurred by discrete grid search and overfitting problems. As a consequence, we developed an ensemble manifold regularization (EMR) framework to approximate the intrinsic manifold by combining several initial guesses. Algorithmically, we designed EMR very carefully so that it (a) learns both the composite manifold and the semi-supervised classifier jointly; (b) is fully automatic for learning the intrinsic manifold hyperparameters implicitly; (c) is conditionally optimal for intrinsic manifold approximation under a mild and reasonable assumption; and (d) is scalable for a large number of candidate manifold hyperparameters, from both time and space perspectives. Extensive experiments over both synthetic and real datasets show the effectiveness of the proposed framework. In this letter, we propose a novel low-rank-represen-tation (LRR) based manifold-regularization approach for zero-shot learning (ZSL). Most existing regularization-based ZSL approaches perform the alignment between visual feature space and semantic space based on the affinity matrix constructed from the test instances. The affinity matrix plays a significant role in exploiting the manifold structures of visual feature space, hence we propose to use the LRR to guide the affinity-matrix construction by exploring the subspace structures of data. Considering the locality and similarity information among data, we incorporate a Laplacian regularization term to the LRR framework to ensure that the learned affinity matrix can capture the local geometric structures in data. We also explicitly impose the nonnegative sparse constraint on the affinity matrix to facilitate the learning of local manifold structures. Moreover, we use an effective manifold-regularization methodology to learn discriminative semantic representations of test instances, leading to significant improvements in classification performance over the unseen classes. Extensive experiments on three benchmark datasets demonstrate that the proposed approach outperforms the state of the arts. In this paper, we propose a novel manifold alignment method by learning the underlying common manifold with supervision of corresponding data pairs from different observation sets. Different from the previous algorithms of semi-supervised manifold alignment, our method learns the explicit corresponding projections from each original observation space to the common embedding space everywhere. Benefiting from this property, our method could process new test data directly rather than re-alignment. Furthermore, our approach doesn’t have any assumption on the data structures, thus it could handle more complex cases and get better results compared with previous work. In the proposed algorithm, manifold alignment is formulated as a minimization problem with proper constraints, which could be solved in an analytical manner with closed-form solution. Experimental results on pose manifold alignment of different objects and faces demonstrate the effectiveness of our proposed method. We consider semi-supervised regression when the predictor variables are drawn from an unknown manifold. A simple two step approach to this problem is to: (i) estimate the manifold geodesic distance between any pair of points using both the labeled and unlabeled instances; and (ii) apply a k nearest neighbor regressor based on these distance estimates. We prove that given sufficiently many unlabeled points, this simple method of geodesic kNN regression achieves the optimal finite-sample minimax bound on the mean squared error, as if the manifold were known. Furthermore, we show how this approach can be efficiently implemented, requiring only O(k N log N) operations to estimate the regression function at all N labeled and unlabeled points. We illustrate this approach on two datasets with a manifold structure: indoor localization using WiFi fingerprints and facial pose estimation. In both cases, geodesic kNN is more accurate and much faster than the popular Laplacian eigenvector regressor. The regularization principals [31] lead approximation schemes to deal with various learning problems, e.g., the regularization of the norm in a reproducing kernel Hilbert space for the ill-posed problem. In this paper, we present a family of subspace learning algorithms based on a new form of regularization, which transfers the knowledge gained in training samples to testing samples. In particular, the new regularization minimizes the Bregman divergence between the distribution of training samples and that of testing samples in the selected subspace, so it boosts the performance when training and testing samples are not independent and identically distributed. To test the effectiveness of the proposed regularization, we introduce it to popular subspace learning algorithms, e.g., principal components analysis (PCA) for cross-domain face modeling; and Fisher's linear discriminant analysis (FLDA), locality preserving projections (LPP), marginal Fisher's analysis (MFA), and discriminative locality alignment (DLA) for cross-domain face recognition and text categorization. Finally, we present experimental evidence on both face image data sets and text data sets, suggesting that the proposed Bregman divergence-based regularization is effective to deal with cross-domain learning problems. The key task in graph-oriented learning is con-structing an informative graph to model the geometrical and discriminant structure of a data manifold. Since traditional graph construction methods are sensitive to noise and less datum-adaptive to changes in density, a new graph construction method so-calledl1-Graph has been proposed [1] recently. A graph construction method needs to have two important properties: sparsity and locality. However, the l1-Graph is strong in sparsity property, but weak in locality. In order to overcome such limitation, we propose a new method of constructing an informative graph using automatic group sparse regularization based on the work ofl1-Graph, which is called as group sparse graph (GroupSp-Graph). The newly developed GroupSp-Graph has the same noise-insensitive property asl1-Graph, and also can successively preserve the group and local information in the graph. In other words, the proposed group sparse graph has both properties of sparsity and locality simultaneously. Furthermore, we integrate the proposed graph with several graph-oriented learning algorithms: spectral em-bedding, spectral clustering, subspace