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Spectra from a single intimate (nonlinear) mixture can be modeled as data points drawn from a smooth manifold. Spectral data sets containing hyperspectral observations of multiple intimate mixtures with some constituent materials in common can, therefore, be modeled as data clouds, in which each point is drawn from a union of manifolds that share a boundary. Two important steps in the processing of such data are to: 1) identify the different mixture manifolds present in the data and 2) invert the nonlinear mixing function by mapping each mixture manifold into some low-dimensional Euclidean space (manifold embedding). The present state-of-the-art algorithms for joint manifold clustering and embedding perform poorly for hyperspectral data, particularly in the embedding task. We propose a novel reconstruction-based algorithm for the improved clustering and the embedding of mixture manifolds. The algorithm attempts to reconstruct each target point as an affine combination of its nearest neighbors with an additional rank penalty on the neighborhood to ensure that only the neighbors on the same manifold as the target point are used in the reconstruction. The reconstruction matrix generated by this technique is both block diagonal and neighborhood-based, leading to improved clustering and embedding. The improved performance of the algorithm against its competitors is exhibited on a variety of simulated and real mixture data sets.

Prediction interval modelling has been proposed in the literature to characterize uncertain phenomena and provide useful information from a decision-making point of view. In most of the reported studies, assumptions about the data distribution are made and/or the models are trained at one step ahead, which can decrease the quality of the interval in terms of the information about the uncertainty modelled for a higher prediction horizon. In this paper, a new prediction interval modelling methodology based on fuzzy numbers is proposed to solve the abovementioned drawbacks. Fuzzy and neural network prediction interval models are developed based on this proposed methodology by minimizing a novel criterion that includes the coverage probability and normalized average width. The fuzzy number concept is considered because the affine combination of fuzzy numbers generates, by definition, prediction intervals that can handle uncertainty without requiring assumptions about the data distribution. The developed models are compared with a covariance-based prediction interval method, and high-quality intervals are obtained, as determined by the narrower interval width of the proposed method. Additionally, the proposed prediction intervals are tested by forecasting up to two days ahead of the load of the Huatacondo microgrid in the north of Chile and the consumption of the residential dwellings in the town of Loughborough, UK. The results show that the proposed models are suitable alternatives to electrical consumption forecasting because they obtain the minimum interval widths that characterize the uncertainty of this type of stochastic process. Furthermore, the information provided by the obtained prediction interval could be used to develop robust energy management systems that, for example, consider the worst-case scenario.

Mixed Oblique Extension Principles (MOEP) provide an important method to construct affine dual frames from refinable functions. This paper addresses MOEP under the setting of reducing subspaces of L 2 ( R d ) . We obtain an MOEP for (non)homogeneous affine dual frames and (non)homogeneous affine Parseval frames.

For a number of years we have been investigating the geometric and algebraic properties of a family of discrete sets of points in Euclidean space generated by a simple binary operation: pairwise affine combination by a xed parameter, which we call xed-parameter extrapolation. By varying the parameter and the set of initial points, a large variety of point sets emerge. To our surprise, many of these sets display aperiodic order and share properties of so-called \quasicrystals" or \quasilattices." Such sets display some ordered crystal-like properties (e.g., generation by a regular set of local rules, such as a nite set of tiles, and possessing a kind of repetitivity), but are \aperiodic" in the sense that they have no translational symmetry. The most famous of such systems are Penrose's aperiodic tilings of the plane [16]. Mathematically, a widely accepted way of capturing the idea of aperiodic order is via the notion of Meyer sets, which we de ne later. Our goal is to classify the sets generated by xed-parameter extrapolation in terms of a number of these properties, including but not limited to aperiodicity, uniform discreteness, and relative density. We also seek to determine exactly which parameter values lead to which types of sets.

Adaptive filters are at the core of many signal processing applications, ranging from acoustic noise supression to echo cancelation, array beamforming, channel equalization, to more recent sensor network applications in surveillance, target localization, and tracking. A trending approach in this direction is to recur to in-network distributed processing in which individual nodes implement adaptation rules and diffuse their estimation to the network. When the a priori knowledge about the filtering scenario is limited or imprecise, selecting the most adequate filter structure and adjusting its parameters becomes a challenging task, and erroneous choices can lead to inadequate performance. To address this difficulty, one useful approach is to rely on combinations of adaptive structures. The combination of adaptive filters exploits to some extent the same divide and conquer principle that has also been successfully exploited by the machine-learning community (e.g., in bagging or boosting). In particular, the problem of combining the outputs of several learning algorithms (mixture of experts) has been studied in the computational learning field under a different perspective: rather than studying the expected performance of the mixture, deterministic bounds are derived that apply to individual sequences and, therefore, reflect worst-case scenarios. These bounds require assumptions different from the ones typically used in adaptive filtering, which is the emphasis of this overview article. We review the key ideas and principles behind these combination schemes, with emphasis on design rules. We also illustrate their performance with a variety of examples.

