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Showing papers on "Metric (mathematics) published in 2012"


Proceedings ArticleDOI
16 Jun 2012
TL;DR: This paper introduces a simple though effective strategy to learn a distance metric from equivalence constraints, based on a statistical inference perspective, which is orders of magnitudes faster than comparable methods.
Abstract: In this paper, we raise important issues on scalability and the required degree of supervision of existing Mahalanobis metric learning methods. Often rather tedious optimization procedures are applied that become computationally intractable on a large scale. Further, if one considers the constantly growing amount of data it is often infeasible to specify fully supervised labels for all data points. Instead, it is easier to specify labels in form of equivalence constraints. We introduce a simple though effective strategy to learn a distance metric from equivalence constraints, based on a statistical inference perspective. In contrast to existing methods we do not rely on complex optimization problems requiring computationally expensive iterations. Hence, our method is orders of magnitudes faster than comparable methods. Results on a variety of challenging benchmarks with rather diverse nature demonstrate the power of our method. These include faces in unconstrained environments, matching before unseen object instances and person re-identification across spatially disjoint cameras. In the latter two benchmarks we clearly outperform the state-of-the-art.

1,762 citations


Book ChapterDOI
05 Nov 2012
TL;DR: Experiments on the VIPeR dataset and the dataset show that the proposed transferred metric learning significantly outperforms directly matching visual features or using a single generic metric learned from the whole training set.
Abstract: Human reidentification is to match persons observed in non-overlapping camera views with visual features for inter-camera tracking. The ambiguity increases with the number of candidates to be distinguished. Simple temporal reasoning can simplify the problem by pruning the candidate set to be matched. Existing approaches adopt a fixed metric for matching all the subjects. Our approach is motivated by the insight that different visual metrics should be optimally learned for different candidate sets. We tackle this problem under a transfer learning framework. Given a large training set, the training samples are selected and reweighted according to their visual similarities with the query sample and its candidate set. A weighted maximum margin metric is online learned and transferred from a generic metric to a candidate-set-specific metric. The whole online reweighting and learning process takes less than two seconds per candidate set. Experiments on the VIPeR dataset and our dataset show that the proposed transferred metric learning significantly outperforms directly matching visual features or using a single generic metric learned from the whole training set.

604 citations


Proceedings Article
03 Dec 2012
TL;DR: A new loss-augmented inference algorithm that is quadratic in the code length and inspired by latent structural SVMs is developed, showing strong retrieval performance on CIFAR-10 and MNIST, with promising classification results using no more than kNN on the binary codes.
Abstract: Motivated by large-scale multimedia applications we propose to learn mappings from high-dimensional data to binary codes that preserve semantic similarity. Binary codes are well suited to large-scale applications as they are storage efficient and permit exact sub-linear kNN search. The framework is applicable to broad families of mappings, and uses a flexible form of triplet ranking loss. We overcome discontinuous optimization of the discrete mappings by minimizing a piecewise-smooth upper bound on empirical loss, inspired by latent structural SVMs. We develop a new loss-augmented inference algorithm that is quadratic in the code length. We show strong retrieval performance on CIFAR-10 and MNIST, with promising classification results using no more than kNN on the binary codes.

562 citations


Book ChapterDOI
07 Oct 2012
TL;DR: This paper proposes to learn a metric from pairs of samples from different cameras, so that even less sophisticated features describing color and texture information are sufficient for finally getting state-of-the-art classification results.
Abstract: Matching persons across non-overlapping cameras is a rather challenging task. Thus, successful methods often build on complex feature representations or sophisticated learners. A recent trend to tackle this problem is to use metric learning to find a suitable space for matching samples from different cameras. However, most of these approaches ignore the transition from one camera to the other. In this paper, we propose to learn a metric from pairs of samples from different cameras. In this way, even less sophisticated features describing color and texture information are sufficient for finally getting state-of-the-art classification results. Moreover, once the metric has been learned, only linear projections are necessary at search time, where a simple nearest neighbor classification is performed. The approach is demonstrated on three publicly available datasets of different complexity, where it can be seen that state-of-the-art results can be obtained at much lower computational costs.

