Two-stage nonnegative sparse representation for large-scale face recognition

Sparse Index Tracking With K-Sparsity or ϵ-Deviation Constraint via ℓ0-Norm Minimization

Sparse index tracking, as one of the passive investment strategies, is to track a benchmark financial index via constructing a portfolio with a few assets in a market index. It can be considered as parameter learning in an adaptive system, in which we periodically update the selected assets and their investment percentages based on the sliding window approach. However, many existing algorithms for sparse index tracking cannot explicitly and directly control the number of assets or the tracking error. This article formulates sparse index tracking as two constrained optimization problems and then proposes two algorithms, namely, nonnegative orthogonal matching pursuit with projected gradient descent (NNOMP-PGD) and alternating direction method of multipliers for l0 -norm (ADMM- l0 ). The NNOMP-PGD aims at minimizing the tracking error subject to the number of selected assets less than or equal to a predefined number. With the NNOMP-PGD, investors can directly and explicitly control the number of selected assets. The ADMM- l0 aims at minimizing the number of selected assets subject to the tracking error that is upper bounded by a preset threshold. It can directly and explicitly control the tracking error. The convergence of the two proposed algorithms is also presented. With our algorithms, investors can explicitly and directly control the number of selected assets or the tracking error of the resultant portfolio. In addition, numerical experiments demonstrate that the proposed algorithms outperform the existing approaches.

A study of sparse representation-based classification for biometric verification based on both handcrafted and deep learning features

Biometric verification is generally considered a one-to-one matching task. In contrast, in this paper, we argue that the one-to-many competitive matching via sparse representation-based classification (SRC) can bring enhanced verification security and accuracy. SRC-based verification introduces non-target subjects to construct dynamic dictionary together with the client claimed and encodes the submitted feature. Owing to the sparsity constraint, a client can only be accepted when it defeats almost all non-target classes and wins a convincing sparsity-based matching score. This will make the verification more secure than those using one-to-one matching. However, intense competition may also lead to extremely inferior genuine scores when data degeneration occurs. Motivated by the latent benefits and concerns, we study SRC-based verification using two sparsity-based matching measures, three biometric modalities (i.e., face, palmprint, and ear) and their multimodal combinations based on both handcrafted and deep learning features. We finally approach a comprehensive study of SRC-based verification, including its methodology, characteristics, merits, challenges and the directions to resolve. Extensive experimental results demonstrate the superiority of SRC-based verification, especially when using multimodal fusion and advanced deep learning features. The concerns about its efficiency in large-scale user applications can be readily solved using a simple dictionary shrinkage strategy based on cluster analysis and random selection of non-target subjects.

Projective Double Reconstructions Based Dictionary Learning Algorithm for Cross-Domain Recognition

Dictionary learning plays a significant role in the field of machine learning. Existing works mainly focus on learning dictionary from a single domain. In this paper, we propose a novel projective double reconstructions (PDR) based dictionary learning algorithm for cross-domain recognition. Owing the distribution discrepancy between different domains, the label information is hard utilized for improving discriminability of dictionary fully. Thus, we propose a more flexible label consistent term and associate it with each dictionary item, which makes the reconstruction coefficients have more discriminability as much as possible. Due to the intrinsic correlation between cross-domain data, the data should be reconstructed with each other. Based on this consideration, we further propose a projective double reconstructions scheme to guarantee that the learned dictionary has the abilities of data itself reconstruction and data crossreconstruction. This also guarantees that the data from different domains can be boosted mutually for obtaining a good data alignment, making the learned dictionary have more transferability. We integrate the double reconstructions, label consistency constraint and classifier learning into a unified objective and its solution can be obtained by proposed optimization algorithm that is more efficient than the conventional l1 optimization based dictionary learning methods. The experiments show that the proposed PDR not only greatly reduces the time complexity for both training and testing, but also outperforms over the stateof- the-art methods.

