Recommendation system is a common demand in daily life and matrix completion is a widely adopted technique for this task. However, most matrix completion methods lack semantic interpretation and usually result in weak-semantic recommendations. To this end, this paper proposes a @math emantic @math nalysis approach for @math ecommendation systems @math , which applies a two-level hierarchical generative process that assigns semantic properties and categories for user and item. @math learns semantic representations of users items merely from user ratings on items, which offers a new path to recommendation by semantic matching with the learned representations. Extensive experiments demonstrate @math outperforms other state-of-the-art baselines substantially.
is formulated as a regression problem, such as regression for graph GRALS @cite_24 , blind regression @cite_21 , Riemannian manifold based regression @cite_13 , and others @cite_8 .
@cite_24 @cite_21 @cite_13 @cite_8
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We study the problem of discovering a manifold that best preserves information relevant to a nonlinear regression. Solving this problem involves extending and uniting two threads of research. On the one hand, the literature on sufficient dimension reduction has focused on methods for finding the best linear subspace for nonlinear regression; we extend this to manifolds. On the other hand, the literature on manifold learning has focused on unsupervised dimensionality reduction; we extend this to the supervised setting. Our approach to solving the problem involves combining the machinery of kernel dimension reduction with Laplacian eigenmaps. Specifically, we optimize cross-covariance operators in kernel feature spaces that are induced by the normalized graph Laplacian. The result is a highly flexible method in which no strong assumptions are made on the regression function or on the distribution of the covariates. We illustrate our methodology on the analysis of global temperature data and image manifolds. In this paper, we treat the problem of continuous pose estimation for object categories as a regression problem on the basis of only 2D training information. While regression is a natural framework for continuous problems, regression methods so far achieved inferior results with respect to 3D-based and 2D-based classification-and-refinement approaches. This may be attributed to their weakness to high intra-class variability as well as to noisy matching procedures and lack of geometrical constraints. We propose to apply regression to Fisher-encoded vectors computed from large cells by learning an array of Fisher regressors. Fisher encoding makes our algorithm flexible to variations in class appearance, while the array structure permits to indirectly introduce spatial context information in the approach. We formulate our problem as a MAP inference problem, where the likelihood function is composed of a generative term based on the prediction error generated by the ensemble of Fisher regressors as well as a discriminative term based on SVM classifiers. We test our algorithm on three publicly available datasets that envisage several difficulties, such as high intra-class variability, truncations, occlusions, and motion blur, obtaining state-of-the-art results. 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. 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.