472 citations


Proceedings ArticleDOI
16 Jun 2012
TL;DR: A novel discriminative learning approach to image set classification by modeling the image set with its natural second-order statistic, i.e. covariance matrix, which shows the superiority of this method over state-of-the-art ones in both accuracy and efficiency, but also its stability to two real challenges: noisy set data and varying set size.
Abstract: We propose a novel discriminative learning approach to image set classification by modeling the image set with its natural second-order statistic, i.e. covariance matrix. Since nonsingular covariance matrices, a.k.a. symmetric positive definite (SPD) matrices, lie on a Riemannian manifold, classical learning algorithms cannot be directly utilized to classify points on the manifold. By exploring an efficient metric for the SPD matrices, i.e., Log-Euclidean Distance (LED), we derive a kernel function that explicitly maps the covariance matrix from the Riemannian manifold to a Euclidean space. With this explicit mapping, any learning method devoted to vector space can be exploited in either its linear or kernel formulation. Linear Discriminant Analysis (LDA) and Partial Least Squares (PLS) are considered in this paper for their feasibility for our specific problem. We further investigate the conventional linear subspace based set modeling technique and cast it in a unified framework with our covariance matrix based modeling. The proposed method is evaluated on two tasks: face recognition and object categorization. Extensive experimental results show not only the superiority of our method over state-of-the-art ones in both accuracy and efficiency, but also its stability to two real challenges: noisy set data and varying set size.

445 citations


Journal ArticleDOI
TL;DR: A new performance indicator, Δp, is defined, which can be viewed as an “averaged Hausdorff distance” between the outcome set and the Pareto front and which is composed of (slight modifications of) the well-known indicators generational distance (GD) and inverted generational Distance (IGD).
Abstract: The Hausdorff distance dH is a widely used tool to measure the distance between different objects in several research fields. Possible reasons for this might be that it is a natural extension of the well-known and intuitive distance between points and/or the fact that dH defines in certain cases a metric in the mathematical sense. In evolutionary multiobjective optimization (EMO) the task is typically to compute the entire solution set-the so-called Pareto set-respectively its image, the Pareto front. Hence, dH should, at least at first sight, be a natural choice to measure the performance of the outcome set in particular since it is related to the terms spread and convergence as used in EMO literature. However, so far, dH does not find the general approval in the EMO community. The main reason for this is that dH penalizes single outliers of the candidate set which does not comply with the use of stochastic search algorithms such as evolutionary strategies. In this paper, we define a new performance indicator, Δp, which can be viewed as an “averaged Hausdorff distance” between the outcome set and the Pareto front and which is composed of (slight modifications of) the well-known indicators generational distance (GD) and inverted generational distance (IGD). We will discuss theoretical properties of Δp (as well as for GD and IGD) such as the metric properties and the compliance with state-of-theart multiobjective evolutionary algorithms (MOEAs), and will further on demonstrate by empirical results the potential of Δp as a new performance indicator for the evaluation of MOEAs.

425 citations


Journal ArticleDOI
16 Jun 2012
TL;DR: This paper proposes a new neighborhood repulsed metric learning (NRML) method for kinship verification, and proposes a multiview NRM-L method to seek a common distance metric to make better use of multiple feature descriptors to further improve the verification performance.
Abstract: Kinship verification from facial images is an interesting and challenging problem in computer vision, and there are very limited attempts on tackle this problem in the literature. In this paper, we propose a new neighborhood repulsed metric learning (NRML) method for kinship verification. Motivated by the fact that interclass samples (without a kinship relation) with higher similarity usually lie in a neighborhood and are more easily misclassified than those with lower similarity, we aim to learn a distance metric under which the intraclass samples (with a kinship relation) are pulled as close as possible and interclass samples lying in a neighborhood are repulsed and pushed away as far as possible, simultaneously, such that more discriminative information can be exploited for verification. To make better use of multiple feature descriptors to extract complementary information, we further propose a multiview NRML (MNRML) method to seek a common distance metric to perform multiple feature fusion to improve the kinship verification performance. Experimental results are presented to demonstrate the efficacy of our proposed methods. Finally, we also test human ability in kinship verification from facial images and our experimental results show that our methods are comparable to that of human observers.

