Recent years have witnessed extensive attention in binary code learning, a.k.a. hashing, for nearest neighbor search problems. It has been seen that high-dimensional data points can be quantized into binary codes to give an efficient similarity approximation via Hamming distance. Among existing schemes, ranking-based hashing is recent promising that targets at preserving ordinal relations of ranking in the Hamming space to minimize retrieval loss. However, the size of the ranking tuples, which shows the ordinal relations, is quadratic or cubic to the size of training samples. By given a large-scale training data set, it is very expensive to embed such ranking tuples in binary code learning. Besides, it remains a dificulty to build ranking tuples efficiently for most ranking-preserving hashing, which are deployed over an ordinal graph-based setting. To handle these problems, we propose a novel ranking-preserving hashing method, dubbed Ordinal Constraint Hashing (OCH), which efficiently learns the optimal hashing functions with a graph-based approximation to embed the ordinal relations. The core idea is to reduce the size of ordinal graph with ordinal constraint projection, which preserves the ordinal relations through a small data set (such as clusters or random samples). In particular, to learn such hash functions effectively, we further relax the discrete constraints and design a specific stochastic gradient decent algorithm for optimization. Experimental results on three large-scale visual search benchmark datasets, i.e. LabelMe, Tiny100K and GIST1M, show that the proposed OCH method can achieve superior performance over the state-of-the-arts approaches.
@cite_1 aims to learn a set of hash functions as Eq. . Such functions can preserve the ordinal relations between @math and @math , where @math is the dissimilarity between the @math -th and the @math -th item. Its goal is to make sure the ordinal relation can be preserved in the produced Hamming space: @math To embed such ordinal relations, OEH first constructs a directed unweighted ordinal graph @math , where each node @math is the dissimilar degree @math , and each directed edge is defined via @math . Then the objective function is to minimize the inconsistency between the given ordinal relation graph and the ones generated from the corresponding hash codes. At last, by using the landmark-based ordinal graph, the quartic ordinal relation is transformed to the triplet ordinal relation, which transforms the target of OEH to @math where @math and @math are the landmark points.
@cite_1
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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. Binary code learning, a.k.a., hashing, has been recently popular due to its high efficiency in large-scale similarity search and recognition. It typically maps high-dimensional data points to binary codes, where data similarity can be efficiently computed via rapid Hamming distance. Most existing unsupervised hashing schemes pursue binary codes by reducing the quantization error from an original real-valued data space to a resulting Hamming space. On the other hand, most existing supervised hashing schemes constrain binary code learning to correlate with pairwise similarity labels. However, few methods consider ordinal relations in the binary code learning process, which serve as a very significant cue to learn the optimal binary codes for similarity search. In this paper, we propose a novel hashing scheme, dubbed Ordinal Embedding Hashing (OEH), which embeds given ordinal relations among data points to learn the ranking-preserving binary codes. The core idea is to construct a directed unweighted graph to capture the ordinal relations, and then train the hash functions using this ordinal graph to preserve the permutation relations in the Hamming space. To learn such hash functions effectively, we further relax the discrete constraints and design a stochastic gradient decent algorithm to obtain the optimal solution. Experimental results on two large-scale benchmark datasets demonstrate that the proposed OEH method can achieve superior performance over the state-of-the-arts approaches. At last, the evaluation on query by humming dataset demonstrates the OEH also has good performance for music retrieval by using user's humming or singing. We consider succinct representations of labeled ordinal trees that support a rich set of operations. Our new representations support a much broader collection of operations than previous work. In our approach, labels of nodes are stored in a preorder label sequence, which can be compressed using any succinct representation of strings that supports @math access , @math rank and @math select operations. Thus, we present a framework for succinct representations of labeled ordinal trees that is able to handle large alphabets. This answers an open problem presented by , which asks for representations of labeled ordinal trees that remain space-efficient for large alphabets. We further extend our work and present the first succinct representations for dynamic labeled ordinal trees that support several label-based operations including finding the level ancestor with a given label. Edge and vertex connectivity are fundamental concepts in graph theory. While they have been thoroughly studied in the case of undirected graphs, surprisingly not much has been investigated for directed graphs. In this paper we study @math -edge connectivity problems in directed graphs and, in particular, we consider the computation of the following natural relation: We say that two vertices @math and @math are @math -edge-connected if there are two edge-disjoint paths from @math to @math and two edge-disjoint paths from @math to @math . This relation partitions the vertices into blocks such that all vertices in the same block are @math -edge-connected. Differently from the undirected case, those blocks do not correspond to the @math -edge-connected components of the graph. We show how to compute this relation in linear time so that we can report in constant time if two vertices are @math -edge-connected. We also show how to compute in linear time a sparse certificate for this relation, i.e., a subgraph of the input graph that has @math edges and maintains the same @math -edge-connected blocks as the input graph.