Laplacian-Uniform Mixture-Driven Iterative Robust Coding With Applications to Face Recognition Against Dense Errors

Outliers due to occlusion, pixel corruption, and so on pose serious challenges to face recognition despite the recent progress brought by sparse representation. In this article, we show that robust statistics implemented by the state-of-the-art methods are insufficient for robustness against dense gross errors. By modeling the distribution of coding residuals with a Laplacian-uniform mixture, we obtain a sparse representation that is significantly more robust than the previous methods. The nonconvex error term of the implemented objective function is nondifferentiable at zero and cannot be properly addressed by the usual iteratively reweighted least-squares formulation. We show that an iterative robust coding algorithm can be derived by local linear approximation of the nonconvex error term, which is both effective and efficient. With iteratively reweighted l₁ minimization of the error term, the proposed algorithm is capable of handling the sparsity assumption of the coding errors more appropriately than the previous methods. Notably, it has the distinct property of addressing error detection and error correction cooperatively in the robust coding process. The proposed method demonstrates significantly improved robustness for face recognition against dense gross errors, either contiguous or discontiguous, as verified by extensive experiments.

Distributed Training for Multi-Layer Neural Networks by Consensus

Over the past decade, there has been a growing interest in large-scale and privacy-concerned machine learning, especially in the situation where the data cannot be shared due to privacy protection or cannot be centralized due to computational limitations. Parallel computation has been proposed to circumvent these limitations, usually based on the master-slave and decentralized topologies, and the comparison study shows that a decentralized graph could avoid the possible communication jam on the central agent but incur extra communication cost. In this brief, a consensus algorithm is designed to allow all agents over the decentralized graph to converge to each other, and the distributed neural networks with enough consensus steps could have nearly the same performance as the centralized training model. Through the analysis of convergence, it is proved that all agents over an undirected graph could converge to the same optimal model even with only a single consensus step, and this can significantly reduce the communication cost. Simulation studies demonstrate that the proposed distributed training algorithm for multi-layer neural networks without data exchange could exhibit comparable or even better performance than the centralized training model.

Data Augmentation-Based Joint Learning for Heterogeneous Face Recognition

Heterogeneous face recognition (HFR) is the process of matching face images captured from different sources. HFR plays an important role in security scenarios. However, HFR remains a challenging problem due to the considerable discrepancies (i.e., shape, style, and color) between cross-modality images. Conventional HFR methods utilize only the information involved in heterogeneous face images, which is not effective because of the substantial differences between heterogeneous face images. To better address this issue, this paper proposes a data augmentation-based joint learning (DA-JL) approach. The proposed method mutually transforms the cross-modality differences by incorporating synthesized images into the learning process. The aggregated data augments the intraclass scale, which provides more discriminative information. However, this method also reduces the interclass diversity (i.e., discriminative information). We develop the DA-JL model to balance this dilemma. Finally, we obtain the similarity score between heterogeneous face image pairs through the log-likelihood ratio. Extensive experiments on a viewed sketch database, forensic sketch database, near-infrared image database, thermal-infrared image database, low-resolution photo database, and image with occlusion database illustrate that the proposed method achieves superior performance in comparison with the state-of-the-art methods.

A New Correntropy-Based Conjugate Gradient Backpropagation Algorithm for Improving Training in Neural Networks

Mean square error (MSE) is the most prominent criterion in training neural networks and has been employed in numerous learning problems. In this paper, we suggest a group of novel robust information theoretic backpropagation (BP) methods, as correntropy-based conjugate gradient BP (CCG-BP). CCG-BP algorithms converge faster than the common correntropy-based BP algorithms and have better performance than the common CG-BP algorithms based on MSE, especially in nonGaussian environments and in cases with impulsive noise or heavy-tailed distributions noise. In addition, a convergence analysis of this new type of method is particularly considered. Numerical results for several samples of function approximation, synthetic function estimation, and chaotic time series prediction illustrate that our new BP method is more robust than the MSE-based method in the sense of impulsive noise, especially when SNR is low.

Robust Adaptive Embedded Label Propagation With Weight Learning for Inductive Classification

We propose a robust inductive semi-supervised label prediction model over the embedded representation, termed adaptive embedded label propagation with weight learning (AELP-WL), for classification. AELP-WL offers several properties. First, our method seamlessly integrates the robust adaptive embedded label propagation with adaptive weight learning into a unified framework. By minimizing the reconstruction errors over embedded features and embedded soft labels jointly, our AELP-WL can explicitly ensure the learned weights to be joint optimal for representation and classification, which differs from most existing LP models that perform weight learning separately by an independent step before label prediction. Second, existing models usually precalculate the weights over the original samples that may contain unfavorable features and noise decreasing performance. To this end, our model adds a constraint that decomposes original data into a sparse component encoding embedded noise-removed sparse representations of samples and a sparse error part fitting noise, and then performs the adaptive weight learning over the embedded sparse representations. Third, our AELP-WL computes the projected soft labels by trading-off the manifold smoothness and label fitness errors over the adaptive weights and the embedded representations for enhancing the label estimation power. By including a regressive label approximation error for simultaneous minimization to correlate sample features with the embedded soft labels, the out-of-sample issue is naturally solved. By minimizing the reconstruction errors over features and embedded soft labels, classification error and label approximation error jointly, state-of-the-art results are delivered.

