A Model for Learning Topographically Organized Parts-Based Representations of Objects in Visual Cortex: Topographic Nonnegative Matrix Factorization

Spectral un mixing has been an important technique for hyperspectral imagery processing. In spectral unmixing methods that are based on the original non-negative matrix factorization (NMF) model, unmixing accuracy is limited by the lack of appropriate constraints that derived from the inherent properties of hyperspectral images. It was shown that the correlation analysis can be extended for NMF-based spectral unmixing model, based on the advantage that the spatial and spectral correlation features can provide necessary constraints for NMF to overcome related problems and obtain better performance. However, many correlation constrained NMF models prefer adopting only spatial correlation, the neglect of spectral correlation may limit the accuracy and application scope. This letter presents a novel method of imposing spatial and spectral correlation constraints simultaneously on the NMF-based unmixing model by adopting Markov Random Field (MRF) and complexity pursuit respectively. The related issues, including reducing the deviation of endmembers' spatial energy distribution and substituting the predictability of signal for piecewise smoothness of spectra, were studied together. Experiments showed that the new proposed NMF model was superior to some existing NMF models with only spatial correlation constraint in terms of estimation accuracy of endmember spectra and fractional abundances.

Nonnegative matrix factorization (NMF) is the problem of approximating an input nonnegative matrix, $V$, as the product of two smaller nonnegative matrices, $W$ and $H$. In this paper, we introduce a general framework to design multiplicative updates (MU) for NMF based on $\beta$-divergences ($\beta$-NMF) with disjoint equality constraints, and with penalty terms in the objective function. By disjoint, we mean that each variable appears in at most one equality constraint. Our MU satisfy the set of constraints after each update of the variables during the optimization process, while guaranteeing that the objective function decreases monotonically. We showcase this framework on three NMF models, and show that it competes favorably the state of the art: (1)~$\beta$-NMF with sum-to-one constraints on the columns of $H$, (2) minimum-volume $\beta$-NMF with sum-to-one constraints on the columns of $W$, and (3) sparse $\beta$-NMF with $\ell_2$-norm constraints on the columns of $W$.

Non-negative Matrix Factorization (NMF) has been frequently applied to image feature extraction and clustering. Especially in image clustering tasks, it can achieve the similar or better performance than most of the matrix factorization algorithms due to its parts-based representations in the brain. However, the features extracted by NMF are not sparse and localized enough and the error of factorization is not small enough. Semi-tensor product of matrices (STP) is a novel operation of matrix multiplication, it is a generalization of the conventional matrix product for allowing the dimensions of factor matrices to be unequal. STP can manage the data hierarchically and the inverse process of STP can separate the data hierarchically. Based on this character of STP, we propose the Non-Negative Matrix Semi-Tensor Factorization (NMSTF). In this algorithm, we use the inverse process of Semi-Tensor Product of matrices for non-negative matrix factorization. This algorithm effectively optimizes the above two problems in NMF. While achieving similar even better performance on image clustering tasks, the size of features extracted by STNMF is at least 50 % smaller than the ones’ extracted by NMF and the error of factorization reduces 30 % in average.

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.

The vehicle recognition consists of two steps; the vehicle region detection step and vehicle identification step based on the feature extracted from the detected region. Among the linear transformations, the non-negative matrix factorization (NMF) and non-negative tensor factorization (NTF) can be used in part-based representation. So we can utilize the local features of a car as a basis vector. In this paper, we propose a feature extraction using NMF and NTF suitable for the vehicle recognition, and verify the recognition rate. We show that the proposed feature is robust in the urban scene where occlusions are frequently occur.

Hyperspectral unmixing is an important technique for analyzing remote sensing images which aims to obtain a collection of endmembers and their corresponding abundances. In recent years, non-negative matrix factorization (NMF) has received extensive attention due to its good adaptability for mixed data with different degrees. The majority of existing NMF-based unmixing methods are developed by incorporating additional constraints into the standard NMF based on the spectral and spatial information of hyperspectral images. However, they neglect to exploit the nature of imbalanced pixels included in the data, which may cause the pixels mixed with imbalanced endmembers to be ignored, and thus the imbalanced endmembers generally cannot be accurately estimated due to the statistical property of NMF. To exploit the information of imbalanced samples in hyperspectral data during the unmixing procedure, in this paper, a cluster-wise weighted NMF (CW-NMF) method for the unmixing of hyperspectral images with imbalanced data is proposed. Specifically, based on the result of clustering conducted on the hyperspectral image, we construct a weight matrix and introduce it into the model of standard NMF. The proposed weight matrix can provide an appropriate weight value to the reconstruction error between each original pixel and the reconstructed pixel in the unmixing procedure. In this way, the adverse effect of imbalanced samples on the statistical accuracy of NMF is expected to be reduced by assigning larger weight values to the pixels concerning imbalanced endmembers and giving smaller weight values to the pixels mixed by majority endmembers. Besides, we extend the proposed CW-NMF by introducing the sparsity constraints of abundance and graph-based regularization, respectively. The experimental results on both synthetic and real hyperspectral data have been reported, and the effectiveness of our proposed methods has been demonstrated by comparing them with several state-of-the-art methods.