423 citations


01 Sep 2012
TL;DR: The Minimum Rank with Hysteresis Objective Function (MRHOF) is described, an Objective Function that selects routes that minimize a metric, while using hysteresi to reduce churn in response to small metric changes.
Abstract: The Routing Protocol for Low-Power and Lossy Networks (RPL) constructs routes by using Objective Functions that optimize or constrain the routes it selects and uses. This specification describes the Minimum Rank with Hysteresis Objective Function (MRHOF), an Objective Function that selects routes that minimize a metric, while using hysteresis to reduce churn in response to small metric changes. MRHOF works with additive metrics along a route, and the metrics it uses are determined by the metrics that the RPL Destination Information Object (DIO) messages advertise. [STANDARDS-TRACK]

401 citations


Proceedings ArticleDOI
01 Jul 2012
TL;DR: A status-age timeliness metric is formulated and the region of feasible average status ages for a pair of updating sources is found and the existence of an optimal rate at which a source should generate its updates is shown.
Abstract: We examine multiple independent sources providing status updates to a monitor through a first-come-first-served M/M/1 queue. We formulate a status-age timeliness metric and find the region of feasible average status ages for a pair of updating sources. In the presence of interfering traffic with a given offered load, we show the existence of an optimal rate at which a source should generate its updates.

369 citations


Journal Article
TL;DR: A novel metric learning approach called DML-eig is introduced which is shown to be equivalent to a well-known eigen value optimization problem called minimizing the maximal eigenvalue of a symmetric matrix.
Abstract: The main theme of this paper is to develop a novel eigenvalue optimization framework for learning a Mahalanobis metric. Within this context, we introduce a novel metric learning approach called DML-eig which is shown to be equivalent to a well-known eigenvalue optimization problem called minimizing the maximal eigenvalue of a symmetric matrix (Overton, 1988; Lewis and Overton, 1996). Moreover, we formulate LMNN (Weinberger et al., 2005), one of the state-of-the-art metric learning methods, as a similar eigenvalue optimization problem. This novel framework not only provides new insights into metric learning but also opens new avenues to the design of efficient metric learning algorithms. Indeed, first-order algorithms are developed for DML-eig and LMNN which only need the computation of the largest eigenvector of a matrix per iteration. Their convergence characteristics are rigorously established. Various experiments on benchmark data sets show the competitive performance of our new approaches. In addition, we report an encouraging result on a difficult and challenging face verification data set called Labeled Faces in the Wild (LFW).

348 citations


Proceedings ArticleDOI
09 Sep 2012
TL;DR: This paper proposes a new CF approach, Collaborative Less-is-More Filtering (CLiMF), where the model parameters are learned by directly maximizing the Mean Reciprocal Rank (MRR), which is a well-known information retrieval metric for measuring the performance of top-k recommendations.
Abstract: In this paper we tackle the problem of recommendation in the scenarios with binary relevance data, when only a few (k) items are recommended to individual users. Past work on Collaborative Filtering (CF) has either not addressed the ranking problem for binary relevance datasets, or not specifically focused on improving top-k recommendations. To solve the problem we propose a new CF approach, Collaborative Less-is-More Filtering (CLiMF). In CLiMF the model parameters are learned by directly maximizing the Mean Reciprocal Rank (MRR), which is a well-known information retrieval metric for measuring the performance of top-k recommendations. We achieve linear computational complexity by introducing a lower bound of the smoothed reciprocal rank metric. Experiments on two social network datasets demonstrate the effectiveness and the scalability of CLiMF, and show that CLiMF significantly outperforms a naive baseline and two state-of-the-art CF methods.

Proceedings Article
01 Oct 2012
TL;DR: In this paper, the Fisher information metric is used to enable a hyperbolic structure on the multivariate normal distributions. But it is not a metric that can be used in statistical manifolds.
Abstract: Information geometry is a new mathematical discipline which applies the methodology of differential geometry to statistics. Therefore, families of exponential distributions are considered as embedded manifolds, called statistical manifolds. This includes so important families like the multivariate normal or the gamma distributions. Fisher information — well known in information theory — becomes a metric on statistical manifolds. The Fisher information metric enables a hyperbolic structure on the multivariate normal distributions. Information geometry offers new methods for hypothesis testings, estimation theory or stochastic filtering. These can be used in engineering areas like signal processing or video processing or finance.