Rectified Gaussian Scale Mixtures and the Sparse Non-Negative Least Squares Problem

In this paper, we develop a Bayesian evidence maximization framework to solve the sparse non-negative least squares problem (S-NNLS). We introduce a family of probability densities referred to as the Rectified Gaussian Scale Mixture (R-GSM), to model the sparsity enforcing prior distribution for the signal of interest. The R-GSM prior encompasses a variety of heavy-tailed distributions such as the rectified Laplacian and rectified Student-t distributions with a proper choice of the mixing density. We utilize the hierarchical representation induced by the R-GSM prior and develop an evidence maximization framework based on the Expectation-Maximization (EM) algorithm. Using the EM-based method, we estimate the hyper-parameters and obtain a point estimate for the solution of interest. We refer to this proposed method as rectified Sparse Bayesian Learning (R-SBL). We provide four EM-based R-SBL variants that offer a range of options to trade-off computational complexity to the quality of the E-step computation. These methods include the Markov Chain Monte Carlo EM, linear minimum mean square estimation, approximate message passing and a diagonal approximation. Using numerical experiments, we show that the proposed R-SBL method outperforms existing S-NNLS solvers in terms of both signal and support recovery, and is very robust against the structure of the design matrix.

Collaborative Random Faces-Guided Encoders for Pose-Invariant Face Representation Learning

Learning discriminant face representation for pose-invariant face recognition has been identified as a critical issue in visual learning systems. The challenge lies in the drastic changes of facial appearances between the test face and the registered face. To that end, we propose a high-level feature learning framework called "collaborative random faces (RFs)-guided encoders" toward this problem. The contributions of this paper are three fold. First, we propose a novel supervised autoencoder that is able to capture the high-level identity feature despite of pose variations. Second, we enrich the identity features by replacing the target values of conventional autoencoders with random signals (RFs in this paper), which are unique for each subject under different poses. Third, we further improve the performance of the framework by incorporating deep convolutional neural network facial descriptors and linking discriminative identity features from different RFs for the augmented identity features. Finally, we conduct face identification experiments on Multi-PIE database, and face verification experiments on labeled faces in the wild and YouTube Face databases, where face recognition rate and verification accuracy with Receiver Operating Characteristic curves are rendered. In addition, discussions of model parameters and connections with the existing methods are provided. These experiments demonstrate that our learning system works fairly well on handling pose variations.

In Defense of Locality-Sensitive Hashing

Hashing-based semantic similarity search is becoming increasingly important for building large-scale content-based retrieval system. The state-of-the-art supervised hashing techniques use flexible two-step strategy to learn hash functions. The first step learns binary codes for training data by solving binary optimization problems with millions of variables, thus usually requiring intensive computations. Despite simplicity and efficiency, locality-sensitive hashing (LSH) has never been recognized as a good way to generate such codes due to its poor performance in traditional approximate neighbor search. We claim in this paper that the true merit of LSH lies in transforming the semantic labels to obtain the binary codes, resulting in an effective and efficient two-step hashing framework. Specifically, we developed the locality-sensitive two-step hashing (LS-TSH) that generates the binary codes through LSH rather than any complex optimization technique. Theoretically, with proper assumption, LS-TSH is actually a useful LSH scheme, so that it preserves the label-based semantic similarity and possesses sublinear query complexity for hash lookup. Experimentally, LS-TSH could obtain comparable retrieval accuracy with state of the arts with two to three orders of magnitudes faster training speed.

