Multi-Output Dependence (MOD) learning is a generalization of standard classification problems that allows for multiple outputs that are dependent on each other. A primary issue that arises in the context of MOD learning is that for any given input pattern there can be multiple correct output patterns. This changes the learning task from function approximation to relation approximation. Previous algorithms do not consider this problem, and thus cannot be readily applied to MOD problems. To perform MOD learning, we introduce the Hierarchical Multi-Output Nearest Neighbor model (HMONN) that employs a basic learning model for each output and a modified nearest neighbor approach to refine the initial results.
While MOD learning is relation approximation, this should not be confused with relational learning. Statistical Relational Learning @cite_7 @cite_9 and Multi-Relational Learning @cite_8 both handle relational data, not relation approximation. These relational learning models learn a function from relational data and handle specially formatted and structured data.
@cite_9 @cite_7 @cite_8
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The problem of cross-modal retrieval, e.g., using a text query to search for images and vice-versa, is considered in this paper. A novel model involving correspondence autoencoder (Corr-AE) is proposed here for solving this problem. The model is constructed by correlating hidden representations of two uni-modal autoencoders. A novel optimal objective, which minimizes a linear combination of representation learning errors for each modality and correlation learning error between hidden representations of two modalities, is used to train the model as a whole. Minimization of correlation learning error forces the model to learn hidden representations with only common information in different modalities, while minimization of representation learning error makes hidden representations are good enough to reconstruct input of each modality. A parameter @math is used to balance the representation learning error and the correlation learning error. Based on two different multi-modal autoencoders, Corr-AE is extended to other two correspondence models, here we called Corr-Cross-AE and Corr-Full-AE. The proposed models are evaluated on three publicly available data sets from real scenes. We demonstrate that the three correspondence autoencoders perform significantly better than three canonical correlation analysis based models and two popular multi-modal deep models on cross-modal retrieval tasks. The aim of statistical relational learning is to learn statistical models from relational or graph-structured data. Three main statistical relational learning paradigms include weighted rule learning, random walks on graphs, and tensor factorization. These paradigms have been mostly developed and studied in isolation for many years, with few works attempting at understanding the relationship among them or combining them. In this paper, we study the relationship between the path ranking algorithm (PRA), one of the most well-known relational learning methods in the graph random walk paradigm, and relational logistic regression (RLR), one of the recent developments in weighted rule learning. We provide a simple way to normalize relations and prove that relational logistic regression using normalized relations generalizes the path ranking algorithm. This result provides a better understanding of relational learning, especially for the weighted rule learning and graph random walk paradigms. It opens up the possibility of using the more flexible RLR rules within PRA models and even generalizing both by including normalized and unnormalized relations in the same model. We propose a modular framework for multi-relational learning via tensor decomposition. In our learning setting, the training data contains multiple types of relationships among a set of objects, which we represent by a sparse three-mode tensor. The goal is to predict the values of the missing entries. To do so, we model each relationship as a function of a linear combination of latent factors. We learn this latent representation by computing a low-rank tensor decomposition, using quasi-Newton optimization of a weighted objective function. Sparsity in the observed data is captured by the weighted objective, leading to improved accuracy when training data is limited. Exploiting sparsity also improves efficiency, potentially up to an order of magnitude over unweighted approaches. In addition, our framework accommodates arbitrary combinations of smooth, task-specific loss functions, making it better suited for learning different types of relations. For the typical cases of real-valued functions and binary relations, we propose several loss functions and derive the associated parameter gradients. We evaluate our method on synthetic and real data, showing significant improvements in both accuracy and scalability over related factorization techniques. Relational learning is concerned with predicting unknown values of a relation, given a database of entities and observed relations among entities. An example of relational learning is movie rating prediction, where entities could include users, movies, genres, and actors. Relations encode users' ratings of movies, movies' genres, and actors' roles in movies. A common prediction technique given one pairwise relation, for example a #users x #movies ratings matrix, is low-rank matrix factorization. In domains with multiple relations, represented as multiple matrices, we may improve predictive accuracy by exploiting information from one relation while predicting another. To this end, we propose a collective matrix factorization model: we simultaneously factor several matrices, sharing parameters among factors when an entity participates in multiple relations. Each relation can have a different value type and error distribution; so, we allow nonlinear relationships between the parameters and outputs, using Bregman divergences to measure error. We extend standard alternating projection algorithms to our model, and derive an efficient Newton update for the projection. Furthermore, we propose stochastic optimization methods to deal with large, sparse matrices. Our model generalizes several existing matrix factorization methods, and therefore yields new large-scale optimization algorithms for these problems. Our model can handle any pairwise relational schema and a wide variety of error models. We demonstrate its efficiency, as well as the benefit of sharing parameters among relations.