The goal of our paper is to semantically edit parts of an image matching a given text that describes desired attributes (e.g., texture, colour, and background), while preserving other contents that are irrelevant to the text. To achieve this, we propose a novel generative adversarial network (ManiGAN), which contains two key components: text-image affine combination module (ACM) and detail correction module (DCM). The ACM selects image regions relevant to the given text and then correlates the regions with corresponding semantic words for effective manipulation. Meanwhile, it encodes original image features to help reconstruct text-irrelevant contents. The DCM rectifies mismatched attributes and completes missing contents of the synthetic image. Finally, we suggest a new metric for evaluating image manipulation results, in terms of both the generation of new attributes and the reconstruction of text-irrelevant contents. Extensive experiments on the CUB and COCO datasets demonstrate the superior performance of the proposed method. Code is available at https://github.com/mrlibw/ManiGAN., Comment: CVPR 2020

Diffusion adaptation is a powerful strategy for distributed estimation and learning over networks. Motivated by the concept of combining adaptive filters, this work proposes a combination framework that aggregates the operation of multiple diffusion strategies for enhanced performance. By assigning a combination coefficient to each node, and using an adaptation mechanism to minimize the network error, we obtain a combined diffusion strategy that benefits from the best characteristics of all component strategies simultaneously in terms of excess-mean-square error (EMSE). Analyses of the universality are provided to show the superior performance of affine combination scheme and to characterize its behavior in the mean and mean-square sense. Simulation results are presented to demonstrate the effectiveness of the proposed strategies, as well as the accuracy of theoretical findings., 31 pages

We describe invariants of centers of ellipse-inscribed triangle families with two vertices fixed to the ellipse boundary and a third one which sweeps it. We prove that: (i) if a triangle center is a fixed affine combination of barycenter and orthocenter, its locus is an ellipse; (ii) and that over the family of said affine combinations, the centers of said loci sweep a line; (iii) over the family of parallel fixed vertices, said loci rigidly translate along a second line. Additionally, we study invariants of the envelope of elliptic loci over combinations of two fixed vertices on the ellipse., Comment: 17 pages, 10 figures, 3 tables, 6 video links

Combining diffusion strategies with complementary properties enables enhanced performance when they can be run simultaneously. In this article, we first propose two schemes for the convex combination of two diffusion strategies, namely, the power-normalized scheme and the sign-regressor scheme. Then, we conduct theoretical analysis for one of the schemes, i.e., the power-normalized one. An analysis of universality shows that it cannot perform worse than any of its component strategies in terms of the excess mean-square-error (EMSE) at steady state, and sometimes even better. An analysis of stability also reveals that it is more stable than affine combination schemes already proposed by the authors in the literature. Next, several adjustments are proposed to further improve the performance of convex combination schemes. A discussion about the computational and communication complexity is provided, as well as a comparison between convex and affine combination schemes. Finally, simulation results are shown to demonstrate their effectiveness, the accuracy of the theoretical results, and the improved stability of the convex power-normalized scheme over the affine one.

Motivated to apply sustainable supply chain principles to air-pollution control systems, this paper presents a dynamic inventory-management approach where substitution is possible to maintain these systems’ equipment. An air-pollution control system’s subsequent reliability depends on the replacement equipment selected. The corresponding problem is formulated as a stochastic dynamic program. Because the state and action space are prohibitively large, the approximate policy iteration algorithm is adapted to generate high-quality solutions. Therefore, this work replaces the value function with an affine combination of nonlinear basis functions and shows that a relaxation of the policy improvement step requires the solving of a mixed integer linear program. This approach helps in the designing of an algorithm that improves the quality of the approximation by solving a convex optimization problem. To assess the quality of resulting solutions, a lower bound is developed by considering a relaxation of the problem. In addition, two classes of heuristics are proposed based on a rolling-horizon two-stage stochastic programming formulation of the problem and a standard base-stock ordering policy. The performance of proposed policies is tested on a variety of settings, and results show that the approximate dynamic programming policies are near-optimal in the settings of interest and significantly outperform available benchmarks. The following analysis reveals that the proposed inventory replenishment policies resemble a base-stock policy with occasional deviations, and assignment and substitution decisions are determined by balancing the reliability with ordering, holding, and shortage costs. The online supplement is available at https://doi.org/10.1287/ijoc.2017.0794 .