Robust Image Regression Based on the Extended Matrix Variate Power Exponential Distribution of Dependent Noise

Dealing with partial occlusion or illumination is one of the most challenging problems in image representation and classification. In this problem, the characterization of the representation error plays a crucial role. In most current approaches, the error matrix needs to be stretched into a vector and each element is assumed to be independently corrupted. This ignores the dependence between the elements of error. In this paper, it is assumed that the error image caused by partial occlusion or illumination changes is a random matrix variate and follows the extended matrix variate power exponential distribution. This has the heavy tailed regions and can be used to describe a matrix pattern of l×m dimensional observations that are not independent. This paper reveals the essence of the proposed distribution: it actually alleviates the correlations between pixels in an error matrix E and makes E approximately Gaussian. On the basis of this distribution, we derive a Schatten p -norm-based matrix regression model with L_q regularization. Alternating direction method of multipliers is applied to solve this model. To get a closed-form solution in each step of the algorithm, two singular value function thresholding operators are introduced. In addition, the extended Schatten p -norm is utilized to characterize the distance between the test samples and classes in the design of the classifier. Extensive experimental results for image reconstruction and classification with structural noise demonstrate that the proposed algorithm works much more robustly than some existing regression-based methods.

Rate of Convergence of the FOCUSS Algorithm

Focal underdetermined system solver (FOCUSS) is a powerful method for basis selection and sparse representation, where it employs the [Formula: see text]-norm with p ∈ (0,2) to measure the sparsity of solutions. In this paper, we give a systematical analysis on the rate of convergence of the FOCUSS algorithm with respect to p ∈ (0,2) . We prove that the FOCUSS algorithm converges superlinearly for and linearly for usually, but may superlinearly in some very special scenarios. In addition, we verify its rates of convergence with respect to p by numerical experiments.

Robust and Low-Rank Representation for Fast Face Identification With Occlusions

In this paper, we propose an iterative method to address the face identification problem with block occlusions. Our approach utilizes a robust representation based on two characteristics in order to model contiguous errors (e.g., block occlusion) effectively. The first fits to the errors a distribution described by a tailored loss function. The second describes the error image as having a specific structure (resulting in low-rank in comparison with image size). We will show that this joint characterization is effective for describing errors with spatial continuity. Our approach is computationally efficient due to the utilization of the alternating direction method of multipliers. A special case of our fast iterative algorithm leads to the robust representation method, which is normally used to handle non-contiguous errors (e.g., pixel corruption). Extensive results on representative face databases (in constrained and unconstrained environments) document the effectiveness of our method over existing robust representation methods with respect to both identification rates and computational time.

A Locality-Constrained and Label Embedding Dictionary Learning Algorithm for Image Classification

Locality and label information of training samples play an important role in image classification. However, previous dictionary learning algorithms do not take the locality and label information of atoms into account together in the learning process, and thus their performance is limited. In this paper, a discriminative dictionary learning algorithm, called the locality-constrained and label embedding dictionary learning (LCLE-DL) algorithm, was proposed for image classification. First, the locality information was preserved using the graph Laplacian matrix of the learned dictionary instead of the conventional one derived from the training samples. Then, the label embedding term was constructed using the label information of atoms instead of the classification error term, which contained discriminating information of the learned dictionary. The optimal coding coefficients derived by the locality-based and label-based reconstruction were effective for image classification. Experimental results demonstrated that the LCLE-DL algorithm can achieve better performance than some state-of-the-art algorithms.

Tensor LRR and Sparse Coding-Based Subspace Clustering

Subspace clustering groups a set of samples from a union of several linear subspaces into clusters, so that the samples in the same cluster are drawn from the same linear subspace. In the majority of the existing work on subspace clustering, clusters are built based on feature information, while sample correlations in their original spatial structure are simply ignored. Besides, original high-dimensional feature vector contains noisy/redundant information, and the time complexity grows exponentially with the number of dimensions. To address these issues, we propose a tensor low-rank representation (TLRR) and sparse coding-based (TLRRSC) subspace clustering method by simultaneously considering feature information and spatial structures. TLRR seeks the lowest rank representation over original spatial structures along all spatial directions. Sparse coding learns a dictionary along feature spaces, so that each sample can be represented by a few atoms of the learned dictionary. The affinity matrix used for spectral clustering is built from the joint similarities in both spatial and feature spaces. TLRRSC can well capture the global structure and inherent feature information of data and provide a robust subspace segmentation from corrupted data. Experimental results on both synthetic and real-world data sets show that TLRRSC outperforms several established stateof- the-art methods.

