Hash learning aims to map multimedia data into Hamming space, in which the data point is represented by low-dimensional binary codes and the similarity relationships are preserved. Despite existing hash learning methods have been effectively used in data retrieval tasks for its merits of low memory cost and high computational efficiency, there still remain two major technical challenges. Firstly, due to the discrete constraints of hash codes, traditional hash methods typically use relaxation strategy to learn real-value features and then quantize them into binary codes through a sign function, resulting in significant quantization errors. Secondly, hash codes are usually low-dimensional, which would be inadequate to preserve either the information of each data point or the relationship between two. These two challenges would greatly limit the retrieval performance of learned hash codes. To solve these problems, we introduce a novel quantization method called scaling factor quantization to enhance hash learning. Unlike traditional hashing methods, we propose to map the data into two parts, i.e., hash codes and scaling factors, to learn the representative codes for the use of retrieval. Specifically, we design a multi-output branch network structure, i.e., Deep Scaling factor Quantization Network (DSQN) and an iterative training strategy for DSQN to learn the two parts of mapping. Comprehensive experiments conducted on three benchmark datasets demonstrate that the hash codes and scaling factors learned by DSQN significantly improve retrieval accuracy compared to existing hash learning methods.

The performance of k-means clustering is often degenerate when dealing with high-dimensional and noisy scenarios. In this study, an end-to-end robust clustering method with low-rank linear embedding techniques (RCLR) is presented in conjunction with k-means. Sparse coefficients and a space projection matrix can be simultaneously learned. The global structures and local neighborhood properties are well captured in the learning procedures. Both the processes of clustering and dimensionality reduction are realized at the same time. The notions of clustering, dimensionality reduction, low-rank representation, and local property preservation are seamlessly integrated into a unified model. The limitation of error accumulation encountered in the previous two-stage clustering framework involving low-rank representation can be alleviated. This is the first attempt to introduce both the global and local geometrical structures into k-means directly, as well L2,1-norm is used as a basic metric instead of the conventional F-norm to further improve the robustness and interpretation of the model. The superiority of the proposed RCLR method is demonstrated by extensive experiments completed on various well-known benchmark datasets.

Shadowed sets provide a meaningful description of information granules by abstracting the corresponding fuzzy sets into three categories: full acceptance, full rejection, and uncertain (represented by shadows). One of the main motivating points to derive shadowed sets from fuzzy sets is the determination and explanation of the separation thresholds based on a specific optimization mechanism. The available optimization objective functions are mainly discussed on semantic interpretations and their mathematical properties; constructive algorithms for optimal solutions have rarely been reported. In this paper, the continuous and convex properties of Pedrycz's optimization objective function to construct shadowed sets, as well as the existence and uniqueness of solution points, are analyzed in detail. It is demonstrated that different approximation region partitions would be generated even under the same optimization model, which requires further criteria to make the constructed shadowed sets well-defined. To address this limitation, the notions of passive and active constrained shadowed sets are introduced. A fast algorithm to obtain the proposed constrained shadowed sets is also designed based on the analyzed mathematical properties. Its performance is then illustrated by some typical fuzzy sets and some real data from the UCI repository.