We present an algorithmic and visual grouping of participants and eye-tracking metrics derived from recorded eye-tracking data. Our method utilizes two well-established visualization concepts. First, parallel coordinates are used to provide an overview of the used metrics, their interactions, and similarities, which helps select suitable metrics that describe characteristics of the eye-tracking data. Furthermore, parallel coordinates plots enable an analyst to test the effects of creating a combination of a subset of metrics resulting in a newly derived eye-tracking metric. Second, a similarity matrix visualization is used to visually represent the affine combination of metrics utilizing an algorithmic grouping of subjects that leads to distinct visual groups of similar behavior. To keep the diagrams of the matrix visualization simple and understandable, we visually encode our eye-tracking data into the cells of a similarity matrix of participants. The algorithmic grouping is performed with a clustering based on the affine combination of metrics, which is also the basis for the similarity value computation of the similarity matrix. To illustrate the usefulness of our visualization, we applied it to an eye-tracking data set involving the reading behavior of metro maps of up to 40 participants. Finally, we discuss limitations and scalability issues of the approach focusing on visual and perceptual issues., 17 pages, 15 figures

Subspace clustering is an important problem in machine learning with many applications in computer vision and pattern recognition. Prior work has studied this problem using algebraic, iterative, statistical, low-rank and sparse representation techniques. While these methods have been applied to both linear and affine subspaces, theoretical results have only been established in the case of linear subspaces. For example, algebraic subspace clustering (ASC) is guaranteed to provide the correct clustering when the data points are in general position and the union of subspaces is transversal . In this paper we study in a rigorous fashion the properties of ASC in the case of affine subspaces. Using notions from algebraic geometry, we prove that the homogenization trick , which embeds points in a union of affine subspaces into points in a union of linear subspaces, preserves the general position of the points and the transversality of the union of subspaces in the embedded space, thus establishing the correctness of ASC for affine subspaces.

We study the regularity properties of the value function associated with an affine optimal control problem with quadratic cost plus a potential, for a fixed final time and initial point. Without assuming any condition on singular minimizers, we prove that the value function is continuous on an open and dense subset of the interior of the attainable set. As a byproduct we obtain that it is actually smooth on a possibly smaller set, still open and dense.

We introduce the two-parameter quantum affine algebra U r , s ( gl ˆ n ) via the RTT realization. The Drinfeld realization is given and the type A quantum affine algebra is proved to be a special subalgebra of our extended algebra.

In this paper, we introduce an affine scaling cubic regularization algorithm for solving optimization problem without available derivatives subject to bound constraints employing a polynomial interpolation approach to handle the unavailable derivatives of the original objective function. We first define an affine scaling cubic model of the approximate objective function which is obtained by the polynomial interpolation approach with an affine scaling method. At each iteration a candidate search direction is determined by solving the affine scaling cubic regularization subproblem and the new iteration is strictly feasible by way of an interior backtracking technique. The global convergence and local superlinear convergence of the proposed algorithm are established under some mild conditions. Preliminary numerical results are reported to show the effectiveness of the proposed algorithm.

We prove that generically, for a self-affine set in $\mathbb{R}^d$, removing one of the affine maps which defines the set results in a strict reduction of the Hausdorff dimension. This gives a partial positive answer to a folklore open question., Comment: arXiv admin note: text overlap with arXiv:1609.07360

Nonlinear single-input affine systems represented in a canonical (normal) form are considered. The control resource is assumed to be constrained. For a closed-loop system obtained by applying a linearizing feedback, the problem of finding an estimate of the attraction domain is set. A method for constructing an ellipsoidal estimate that is based on results of absolute stability theory is suggested. Construction of the estimate reduces to solving a system of linear matrix inequalities. The discussion is illustrated by numerical examples.

In this paper we introduce a notion of a para-complex affine hypersphere. We give a complete local classification of such hypersurfaces and give several examples. It turns out that every para-complex affine hypersphere can be constructed from (real) affine hyperspheres. As an application, we classify all 2-dimensional para-complex affine hyperspheres.