Approximate Orthogonal Sparse Embedding for Dimensionality Reduction

Locally linear embedding (LLE) is one of the most well-known manifold learning methods. As the representative linear extension of LLE, orthogonal neighborhood preserving projection (ONPP) has attracted widespread attention in the field of dimensionality reduction. In this paper, a unified sparse learning framework is proposed by introducing the sparsity or L1-norm learning, which further extends the LLE-based methods to sparse cases. Theoretical connections between the ONPP and the proposed sparse linear embedding are discovered. The optimal sparse embeddings derived from the proposed framework can be computed by iterating the modified elastic net and singular value decomposition. We also show that the proposed model can be viewed as a general model for sparse linear and nonlinear (kernel) subspace learning. Based on this general model, sparse kernel embedding is also proposed for nonlinear sparse feature extraction. Extensive experiments on five databases demonstrate that the proposed sparse learning framework performs better than the existing subspace learning algorithm, particularly in the cases of small sample sizes.

Bidirectional Active Learning: A Two-Way Exploration Into Unlabeled and Labeled Data Set

In practical machine learning applications, human instruction is indispensable for model construction. To utilize the precious labeling effort effectively, active learning queries the user with selective sampling in an interactive way. Traditional active learning techniques merely focus on the unlabeled data set under a unidirectional exploration framework and suffer from model deterioration in the presence of noise. To address this problem, this paper proposes a novel bidirectional active learning algorithm that explores into both unlabeled and labeled data sets simultaneously in a two-way process. For the acquisition of new knowledge, forward learning queries the most informative instances from unlabeled data set. For the introspection of learned knowledge, backward learning detects the most suspiciously unreliable instances within the labeled data set. Under the two-way exploration framework, the generalization ability of the learning model can be greatly improved, which is demonstrated by the encouraging experimental results.

Compact and Discriminative Descriptor Inference Using Multi-Cues

Feature descriptors around local interest points are widely used in human action recognition both for images and videos. However, each kind of descriptors describes the local characteristics around the reference point only from one cue. To enhance the descriptive and discriminative ability from multiple cues, this paper proposes a descriptor learning framework to optimize the descriptors at the source by learning a projection from multiple descriptors' spaces to a new Euclidean space. In this space, multiple cues and characteristics of different descriptors are fused and complemented for each other. In order to make the new descriptor more discriminative, we learn the multi-cue projection by the minimization of the ratio of within-class scatter to between-class scatter, and therefore, the discriminative ability of the projected descriptor is enhanced. In the experiment, we evaluate our framework on the tasks of action recognition from still images and videos. Experimental results on two benchmark image and two benchmark video data sets demonstrate the effectiveness and better performance of our method.

Matrix variate distribution-induced sparse representation for robust image classification

Sparse representation learning has been successfully applied into image classification, which represents a given image as a linear combination of an over-complete dictionary. The classification result depends on the reconstruction residuals. Normally, the images are stretched into vectors for convenience, and the representation residuals are characterized by l2 -norm or l1 -norm, which actually assumes that the elements in the residuals are independent and identically distributed variables. However, it is hard to satisfy the hypothesis when it comes to some structural errors, such as illuminations, occlusions, and so on. In this paper, we represent the image data in their intrinsic matrix form rather than concatenated vectors. The representation residual is considered as a matrix variate following the matrix elliptically contoured distribution, which is robust to dependent errors and has long tail regions to fit outliers. Then, we seek the maximum a posteriori probability estimation solution of the matrix-based optimization problem under sparse regularization. An alternating direction method of multipliers (ADMMs) is derived to solve the resulted optimization problem. The convergence of the ADMM is proven theoretically. Experimental results demonstrate that the proposed method is more effective than the state-of-the-art methods when dealing with the structural errors.

Nuclear norm-based 2-DPCA for extracting features from images

The 2-D principal component analysis (2-DPCA) is a widely used method for image feature extraction. However, it can be equivalently implemented via image-row-based principal component analysis. This paper presents a structured 2-D method called nuclear norm-based 2-DPCA (N-2-DPCA), which uses a nuclear norm-based reconstruction error criterion. The nuclear norm is a matrix norm, which can provide a structured 2-D characterization for the reconstruction error image. The reconstruction error criterion is minimized by converting the nuclear norm-based optimization problem into a series of F-norm-based optimization problems. In addition, N-2-DPCA is extended to a bilateral projection-based N-2-DPCA (N-B2-DPCA). The virtue of N-B2-DPCA over N-2-DPCA is that an image can be represented with fewer coefficients. N-2-DPCA and N-B2-DPCA are applied to face recognition and reconstruction and evaluated using the Extended Yale B, CMU PIE, FRGC, and AR databases. Experimental results demonstrate the effectiveness of the proposed methods.

