Determining a suitable metric when using non-negative matrix factorization

Nonnegative Matrix Factorization (NMF) approximates a given data matrix as a product of two low rank nonnegative matrices, usually by minimizing the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> or the KL distance between the data matrix and the matrix product. This factorization was shown to be useful for several important computer vision applications. We propose here a new NMF algorithm that minimizes the Earth Mover's Distance (EMD) error between the data and the matrix product. We propose an iterative NMF algorithm (EMD NMF) and prove its convergence. The algorithm is based on linear programming. We discuss the numerical difficulties of the EMD NMF and propose an efficient approximation. Naturally, the matrices obtained with EMD NMF are different from those obtained with L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> NMF. We discuss these differences in the context of two challenging computer vision tasks - texture classification and face recognition - and demonstrate the advantages of the proposed method.

One of the issues in 3-D shape retrieval arises from the fact that users suffer inherent difficulty in generating a 3-D query. Therefore, the demands on the user-friendly querying interface such as query by sketch and query by editing are increasing. In this paper, we propose a new shape retrieval system using the attributed relational graph (ARG) composed of nodes with unary attributes and edges with binary relations for interactive searching and comparing 3-D objects. As similar to other systems, the ARG-based retrieval system consists of two components: the ARG generation via feature extraction and its similarity matching. For the ARG generation, we adopt a morphological shape decomposition scheme as a part-based representation. More specifically, the part-based representation can be generated from the decomposition result and encoded into the ARG. In the case of similarity matching, we consider various ARG matching algorithms such as earth mover's distance, spectral analysis, relaxation and modified nested earth mover's distance. Finally, we have conducted the experiments for the ARG with various matching algorithms and the Shape3D, i.e.an international standard for description and browsing of 3-D object database in MPEG-7, using the database of 480 3-D objects with 33 categories in terms of the bull's eye performance, the average normalized modified retrieval rate, and the precision/recall curve.

Introducing a weighted non-negative matrix factorization for image classification

A novel approach for assessing the extent of electron displacement in optical transitions is proposed by implementing the Earth Mover's Distance (EMD) method, which quantifies the spatial dissimilarity between ground and excited state electron density distributions. In contrast to previous descriptors, this index provides a representative and intuitively understandable distance under a robust and computationally efficient scheme for all possible forms of locality, even in the most difficult to dissect topological cases. The theoretical differences among the existing indices and our method are first illustrated with the help of a simplified model system, followed by a benchmarking of several partial atomic charge models using experimentally relevant push-pull compounds with diverse symmetries. These same molecules are finally employed to further demonstrate the principal advantages of the EMD index and its capabilities in rationalizing charge transfer phenomena.

Earth Mover's Distance (EMD), as a similarity measure, has received a lot of attention in the fields of multimedia and probabilistic databases, computer vision, image retrieval, machine learning, etc. EMD on multidimensional histograms provides better distinguishability between the objects approximated by the histograms (e.g., images), compared to classic measures like Euclidean distance. Despite its usefulness, EMD has a high computational cost; therefore, a number of effective filtering methods have been proposed, to reduce the pairs of histograms for which the exact EMD has to be computed, during similarity search. Still, EMD calculations in the refinement step remain the bottleneck of the whole similarity search process. In this paper, we focus on optimizing the refinement phase of EMD-based similarity search by (i) adapting an efficient min-cost flow algorithm (SIA) for EMD computation, (ii) proposing a dynamic distance bound, which can be used to terminate an EMD refinement early, and (iii) proposing a dynamic refinement order for the candidates which, paired with a concurrent EMD refinement strategy, reduces the amount of needless computations. Our proposed techniques are orthogonal to and can be easily integrated with the state-of-the-art filtering techniques, reducing the cost of EMD-based similarity queries by orders of magnitude.

The Earth Mover''s Distance (EMD) between two finite distributions of weight is proportional to the minimum amount of work required to transform one distribution into the other. Current content-based retrieval work in the Stanford Vision Laboratory uses the EMD as a common framework for measuring image similarity with respect to color, texture, and shape content. In this report, we present some fast to compute lower bounds on the EMD which may allow a system to avoid exact, more expensive EMD computations during query processing. The effectiveness of the lower bounds is tested in a color-based retrieval system. In addition to the lower bound work, we also show how to compute the EMD under translation. In this problem, the points in one distribution are free to translate, and the goal is to find a translation that minimizes the EMD to the other distribution.