A mental health trial is analyzed using a dose–response model, in which the number of sessions attended by the patients is deemed indicative of the dose of psychotherapeutic treatment. Here, the parameter of interest is the difference in causal treatment effects between the subpopulations that take part in different numbers of therapy sessions. For this data set, interactions between random treatment allocation and prognostic baseline variables provide the requisite instrumental variables. While the corresponding two‐stage least squares (TSLS) estimator tends to have smaller bias than the ordinary least squares (OLS) estimator; the TSLS suffers from larger variance. It is therefore appealing to combine the desirable properties of the OLS and TSLS estimators. Such a trade‐off is achieved through an affine combination of these two estimators, using mean squared error as a criterion. This produces the semi‐parametric Stein‐like (SPSL) estimator as introduced by Judge and Mittelhammer (2004). The SPSL estimator is used in conjunction with multiple imputation with chained equations, to provide an estimator that can exploit all available information. Simulated data are also generated to illustrate the superiority of the SPSL estimator over its OLS and TSLS counterparts. A package entitled SteinIV implementing these methods has been made available through the R platform. © 2017 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.

In this paper we study fundamental equations of geodesic mappings of manifolds with affine connection onto symmetric manifolds. We obtain fundamental equations of this problem. At the end of our paper we demonstrate example of studied mappings.

We study affine Bessel sequences in connection with the spectral theory and the multishift structure in Hilbert space. We construct a non-Besselian affine system fun(x)g1 n=0 generated by continuous periodic function u(x). The result is based on Nikishin?s example concerning convergence in measure. We also show that affine systems fun(x)g1 n=0 generated by any Lipchitz function u(x) are Besselian.

This paper proposes a robust set-membership affine projection algorithm with coefficient vector reuse (RSM-APA-CVR) for high background noise environment. In the proposed algorithm, the sum of the squared $$L_{2}$$ norms of the differences between the updated weight vector and past weight vectors is minimized and a new robust error bound is designed. Simulation results in acoustic echo cancellation context show that the proposed algorithm has faster convergence rate and smaller steady-state misalignment as compared to the conventional RSM-APA.

Affine invariant measures are powerful tools to develop robust shape descriptors that can be applied, for example, to shape matching, shape retrieval, or symmetry detection problems. In this work we introduce estimators for the affine structure of surfaces represented by triangle meshes, i.e. affine co-normal and normal vectors, affine curvature tensors, affine mean and Gaussian curvatures, and affine principal directions and curvatures. The proposed method estimates the affine normal using a finite differences scheme together with a least-squares approximation, followed by a weighted average strategy to approach discrete affine curvature tensors. When compared to the exact geometric measures of analytic models, experiments on regular meshes obtain small error, which decreases for finer meshes, and outperforms the state-of-the-art method in some cases. Experiments to evaluate affine invariance show that the difference between measures before and after equi-affine transformations remains small even after large deformations.

We propose a composite step method, designed for equality constrained optimization with partial differential equations. Focus is laid on the construction of a globalization scheme, which is based on cubic regularization of the objective and an affine covariant damped Newton method for feasibility. We show finite termination of the inner loop and fast local convergence of the algorithm. Numerical results are shown for optimal control problems subject to a nonlinear heat equation.

A generalized polygon is an ordered set of vertices. This notion generalizes the concept of the boundary of a polygonal shape because self-intersections are allowed. In this paper we study the problem of matching generalized polygons under affine transformations. Our approach is based on invariants. Firstly we associate an ordered set of complex numbers with each polygon and construct a collection of complex scalar functions on the space of plane polygons. These invariant functions are defined as quotients of the so-called Fourier descriptors, also known as discrete Fourier transforms.Each one of these functions is invariant under similarity transformations; that is, the function associates the same complex number to similar polygons. Moreover, if two polygons are affine related (one of them is the image of the other under an affine transformation), the pseudo-hyperbolic distance between their associated values is a constant that depends only on the affine transformation involved, but independent of the polygons.More formally, given a collection { Z 1 , Z 2 , ź , Z m } of n-sided polygons in the plane and a query polygon W, we give algorithms to find all Z ź such that f ( Z ź ) = W + Δ W , where f is an unknown affine transformation and Δ W = ( Δ w 1 , ź , Δ w n ) with | Δ w k | ź ź , where ź is certain tolerance.