Journal ArticleDOI
TL;DR: A sparse neighbor selection scheme for SR reconstruction is proposed that can achieve competitive SR quality compared with other state-of-the-art baselines and develop an extended Robust-SL0 algorithm to simultaneously find the neighbors and to solve the reconstruction weights.
Abstract: Until now, neighbor-embedding-based (NE) algorithms for super-resolution (SR) have carried out two independent processes to synthesize high-resolution (HR) image patches. In the first process, neighbor search is performed using the Euclidean distance metric, and in the second process, the optimal weights are determined by solving a constrained least squares problem. However, the separate processes are not optimal. In this paper, we propose a sparse neighbor selection scheme for SR reconstruction. We first predetermine a larger number of neighbors as potential candidates and develop an extended Robust-SL0 algorithm to simultaneously find the neighbors and to solve the reconstruction weights. Recognizing that the k-nearest neighbor (k-NN) for reconstruction should have similar local geometric structures based on clustering, we employ a local statistical feature, namely histograms of oriented gradients (HoG) of low-resolution (LR) image patches, to perform such clustering. By conveying local structural information of HoG in the synthesis stage, the k-NN of each LR input patch is adaptively chosen from their associated subset, which significantly improves the speed of synthesizing the HR image while preserving the quality of reconstruction. Experimental results suggest that the proposed method can achieve competitive SR quality compared with other state-of-the-art baselines.

Journal ArticleDOI
TL;DR: A new psychovisual quality metric of images is proposed based on recent developments in brain theory and neuroscience, particularly the free-energy principle and can measure correctly the visual quality of some model-based image processing algorithms, for which the competing metrics often contradict with viewers' opinions.
Abstract: In this paper, we propose a new psychovisual quality metric of images based on recent developments in brain theory and neuroscience, particularly the free-energy principle. The perception and understanding of an image is modeled as an active inference process, in which the brain tries to explain the scene using an internal generative model. The psychovisual quality is thus closely related to how accurately visual sensory data can be explained by the generative model, and the upper bound of the discrepancy between the image signal and its best internal description is given by the free energy of the cognition process. Therefore, the perceptual quality of an image can be quantified using the free energy. Constructively, we develop a reduced-reference free-energy-based distortion metric (FEDM) and a no-reference free-energy-based quality metric (NFEQM). The FEDM and the NFEQM are nearly invariant to many global systematic deviations in geometry and illumination that hardly affect visual quality, for which existing image quality metrics wrongly predict severe quality degradation. Although with very limited or even without information on the reference image, the FEDM and the NFEQM are highly competitive compared with the full-reference SSIM image quality metric on images in the popular LIVE database. Moreover, FEDM and NFEQM can measure correctly the visual quality of some model-based image processing algorithms, for which the competing metrics often contradict with viewers' opinions.

Journal ArticleDOI
TL;DR: The gray distance is much better than the Minkowski distance at both capturing the proximity relationship (or nearness) of two instances and dealing with mixed attributes, and experimental results show that the GkNN algorithm is much more efficient than existent kNN imputation methods.

Proceedings ArticleDOI
16 Jun 2012
TL;DR: The proposed metric provides more accurate indications of the performance of action recognition algorithms for games and other similar applications since it takes into consideration restrictions related to time and consecutive repetitions.
Abstract: In this paper a novel evaluation framework for measuring the performance of real-time action recognition methods is presented. The evaluation framework will extend the time-based event detection metric to model multiple distinct action classes. The proposed metric provides more accurate indications of the performance of action recognition algorithms for games and other similar applications since it takes into consideration restrictions related to time and consecutive repetitions. Furthermore, a new dataset, G3D for real-time action recognition in gaming containing synchronised video, depth and skeleton data is provided. Our results indicate the need of an advanced metric especially designed for games and other similar real-time applications.