Supplier selection is a critical decision-making process in supply chain management, significantly impacting operational efficiency and cost-effectiveness. Traditional methods often struggle to handle the inherent uncertainty and vagueness involved in evaluating supplier criteria. This paper proposes a novel approach for supplier selection using Intuitionistic Trapezoidal Fuzzy Numbers (ITFNs), combined with Earth Mover's Distance (EMD) and Fuzzy Technique for Order Preference by Similarity to Ideal Solution (Fuzzy TOPSIS), to effectively model and address these uncertainties. ITFNs allow for the representation of both membership and non-membership degrees, enhancing the ability to capture imprecision in supplier evaluations. The Earth Mover's Distance, a powerful metric from transportation theory, is employed to rank the intuitionistic trapezoidal fuzzy numbers and compare suppliers based on multiple criteria using fuzzy TOPSIS. The proposed model is demonstrated through a case study, showcasing its ability to provide more flexible and accurate supplier rankings compared to conventional methods. This approach offers valuable insights for decision-makers aiming to optimize supplier selection in complex and uncertain environments, contributing to more resilient supply chains.

이 논문은 고차원의 데이터를 저 차원으로 줄이는 방법 중 하나인 특징추출에 대한 방법들의 특성을 비교한다. 비교대상 방법은 전통적인 PCA(Principal Component Analysis)방법과 시각피질의 특성을 보인다고 알려진 ICA(Independent Component Analysis), 국소기반인식을 구현한 NMF(Non-negative Matrix Factorization), 그리고 이의 성능을 개선한 sNMF(Sparse NMF)로 정하였다. 추출된 특징들의 특성을 시각적으로 확인하기 위하여 필기체 숫자 영상을 대상으로 특징추출을 수행하였으며, 인식기에 적용한 효과의 확인을 위하여 추출된 특징을 다층퍼셉트론에 학습시켜보았다. 각 방법의 특성을 비교한 결과는 응용하고자 하는 문제에서 어떤 특징을 추출하기 원하느냐에 따라 특징추출 방법을 선정할 때 유용할 것이다. In this paper, feature extraction methods, which is one field of reducing dimensions of high-dimensional data, are empirically investigated. We selected the traditional PCA(Principal Component Analysis), ICA(Independent Component Analysis), NMF(Non-negative Matrix Factorization), and sNMF(Sparse NMF) for comparisons. ICA has a similar feature with the simple cell of V1. NMF implemented a "parts-based representation in the brain" and sNMF is a improved version of NMF. In order to visually investigate the extracted features, handwritten digits are handled. Also, the extracted features are used to train multi-layer perceptrons for recognition test. The characteristic of each feature extraction method will be useful when applying feature extraction methods to many real-world problems.

Color descriptors are one of the important features used in content-based image retrieval. The Dominant Color Descriptor (DCD) represents a few perceptually dominant colors in an image through color quantization. For image retrieval based on DCD, the earth mover's distance and the optimal color composition distance are proposed to measure the dissimilarity between two images. Although providing good retrieval results, both methods are too time-consuming to be used in a large image database. To solve the problem, we propose a new distance function that calculates an approximate earth mover's distance in linear time. To calculate the dissimilarity in linear time, the proposed approach employs the space-filling curve for multidimensional color space. To improve the accuracy, the proposed approach uses multiple curves and adjusts the color positions. As a result, our approach achieves order-of-magnitude time improvement but incurs small errors. We have performed extensive experiments to show the effectiveness and efficiency of the proposed approach. The results reveal that our approach achieves almost the same results with the EMD in linear time.

In this paper, we propose a novel approach to learning robust ground distance functions of the Earth Mover's distance to make it appropriate for quantifying the partial similarity between two feature-sets. First, we define the ground distance as a monotonic transformation of commonly used feature-to-feature base distance (or similarity) measures, so that in computing the Earth Mover's distance, the algorithm could better turn its focus on the feature pairs that are correctly matched, while being less affected by irrelevant ones. As a result, the proposed method is especially suited for 3D partial shape retrieval where occlusion and clutter are serious problems. We prove that when the transformation satisfies certain conditions, the metric property of the base distance is sufficient to guarantee the ground distance is a metric (and so is the Earth Mover's distance), which makes fast shape retrieval on large databases technically possible. Second, we propose a discriminative learning framework to optimize the transformation function based on the real Adaboost algorithm. The optimization is performed in the space of the piecewise constant approximations of the transformation without making any parametric assumption. Finally, extensive experiments on 3D partial shape retrieval convincingly demonstrate the effectiveness of the proposed techniques.