learning and manifold regularized non-negative matrix factorization. The empirical studies on benchmark data sets show that the proposed algo-rithms achieve considerable improvement over classic graph constructing methods and the l1-Graph method in various learning task. The classic Generative Adversarial Net and its variants can be roughly categorized into two large families: the unregularized versus regularized GANs. By relaxing the non-parametric assumption on the discriminator in the classic GAN, the regularized GANs have better generalization ability to produce new samples drawn from the real distribution. It is well known that the real data like natural images are not uniformly distributed over the whole data space. Instead, they are often restricted to a low-dimensional manifold of the ambient space. Such a manifold assumption suggests the distance over the manifold should be a better measure to characterize the distinct between real and fake samples. Thus, we define a pullback operator to map samples back to their data manifold, and a manifold margin is defined as the distance between the pullback representations to distinguish between real and fake samples and learn the optimal generators. We justify the effectiveness of the proposed model both theoretically and empirically. Previous efforts in hashing intend to preserve data variance or pairwise affinity, but neither is adequate in capturing the manifold structures hidden in most visual data. In this paper, we tackle this problem by reconstructing the locally linear structures of manifolds in the binary Hamming space, which can be learned by locality-sensitive sparse coding. We cast the problem as a joint minimization of reconstruction error and quantization loss, and show that, despite its NP-hardness, a local optimum can be obtained efficiently via alternative optimization. Our method distinguishes itself from existing methods in its remarkable ability to extract the nearest neighbors of the query from the same manifold, instead of from the ambient space. On extensive experiments on various image benchmarks, our results improve previous state-of-the-art by 28-74 typically, and 627 on the Yale face data. Many applications in computer vision and pattern recognition involve drawing inferences on certain manifold-valued parameters. In order to develop accurate inference algorithms on these manifolds we need to a) understand the geometric structure of these manifolds b) derive appropriate distance measures and c) develop probability distribution functions (pdf) and estimation techniques that are consistent with the geometric structure of these manifolds. In this paper, we consider two related manifolds - the Stiefel manifold and the Grassmann manifold, which arise naturally in several vision applications such as spatio-temporal modeling, affine invariant shape analysis, image matching and learning theory. We show how accurate statistical characterization that reflects the geometry of these manifolds allows us to design efficient algorithms that compare favorably to the state of the art in these very different applications. In particular, we describe appropriate distance measures and parametric and non-parametric density estimators on these manifolds. These methods are then used to learn class conditional densities for applications such as activity recognition, video based face recognition and shape classification. a b s t r a c t Constructing an informative and discriminative graph plays an important role in various pattern recogni- tion tasks such as clustering and classification. Among the existing graph-based learning models, low-rank representation (LRR) is a very competitive one, which has been extensively employed in spectral cluster- ing and semi-supervised learning (SSL). In SSL, the graph is composed of both labeled and unlabeled sam- ples, where the edge weights are calculated based on the LRR coefficients. However, most of existing LRR related approaches fail to consider the geometrical structure of data, which has been shown beneficial for discriminative tasks. In this paper, we propose an enhanced LRR via sparse manifold adaption, termed manifold low-rank representation (MLRR), to learn low-rank data representation. MLRR can explicitly take the data local manifold structure into consideration, which can be identified by the geometric sparsity idea; specifically, the local tangent space of each data point was sought by solving a sparse representation objective. Therefore, the graph to depict the relationship of data points can be built once the manifold information is obtained. We incorporate a regularizer into LRR to make the learned coefficients preserve the geometric constraints revealed in the data space. As a result, MLRR combines both the global informa- tion emphasized by low-rank property and the local information emphasized by the identified manifold structure. Extensive experimental results on semi-supervised classification tasks demonstrate that MLRR is an excellent method in comparison with several state-of-the-art graph construction approaches. In this paper, we introduce a feature-preserving denoising algorithm. It is built on the premise that the underlying surface of a noisy mesh is piecewise smooth, and a sharp feature lies on the intersection of multiple smooth surface regions. A vertex close to a sharp feature is likely to have a neighborhood that includes distinct smooth segments. By defining the consistent subneighborhood as the segment whose geometry and normal orientation most consistent with those of the vertex, we can completely remove the influence from neighbors lying on other segments during denoising. Our method identifies piecewise smooth subneighborhoods using a robust density-based clustering algorithm based on shared nearest neighbors. In our method, we obtain an initial estimate of vertex normals and curvature tensors by robustly fitting a local quadric model. An anisotropic filter based on optimal estimation theory is further applied to smooth the normal field and the curvature tensor field. This is followed by second-order bilateral filtering, which better preserves curvature details and alleviates volume shrinkage during denoising. The support of these filters is defined by the consistent subneighborhood of a vertex. We have applied this algorithm to both generic and CAD models, and sharp features, such as edges and corners, are very well preserved.