Journal ArticleDOI
TL;DR: Vector diffusion maps (VDM) as mentioned in this paper is a generalization of diffusion maps and other nonlinear dimensionality reduction methods, such as LLE, ISOMAP, and Laplacian eigenmaps.
Abstract: We introduce vector diffusion maps (VDM), a new mathematical framework for organizing and analyzing massive high-dimensional data sets, images, and shapes. VDMis a mathematical and algorithmic generalization of diffusion maps and other nonlinear dimensionality reduction methods, such as LLE, ISOMAP, and Laplacian eigenmaps. While existing methods are either directly or indirectly related to the heat kernel for functions over the data, VDM is based on the heat kernel for vector fields. VDM provides tools for organizing complex data sets, embedding them in a low-dimensional space, and interpolating and regressing vector fields over the data. In particular, it equips the data with a metric, which we refer to as the vector diffusion distance. In the manifold learning setup, where the data set is distributed on a low-dimensional manifold embedded in , we prove the relation between VDM and the connection Laplacian operator for vector fields over the manifold. © 2012 Wiley Periodicals, Inc.

Posted Content
TL;DR: In this paper, the authors present metrics for measuring the similarity of states in a finite Markov decision process (MDP) based on the notion of bisimulation, with an aim towards solving discounted infinite horizon reinforcement learning tasks.
Abstract: We present metrics for measuring the similarity of states in a finite Markov decision process (MDP). The formulation of our metrics is based on the notion of bisimulation for MDPs, with an aim towards solving discounted infinite horizon reinforcement learning tasks. Such metrics can be used to aggregate states, as well as to better structure other value function approximators (e.g., memory-based or nearest-neighbor approximators). We provide bounds that relate our metric distances to the optimal values of states in the given MDP.

Proceedings ArticleDOI
01 Sep 2012
TL;DR: This paper proposes a novel face representation based on Local Quantized Patterns that gives state-of-the-art performance without requiring neither a metric learning stage nor a costly labelled training dataset.
Abstract: This paper proposes a novel face representation based on Local Quantized Patterns (LQP). LQP is a generalization of local pattern features that makes use of vector quantization and lookup table to let local pattern features have many more pixels and/or quantization levels without sacrificing simplicity and computational efficiency. Our new LQP face representation not only outperforms any other representation on challenging face datasets but performs equally well in the intensity space and orientation space (obtained by applying gradient or Gabor Filters) and hence is intrinsically robust to illumination variations. Extensive experiments on several challenging face recognition datasets (such as FERET and LFW) show that this representation gives state-of-the-art performance (improving the earlier state-of-the-art by around 3%) without requiring neither a metric learning stage nor a costly labelled training dataset, having the comparison of two faces being made by simply computing the Cosine similarity between their LQP representations in a projected space.

Journal ArticleDOI
TL;DR: In this paper, the fixed point theory in metric-like spaces was initiated and some new fixed point results in partial metric spaces were derived, unifying and generalizing some well-known results in the literature.
Abstract: By a metric-like space, as a generalization of a partial metric space, we mean a pair , where X is a nonempty set and satisfies all of the conditions of a metric except that may be positive for . In this paper, we initiate the fixed point theory in metric-like spaces. As an application, we derive some new fixed point results in partial metric spaces. Our results unify and generalize some well-known results in the literature. MSC:47H10.

01 Sep 2012
TL;DR: In this article, the authors generalized the Wasserstein metric to reaction-diffusion systems with reversible mass-action kinetic and showed that this gradient structure can be generalized to systems including electrostatic interactions and correct energy balance via coupling to the heat equation.
Abstract: In recent years the theory of the Wasserstein metric has opened up new treatments of diffusion equations as gradient systems, where the free energy or entropy take the role of the driving functional and where the space is equipped with the Wasserstein metric. We show on the formal level that this gradient structure can be generalized to reaction–diffusion systems with reversible mass-action kinetic. The metric is constructed using the dual dissipation potential, which is a quadratic functional of all chemical potentials including the mobilities as well as the reaction kinetics. The metric structure is obtained by Legendre transform from the dual dissipation potential.The same ideas extend to systems including electrostatic interactions or a correct energy balance via coupling to the heat equation. We show this by treating the semiconductor equations involving the electron and hole densities, the electrostatic potential, and the temperature. Thus, the models in Albinus et al (2002 Nonlinearity 15 367–83), which stimulated this work, have a gradient structure.