An efficient iterative Earth Mover's Distance (iEMD) algorithm for visual tracking is proposed in this paper. The Earth Mover's Distance (EMD) is used as the similarity measure to search for the optimal template candidates in feature-spatial space in a video sequence. The local sparse representation is used as the appearance model for the iEMD tracker. The maximum-alignment-pooling method is used for constructing a sparse coding histogram which reduces the computational complexity of the EMD optimization. The template update algorithm based on the EMD is also presented. When the camera is mounted on a moving robot, e.g., a flying quadcopter, the camera could experience a sudden and rapid motion leading to large inter-frame movements. To ensure that the tracking algorithm converges, a gyro-aided extension of the iEMD tracker is presented, where synchronized gyroscope information is utilized to compensate for the rotation of the camera. The iEMD algorithm's performance is evaluated using eight publicly available videos from the CVPR 2013 dataset. The performance of the iEMD algorithm is compared with eight state-of-the-art tracking algorithms based on relative percentage overlap. Experimental results show that the iEMD algorithm performs robustly in the presence of illumination variation and deformation. The robustness of this algorithm for large inter-frame displacements is also illustrated.

A novel approach is introduced to define a quantitative measure of closeness between vector fields. The usefulness of this measurement can be seen when comparing computational and experimental flow fields under the same conditions. Furthermore, its applicability can be extended to more cumbersome tasks, such as navigating through a large database, searching for similar topologies. This new measure relies on the use of critical points, which are a key feature in vector field topology. In order to characterize critical points, /spl alpha/ and /spl beta/ parameters are introduced. They are used to form a closed set of eight unique patterns for simple critical points. These patterns are also basic building blocks for higher-order nonlinear vector fields. In order to study and compare a given set of vector fields, a measure of distance between different patterns of critical points is introduced. The basic patterns of critical points are mapped onto a unit circle in /spl alpha/-/spl beta/ space. The concept of the "Earth mover's distance" is used to compute the closeness between various pairs of vector fields, and a nearest-neighbor query is thus produced to illustrate the relationship between the given set of vector fields. This approach quantitatively measures the similarity and dissimilarity between vector fields. It is ideal for data compression of a large flow field, since only the number and types of critical points along with their corresponding /spl alpha/ and /spl beta/ parameters are necessary to reconstruct the whole field. It can also be used to better quantify the changes in time-varying data sets.

Earth mover's distance (EMD) is a distance measure between two distributions, and has been widely used in multimedia information retrieval systems, especially content-based image retrieval systems. When the EMD is applied to image problems based on color or texture, the EMD reflects the human perceptual similarities. Its computations, however, is too expensive to use in large-scale databases. In order to achieve the efficient computation of the EMD during query processing, we have developed ldquofastEMDrdquo, a library for high-speed feature-based similarity retrievals in large databases. This paper introduces techniques that are used in the implementation of the fastEMD and demonstrates the efficiency in extensive experiments.

We present a new algorithm for a robust family of Earth Mover's Distances - EMDs with thresholded ground distances. The algorithm transforms the flow-network of the EMD so that the number of edges is reduced by an order of magnitude. As a result, we compute the EMD by an order of magnitude faster than the original algorithm, which makes it possible to compute the EMD on large histograms and databases. In addition, we show that EMDs with thresholded ground distances have many desirable properties. First, they correspond to the way humans perceive distances. Second, they are robust to outlier noise and quantization effects. Third, they are metrics. Finally, experimental results on image retrieval show that thresholding the ground distance of the EMD improves both accuracy and speed.

Many problems in image representation and classification involve some form of dimensionality reduction. Non-negative matrix factorization (NMF) is a recently proposed unsupervised procedure for learning spatially localized, parts-based subspace representation of objects. Here we present an improvement of the classical NMF by combining with Log-Gabor wavelets to enhance its part-based learning ability. In addition, we compare the new method with principal component analysis (PCA) and locally linear embedding (LLE) proposed recently in Science. Finally, we apply the new method to several real world datasets and achieve good performance in representation and classification.