Many real-world datasets are labeled with natural orders, i.e., ordinal labels. Ordinal regression is a method to predict ordinal labels that finds a wide range of applications in data-rich science domains, such as medical, social and economic sciences. Most existing approaches work well for a single ordinal regression task. However, they ignore the task relatedness when there are multiple related tasks. Multi-task learning (MTL) provides a framework to encode task relatedness, to bridge data from all tasks, and to simultaneously learn multiple related tasks to improve the generalization performance. Even though MTL methods have been extensively studied, there is barely existing work investigating MTL for data with ordinal labels. We tackle multiple ordinal regression problems via sparse and deep multi-task approaches, i.e., two regularized multi-task ordinal regression (RMTOR) models for small datasets and two deep neural networks based multi-task ordinal regression (DMTOR) models for large-scale datasets. The performance of the proposed multi-task ordinal regression models (MTOR) is demonstrated on three real-world medical datasets for multi-stage disease diagnosis. Our experimental results indicate that our proposed MTOR models markedly improve the prediction performance comparing with single-task learning (STL) ordinal regression models.
Ordinal binary decomposition approaches are proposed to decompose the ordinal labels into several binary ones that are then estimated by multiple models @cite_50 @cite_54 . For example, @cite_50 transforms the data from @math -classes ordinal problems to @math ordered binary classification problems and then they are trained in conjunction with a decision tree learner to encode the ordering of the original ranks, i.e., train @math binary classifiers using C4.5 algorithm. Threshold models are proposed based on the idea of approximating the real value predictor followed with partitioning the real line of ordinal values into segments. During the last decade, the two most popular threshold models are support vector machines (SVM) models @cite_45 @cite_44 @cite_56 @cite_22 and generalized linear models for ordinal regression @cite_46 @cite_23 @cite_17 @cite_32 ; the former is to find the hyperplane that separates the segments by maximizing margin using the loss and the latter is to predict the ordinal labels by maximizing the likelihood given the training data.
@cite_22 @cite_54 @cite_32 @cite_56 @cite_44 @cite_45 @cite_50 @cite_23 @cite_46 @cite_17
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We present a reduction framework from ordinal regression to binary classification based on extended examples. The framework consists of three steps: extracting extended examples from the original examples, learning a binary classifier on the extended examples with any binary classification algorithm, and constructing a ranking rule from the binary classifier. A weighted 0 1 loss of the binary classifier would then bound the mislabeling cost of the ranking rule. Our framework allows not only to design good ordinal regression algorithms based on well-tuned binary classification approaches, but also to derive new generalization bounds for ordinal regression from known bounds for binary classification. In addition, our framework unifies many existing ordinal regression algorithms, such as perceptron ranking and support vector ordinal regression. When compared empirically on benchmark data sets, some of our newly designed algorithms enjoy advantages in terms of both training speed and generalization performance over existing algorithms, which demonstrates the usefulness of our framework. In this letter, we propose two new support vector approaches for ordinal regression, which optimize multiple thresholds to define parallel discriminant hyperplanes for the ordinal scales. Both approaches guarantee that the thresholds are properly ordered at the optimal solution. The size of these optimization problems is linear in the number of training samples. The sequential minimal optimization algorithm is adapted for the resulting optimization problems; it is extremely easy to implement and scales efficiently as a quadratic function of the number of examples. The results of numerical experiments on some benchmark and real-world data sets, including applications of ordinal regression to information retrieval, verify the usefulness of these approaches. Binary code learning, a.k.a. hashing, has been successfully applied to the approximate nearest neighbor search in large-scale image collections. The key challenge lies in reducing the quantization error from the original real-valued feature space to a discrete Hamming space. Recent advances in unsupervised hashing advocate the preservation of ranking information, which is achieved by constraining the binary code learning to be correlated with pairwise similarity. However, few unsupervised methods consider the preservation of ordinal relations in the learning process, which serves as a more basic cue to learn optimal binary codes. In this paper, we propose a novel hashing scheme, termed Ordinal Constraint Hashing (OCH), which embeds the ordinal relation among data points to preserve ranking into binary codes. The core idea is to construct an ordinal graph via tensor product, and then train the hash function over this graph to preserve the permutation relations among data points in the Hamming space. Subsequently, an in-depth acceleration scheme, termed Ordinal Constraint Projection (OCP), is introduced, which approximates the @math -pair ordinal graph by @math -pair anchor-based ordinal graph, and reduce the corresponding complexity from @math to @math ( @math ). Finally, to make the optimization tractable, we further relax the discrete constrains and design a customized stochastic gradient decent algorithm on the Stiefel manifold. Experimental results on serval large-scale benchmarks demonstrate that the proposed OCH method can achieve superior performance over the state-of-the-art approaches. In this paper, we aim to learn a mapping (or embedding) from images to a compact binary space in which Hamming distances correspond to a ranking measure for the image retrieval task. We make use of a triplet loss because this has been shown to be most effective for ranking problems. However, training in previous works can be prohibitively expensive due to the fact that optimization is directly performed on the triplet space, where the number of possible triplets for training is cubic in the number of training examples. To address this issue, we propose to formulate high-order binary codes learning as a multi-label classification problem by explicitly separating learning into two interleaved stages. To solve the first stage, we design a large-scale high-order binary codes inference algorithm to reduce the high-order objective to a standard binary quadratic problem such that graph cuts can be used to efficiently infer the binary code which serve as the label of each training datum. In the second stage we propose to map the original image to compact binary codes via carefully designed deep convolutional neural networks (CNNs) and the hashing function fitting can be solved by training binary CNN classifiers. An incremental interleaved optimization strategy is proffered to ensure that these two steps are interactive with each other during training for better accuracy. We conduct experiments on several benchmark datasets, which demonstrate both improved training time (by as much as two orders of magnitude) as well as producing state-of-the-art hashing for various retrieval tasks.