Image hashing is an effective technique for copy detection. This article proposes a robust color image hashing scheme with nonnegative matrix factorization (NMF) and saliency maps. The proposed hashing scheme constructs a weighted feature map via the saliency map and the matrix of color vector angles. It extracts hash from the weighted feature map by using 2-D discrete wavelet transform (DWT), NMF, and ordinal measurements. The use of 2-D DWT achieves an initial data reduction, the NMF provides a parts-based representation, and the ordinal measurements can make a short and robust sequence. These techniques jointly contribute to a robust, short, and discriminative hash. Experiments of classification and copy detection are both done. Comparative results indicate that the proposed hashing scheme outperforms the compared hashing schemes.

This paper presents two single channel speech dereverberation methods to enhance the quality of speech signals that have been recorded in an enclosed space. For both methods, the room acoustics are modeled using a nonnegative approximation of the convolutive transfer function (NCTF), and to additionally exploit the spectral properties of the speech signal, such as the low rank nature of the speech spectrogram, the speech spectrogram is modeled using nonnegative matrix factorization (NMF). Two methods are described to combine the NCTF and NMF models. In the first method, referred to as the integrated method, a cost function is constructed by directly integrating the speech NMF model into the NCTF model, while in the second method, referred to as the weighted method, the NCTF and NMF based cost functions are weighted and summed. Efficient update rules are derived to solve both optimization problems. In addition, an extension of the integrated method is presented, which exploits the temporal dependencies of the speech signal. Several experiments are performed on reverberant speech signals with and without background noise, where the integrated method yields a considerably higher speech quality than the baseline NCTF method and a state of the art spectral enhancement method. Moreover, the experimental results indicate that the weighted method can even lead to a better performance in terms of instrumental quality measures, but that the optimal weighting parameter depends on the room acoustics and the utilized NMF model. Modeling the temporal dependencies in the integrated method was found to be useful only for highly reverberant conditions.

Label consistency-based deep semisupervised NMF for tumor recognition

Feature analysis is a fundamental research area in computer graphics; meanwhile, meaningful and part-aware feature bases are always demanding. This paper proposes a framework for conducting feature analysis on a three-dimensional (3D) model by introducing modified Non-negative Matrix Factorization (NMF) model into the graphical feature space and push forward further applications. By analyzing and utilizing the intrinsic ideas behind NMF, we propose conducting the factorization on feature matrices constructed based on descriptors or graphs, which provides a simple but effective way to raise compressed and scale-aware descriptors. In order to enable part-aware model analysis, we modify the NMF model to be sparse and constrained regarding to both bases and encodings, which gives rise to Sparse and Constrained Non-negative Matrix Factorization (SAC-NMF). Subsequently, by adapting the analytical components (including hidden variables, bases, and encodings) to design descriptors, several applications have been easily but effectively realized. The extensive experimental results demonstrate that the proposed framework has many attractive advantages, such as being efficient, extendable, and so forth.

Sparse coding or independent component analysis (ICA) which is a holistic representation, was successfully applied to elucidate early auditory processing and to the task of sound classification. In contrast, parts-based representation is an alternative way of understanding object recognition in brain. In this paper we employ the non-negative matrix factorization (NMF) [D.D. Lee et al., 1999] which learns parts-based representation in the task of sound classification. Methods of feature extraction from spectro-temporal sounds using the NMF in the absence or presence of noise are explained. Experimental results show that NMF-based features improve the performance of sound classification over ICA-based features.

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.

Nonnegative matrix factorization (NMF) is an effectively parts-based feature representation approach and has achieved good performance in different tasks such as computer vision, clustering and so on. To enhance the discriminative power of NMF in nonnegative feature space, this paper proposes a novel supervised matrix decomposition method, called Class-Cone based Nonnegative Matrix Factorization (CCNMF). We establish a loss function with class-cone regularization which contains the volumes of class-cones and the quantity of between class-cones. To minimize the objective function will leads to small class-cones and large distance between class-cones. This good property is beneficial to the performance of NMF algorithm. We solve the optimization problem using KKT conditions and obtain the updating rules of CCNMF. Our approach is experimentally shown to be convergence and successfully applied to face recognition. Experimental results demonstrate the effectiveness of the proposed CCNMF algorithm.

Nonnegative matrix factorization (NMF) is known as a parts-based linear representation for nonnegative data. This method has been applied for blind source separation (BSS) when the sources are nonnegative. This paper presents a new NMF method for independent component analysis (ICA), which is useful for BSS without the nonnegativity constraint. Using this method, we transform the sources by their cumulative distribution functions (CDFs) and perform the nonparametric quantization to construct a nonnegative matrix where each entry represents the joint probability density of two transformed signals. The NMF procedure is accordingly realized to find the ICA demixing matrix. The independence between sources is maximized towards attaining the uniformity in the joint probability density. In the experiments on the separation of signal and music signals, we show the effectiveness of the proposed NMF-ICA compared to the infomax ICA and FastICA algorithms.

Source apportionment of gaseous pollutants in oil and gas extraction areas: A comparison between positive matrix factorization and self-organizing maps approaches.