This work presents a data driven control strategy able to track a set point without steady-state error. The control sequence is computed as an affine combination of past control signals, which belong to a set of trajectories stored in a process historian database. This affine combination is computed so that the variance of the tracking error is minimised. It is shown that offset free control, that is zero mean tracking error, is achieved under the assumption that the state is measurable, the underlying dynamics are linear and the trajectories of the database share the same error dynamics and are in turn offset free. The proposed strategy learns the underlying controller stored in the database while maintaining its offset free tracking capability in spite of differences in the reference, disturbances and operating conditions. No training phase is required and newly obtained process data can be easily taken into account. The proposed strategy, related to direct weight optimisation learning techniques, is tested on a process control trainer. MINECO-Spain and FEDER Funds project DPI2016-76493-C3-1-R University of Seville(Spain) grant 2014/425

The Straight-Through Estimator (STE) is widely used for back-propagating gradients through the quantization function, but the STE technique lacks a complete theoretical understanding. We propose an alternative methodology called alpha-blending (AB), which quantizes neural networks to low-precision using stochastic gradient descent (SGD). Our method (AB) avoids STE approximation by replacing the quantized weight in the loss function by an affine combination of the quantized weight w_q and the corresponding full-precision weight w with non-trainable scalar coefficient $\alpha$ and $1-\alpha$. During training, $\alpha$ is gradually increased from 0 to 1; the gradient updates to the weights are through the full-precision term, $(1-\alpha)w$, of the affine combination; the model is converted from full-precision to low-precision progressively. To evaluate the method, a 1-bit BinaryNet on CIFAR10 dataset and 8-bits, 4-bits MobileNet v1, ResNet_50 v1/2 on ImageNet dataset are trained using the alpha-blending approach, and the evaluation indicates that AB improves top-1 accuracy by 0.9%, 0.82% and 2.93% respectively compared to the results of STE based quantization., Comment: conference version accepted by IJCAI-2019

This paper investigates affine control systems with jumps for which the ideal If(g1, …, gm) generated by the drift vector field f in the Lie algebra L(f, g1, …, gm) can be imbedded as a kernel of a linear first-order partial differential equation. It will lead us to uncontrollable affine control systems with jumps for which the corresponding reachable sets are included in explicitly described differentiable manifolds.

In this work we give a systemic study of affine translation surfaces in affine 3-dimensional space. Specifically, we obtain the complete classification of minimal affine translation surfaces. Moreover, we consider affine translation surfaces with some natural geometric conditions, such as constant affine mean curvature and constant Gauss–Kronecker curvature. Some characterization results with these geometric conditions are also obtained.

The goal of this paper is to clarify when a stochastic partial differential equation with an affine realization admits affine state processes. This includes a characterization of the set of initial points of the realization. Several examples, as the HJMM equation from mathematical finance, illustrate our results., 27 pages

Sufficient conditions for the stability of a nonlinear affine system subject to interval initial conditions are established. These conditions are based on new estimates of the norms of the solutions of the systems of perturbed equations of motion. This stabilization method is used to analyze an electromechanical system with permanent magnet

In this paper we present an affine combination strategy for two adaptive filters. One filter is designed to handle sparse impulse responses and the other one performs better if impulse response is dispersive. Filter outputs are combined using an adaptive mixing parameter and the resulting output shows a better performance than each of the combining filters separately. We also demonstrate that affine combination results in faster convergence than a convex combination of two adaptive filters.

Optimal Sobolev norms under volume preserving affine transformations are considered. It turns out that this minimal transform is equivalent to the ( p , 2 ) Fisher information matrix defined by Lutwak, Lv, Yang, and Zhang. Furthermore, some analytic inequalities regarding to the L p affine and L p sine energies for the optimal function are investigated.

This paper aims to solve the keypoint correspondence problem in lunar surface images, a typical correspondence task under point ambiguity. Point ambiguity may be caused by repetitive patterns, cluttered scenes, and outliers in the images, which makes the local descriptors less discriminative. In this paper we introduce locally affine-invariance constraints on graphs to tackle the keypoint correspondence problem under point ambiguity. The key idea is that each point can be represented with the affine combination of its neighbors. It is suitable for our problem because it is not only invariant to scale and rotational change, but also more resistant to outliers. Specifically, we introduce the locally affine-invariance constraints into the subgraph matching problem and the common subgraph matching problem. The locally affine-invariance constraint is not directly applicable on common subgraph matching due to its dependency on awareness of selected keypoints. This problem is approximately addressed by solving a series of reliable matching identification and rematching problems. In the experiments, we first apply the proposed method on standard graph matching datasets to evaluate its effectiveness on general correspondence problem under point ambiguity, and second validate the applicability on the lunar surface image dataset.