Proceedings ArticleDOI
18 Sep 2012
TL;DR: This paper addresses the first problem by learning the transition from one camera to the other by learning a Mahalanobis metric using pairs of labeled samples from different cameras, which additionally provides much better generalization properties.
Abstract: Recognizing persons over a system of disjunct cameras is a hard task for human operators and even harder for automated systems In particular, realistic setups show difficulties such as different camera angles or different camera properties Additionally, also the appearance of exactly the same person can change dramatically due to different views (eg, frontal/back) of carried objects In this paper, we mainly address the first problem by learning the transition from one camera to the other This is realized by learning a Mahalanobis metric using pairs of labeled samples from different cameras Building on the ideas of Large Margin Nearest Neighbor classification, we obtain a more efficient solution which additionally provides much better generalization properties To demonstrate these benefits, we run experiments on three different publicly available datasets, showing state-of-the-art or even better results, however, on much lower computational efforts This is in particular interesting since we use quite simple color and texture features, whereas other approaches build on rather complex image descriptions!

Proceedings Article
26 Jun 2012
TL;DR: In this paper, the authors proposed to jointly learn domain-invariant features and discriminative feature space by optimizing an information-theoretic metric as an proxy to the expected misclassification error on the target domain.
Abstract: We study the problem of unsupervised domain adaptation, which aims to adapt classifiers trained on a labeled source domain to an unlabeled target domain. Many existing approaches first learn domain-invariant features and then construct classifiers with them. We propose a novel approach that jointly learn the both. Specifically, while the method identifies a feature space where data in the source and the target domains are similarly distributed, it also learns the feature space discriminatively, optimizing an information-theoretic metric as an proxy to the expected misclassification error on the target domain. We show how this optimization can be effectively carried out with simple gradient-based methods and how hyperparameters can be cross-validated without demanding any labeled data from the target domain. Empirical studies on benchmark tasks of object recognition and sentiment analysis validated our modeling assumptions and demonstrated significant improvement of our method over competing ones in classification accuracies.

Book ChapterDOI
07 Oct 2012
TL;DR: 2PKNN, a two-step variant of the classical K-nearest neighbour algorithm, is proposed that performs comparable to the current state-of-the-art on three challenging image annotation datasets, and shows significant improvements after metric learning.
Abstract: Automatic image annotation aims at predicting a set of textual labels for an image that describe its semantics. These are usually taken from an annotation vocabulary of few hundred labels. Because of the large vocabulary, there is a high variance in the number of images corresponding to different labels ("class-imbalance"). Additionally, due to the limitations of manual annotation, a significant number of available images are not annotated with all the relevant labels ("weak-labelling"). These two issues badly affect the performance of most of the existing image annotation models. In this work, we propose 2PKNN, a two-step variant of the classical K-nearest neighbour algorithm, that addresses these two issues in the image annotation task. The first step of 2PKNN uses "image-to-label" similarities, while the second step uses "image-to-image" similarities; thus combining the benefits of both. Since the performance of nearest-neighbour based methods greatly depends on how features are compared, we also propose a metric learning framework over 2PKNN that learns weights for multiple features as well as distances together. This is done in a large margin set-up by generalizing a well-known (single-label) classification metric learning algorithm for multi-label prediction. For scalability, we implement it by alternating between stochastic sub-gradient descent and projection steps. Extensive experiments demonstrate that, though conceptually simple, 2PKNN alone performs comparable to the current state-of-the-art on three challenging image annotation datasets, and shows significant improvements after metric learning.