Data imputation through the identification of local anomalies

We introduce a comprehensive and statistical framework in a model free setting for a complete treatment of localized data corruptions due to severe noise sources, e.g., an occluder in the case of a visual recording. Within this framework, we propose: 1) a novel algorithm to efficiently separate, i.e., detect and localize, possible corruptions from a given suspicious data instance and 2) a maximum a posteriori estimator to impute the corrupted data. As a generalization to Euclidean distance, we also propose a novel distance measure, which is based on the ranked deviations among the data attributes and empirically shown to be superior in separating the corruptions. Our algorithm first splits the suspicious instance into parts through a binary partitioning tree in the space of data attributes and iteratively tests those parts to detect local anomalies using the nominal statistics extracted from an uncorrupted (clean) reference data set. Once each part is labeled as anomalous versus normal, the corresponding binary patterns over this tree that characterize corruptions are identified and the affected attributes are imputed. Under a certain conditional independency structure assumed for the binary patterns, we analytically show that the false alarm rate of the introduced algorithm in detecting the corruptions is independent of the data and can be directly set without any parameter tuning. The proposed framework is tested over several well-known machine learning data sets with synthetically generated corruptions and experimentally shown to produce remarkable improvements in terms of classification purposes with strong corruption separation capabilities. Our experiments also indicate that the proposed algorithms outperform the typical approaches and are robust to varying training phase conditions.

Robust Exemplar Extraction Using Structured Sparse Coding

Robust exemplar extraction from the noisy sample set is one of the most important problems in pattern recognition. In this brief, we propose a novel approach for exemplar extraction through structured sparse learning. The new model accounts for not only the reconstruction capability and the sparsity, but also the diversity and robustness. To solve the optimization problem, we adopt the alternating directional method of multiplier technology to design an iterative algorithm. Finally, the effectiveness of the approach is demonstrated by experiments of various examples including traffic sign sequences.

On the parallel efficiency and scalability of the correntropy coefficient for image analysis

Background

Similarity measures have application in many scenarios of digital image processing. The correntropy is a robust and relatively new similarity measure that recently has been employed in various engineering applications. Despite other competitive characteristics, its computational cost is relatively high and may impose hard-to-cope time restrictions for high-dimensional applications, including image analysis and computer vision.

Methods

We propose a parallelization strategy for calculating the correntropy on multi-core architectures that may turn the use of this metric viable in such applications. We provide an analysis of its parallel efficiency and scalability.

Results

The simulation results were obtained on a shared memory system with 24 processing cores for input images of different dimensions. We performed simulations of various scenarios with images of different sizes. The aim was to analyze the parallel and serial fraction of the computation of the correntropy coefficient and the influence of these fractions in its speedup and efficiency.

Conclusions

The results indicate that correntropy has a large potential as a metric for image analysis in the multi-core era due to its high parallel efficiency and scalability.

An ordered-patch-based image classification approach on the image Grassmannian manifold

This paper presents an ordered-patch-based image classification framework integrating the image Grassmannian manifold to address handwritten digit recognition, face recognition, and scene recognition problems. Typical image classification methods explore image appearances without considering the spatial causality among distinctive domains in an image. To address the issue, we introduce an ordered-patch-based image representation and use the autoregressive moving average (ARMA) model to characterize the representation. First, each image is encoded as a sequence of ordered patches, integrating both the local appearance information and spatial relationships of the image. Second, the sequence of these ordered patches is described by an ARMA model, which can be further identified as a point on the image Grassmannian manifold. Then, image classification can be conducted on such a manifold under this manifold representation. Furthermore, an appropriate Grassmannian kernel for support vector machine classification is developed based on a distance metric of the image Grassmannian manifold. Finally, the experiments are conducted on several image data sets to demonstrate that the proposed algorithm outperforms other existing image classification methods.