Shape is considered to be one of the most promising tools to represent and recognise an object. In this study, an effective and rigorous shape matching algorithm is developed based on a new descriptor and relaxation labelling technique. For each contour point, the descriptor captures the distribution of all points within the shape region along the vector perpendicular to that from the centroid to the point. In addition to stable affine invariance, the descriptor is robust to noise since it makes use of all points in the shape region. The descriptor distance is used to initialise the contour point matching probability, and relaxation labelling technique is utilised to update the matching probability using a new compatibility coefficient function, which is defined based on the shape projection preserving characteristic. The experiments on synthetic and real remote sensing data are provided to test the performance of the authors’ proposed algorithm. Compared to other four state-of-the-art contour-based shape matching algorithms, their algorithm is more robust and capable of shape matching under affine transformations and noise.

In this paper, the relative p p -affine capacities are introduced, developed, and subsequently applied to the trace theory of affine Sobolev spaces. In particular, we geometrically characterize such a nonnegative Radon measure μ \mu given on an open set O ⊆ R n \mathcal {O}\subseteq \mathbb R^n that naturally induces an embedding of the p p -affine Sobolev class W 0 , d 1 , p ( O ) {W}^{1,p}_{0,d}(\mathcal {O}) into the Lebesgue space L q ( O , μ ) L^q(\mathcal {O},\mu ) (under 1 ≤ p ≤ q > ∞ 1\le p\le q>\infty ) and the exponentially-integrable Lebesgue space exp ⁡ ( ( n ω n 1 n | f | ) n / ( n − 1 ) ) ∈ L 1 ( O , μ ) \exp \big ((n\omega _n^\frac 1n|f|)^{n/(n-1)}\big )\in L^1(\mathcal {O},\mu ) (under p = n p=n ) as well as the Lebesgue space L ∞ ( O , μ ) L^\infty (\mathcal {O},\mu ) (under n > p > ∞ n>p>\infty ) with μ ( O ) > ∞ \mu (\mathcal {O})>\infty . The results discovered here are new and nontrivial.

In this note, we determined the distance signatures of the incidence matrices of affine resolvable designs. This proves a conjecture by Kohei Yamada., Comment: 5 pages

The statistical affine shape theory is set in this work in the context of real normed division algebras. The general densities apply for every field: real, complex, quaternion, octonion, and for any noncentral and non-isotropic elliptical distribution; then the separated published works about real and complex shape distributions can be obtained as corollaries by a suitable selection of the field parameter and univariate integrals involving the generator elliptical function. As a particular case, the complex normal affine density is derived and applied in brain magnetic resonance scans of normal and schizophrenic patients. Statistical affine shape theory in the context of real normed division algebras is studied.Affine shape density is obtained.The complex normal affine density is derived and applied in brain magnetic resonance scans of normal and schizophrenic patients.

The aim in this paper is to construct an affine transformation using the classical physics analogy between the fields of optics and mechanics. Since optics and mechanics both have symplectic structures, the concept of optics can be replaced by that of mechanics and vice versa. We list the four types of eikonal (generating functions). We also introduce a unitary operator for the affine transformation. Using the unitary operator, the kernel (propagator) is calculated and the wavization (quantization) of the Gabor function is discussed. The dynamic properties of the affine transformed Wigner function are also discussed.

We obtain the general solution of the following functional equation f(kx1+x2+???+xk)+f(x1+kx2+???+xk)+???+f(x1+x2+???+kxk)+f(x1)+ f(x2)+???+ f(xk)= 2kf(x1+ x2+???+xk), k ? 2. We establish the Hyers-Ulam-Rassias stability of the above functional equation in the fuzzy normed spaces. More precisely, we show under suitable conditions that a fuzzy q-almost affine mapping can be approximated by an affine mapping. Further, we determine the stability of same functional equation by using fixed point alternative method in fuzzy normed spaces.

The affine rank minimization problem is to find a low-rank matrix satisfying a set of linear equations, which includes the well-known matrix completion problem as a special case and draws much attention in recent years. In this paper, a new model for affine rank minimization problem is proposed. The new model not only enhances the robustness of affine rank minimization problem, but also leads to high nonconvexity. We show that if the classical projected gradient method is applied to solve our new model, the linear convergence rate can be established under some conditions. Some preliminary experiments have been conducted to show the efficiency and effectiveness of our method.