Proceedings ArticleDOI
16 Jun 2012
TL;DR: This paper proposes a fusion algorithm which outputs enhanced metrics by combining multiple given metrics (similarity measures) through diffusion process in an unsupervised way and has a wide range of applications in machine learning and computer vision.
Abstract: Metric learning is a fundamental problem in computer vision. Different features and algorithms may tackle a problem from different angles, and thus often provide complementary information. In this paper, we propose a fusion algorithm which outputs enhanced metrics by combining multiple given metrics (similarity measures). Unlike traditional co-training style algorithms where multi-view features or multiple data subsets are used for classification or regression, we focus on fusing multiple given metrics through diffusion process in an unsupervised way. Our algorithm has its particular advantage when the input similarity matrices are the outputs from diverse algorithms. We provide both theoretical and empirical explanations to our method. Significant improvements over the state-of-the-art results have been observed on various benchmark datasets. For example, we have achieved 100% accuracy (no longer the bull's eye measure) on the MPEG-7 shape dataset. Our method has a wide range of applications in machine learning and computer vision.

Proceedings ArticleDOI
08 Feb 2012
TL;DR: This work proposes novel models which approximately optimize NDCG for the recommendation task, essentially variations on matrix factorization models where the features associated with the users and the items for the ranking task are learned.
Abstract: Typical recommender systems use the root mean squared error (RMSE) between the predicted and actual ratings as the evaluation metric. We argue that RMSE is not an optimal choice for this task, especially when we will only recommend a few (top) items to any user. Instead, we propose using a ranking metric, namely normalized discounted cumulative gain (NDCG), as a better evaluation metric for this task. Borrowing ideas from the learning to rank community for web search, we propose novel models which approximately optimize NDCG for the recommendation task. Our models are essentially variations on matrix factorization models where we also additionally learn the features associated with the users and the items for the ranking task. Experimental results on a number of standard collaborative filtering data sets validate our claims. The results also show the accuracy and efficiency of our models and the benefits of learning features for ranking.

Journal ArticleDOI
TL;DR: In this article, Suzuki's fixed point results from (Suzuki, Proc. Am. Math. Soc. 136:1861-1869, 2008) were extended to the case of metric type spaces and cone metric types.
Abstract: Suzuki’s fixed point results from (Suzuki, Proc. Am. Math. Soc. 136:1861-1869, 2008) and (Suzuki, Nonlinear Anal. 71:5313-5317, 2009) are extended to the case of metric type spaces and cone metric type spaces. Examples are given to distinguish our results from the known ones. MSC:47H10, 54H25.

Journal ArticleDOI
TL;DR: In this paper, the authors prove equivalent conditions for two-sided sub-Gaussian estimates of heat kernels on metric measure spaces and show that these are equivalent to the conditions for heat kernels in the case of two sides.
Abstract: We prove equivalent conditions for two-sided sub-Gaussian estimates of heat kernels on metric measure spaces.

Journal ArticleDOI
TL;DR: In this article, Boyer and Galicki introduced a contact reduction method in the context of Sasakian manifolds, which produces 5-dimentional Sasaki-Einstein manifolds from a 7-sphere.
Abstract: In [9], Boyer and Galicki introduced a contact reduction method in the context of Sasakian manifolds, which produces 5-dimentional Sasaki-Einstein manifolds from a 7-sphere. In this paper, we compute very explicitly the metric obtained from the above mentioned reduction via a projection, $S^3 \times S^3 \to S^2 \times S^3$, and show that this metric is the homogeneous Kobayashi-Tanno metric.

Journal ArticleDOI
TL;DR: A log-euclidean block-division appearance model is developed which captures both the global and local spatial layout information about object appearances and which obtains more accurate results than six state-of-the-art tracking algorithms.
Abstract: Object appearance modeling is crucial for tracking objects, especially in videos captured by nonstationary cameras and for reasoning about occlusions between multiple moving objects. Based on the log-euclidean Riemannian metric on symmetric positive definite matrices, we propose an incremental log-euclidean Riemannian subspace learning algorithm in which covariance matrices of image features are mapped into a vector space with the log-euclidean Riemannian metric. Based on the subspace learning algorithm, we develop a log-euclidean block-division appearance model which captures both the global and local spatial layout information about object appearances. Single object tracking and multi-object tracking with occlusion reasoning are then achieved by particle filtering-based Bayesian state inference. During tracking, incremental updating of the log-euclidean block-division appearance model captures changes in object appearance. For multi-object tracking, the appearance models of the objects can be updated even in the presence of occlusions. Experimental results demonstrate that the proposed tracking algorithm obtains more accurate results than six state-of-the-art tracking algorithms.