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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.

We propose a method based on the collaborative Low-Rank Representation (LRR) and Sparse Subspace Clustering (SSC) to cluster data drawn from multiple linear subspaces in a high-dimensional space. Given a set of data vectors, Collaborative Low-Rank and Sparse Subspace Clustering(CLRS) want to seek a better representation among the candidates that represent all vectors as affine combination of the bases in a dictionary. Both theoretical and experimental results show that CLRS is a promising method for subspace segmentation.

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.

Part I Affine geometry: vector spaces matrices systems of linear equations some linear algebra rank determinants affine space - (I) - (II) geometry of affine planes geometry of affine space linear maps linear maps and matrices, affine changes of coordinates linear operators transformation groups. Part II Euclidean geometry: bilinear and quadratic forms diagonalizing quadratic forms scalar product vector product Euclidean space unitary operators and isometries isometries of the plane and of three-dimensional space the complex case.

This presentation is an overview on the integral arithmetic means of functions and mappings in higher dimensions. The barycenters of sets are represented as special cases of integral arithmetic means. The starting point of this approach are centers of linear combinations. These centers can be considered as the source of affine and convex combinations. Convex combinations can be inserted into the integral method, and thus produce the integral arithmetic means.

Graph-based semi-supervised classification is one of the hottest research areas in machine learning and data mining. These methods usually model an entire dataset as a graph, then utilize the structure information extracted by the graph to help with the classification of unlabeled data. Generally speaking, the performance of graph-based semi-supervised classification methods highly depends on the constructed graphs. In this paper, we propose a new kind of graph construction method based on affine subspace sparse representation. The proposed sparse coding method minimizes the construction error of the input signal, considering three constraints: (1) the input signal being approximately reconstructed by the affine combination of the dictionary; (2) the nonnegativity constraint of the reconstruction coefficients; (3) the sparsity constraint of the reconstruction coefficients. Based on the constraints, we present the ι 0-norm constrained optimization problem for sparse coding; then, we propose the algorithm to solve the problem and further construct the ι 0-graph of data. Finally, under the manifold regularization framework, we propose a new kind of semi-supervised classification method by introducing the regularization term that measures the structure preserving error of the ι 0-graph. The proposed semi-supervised classification method has an explicit multiclass classification function and inherits the strong discriminative information from sparse representation. As a result, it has efficient and effective classification ability. Experimental results on artificial and real-world datasets are provided to show the effectiveness of the proposed method.

In this paper, we propose a semiparametric procedure called the “Model Averaging MArginal Regression” (MAMAR) that is flexible for forecasting of time series. This procedure considers approximating a multivariate regression function by an affine combination of one-dimensional marginal regression functions. The weight parameters involved in the approximation are estimated by least squares on the basis of the first-stage nonparametric kernel estimates of the marginal regressions. Under some mild conditions, we have established asymptotic normality for the estimated weights and the regression function in two cases: Case I considers that the number of the covariates is fixed while Case II allows the number of the covariates depending on the sample size and diverging. As the observations are assumed to be stationary and near epoch dependent, the approach developed is applicable to both the estimation and forecasting issues in time series analysis. Furthermore, the method and result are augmented by a simulation study and illustrated by an application in forecasting the high frequency volatility of the FTSE100 index.

We consider a class of matrix spectral norm approximation problems for finding an affine combination of given matrices having the minimal spectral norm subject to some prescribed linear equality and inequality constraints. These problems arise often in numerical algebra, engineering and other areas, such as finding Chebyshev polynomials of matrices and fastest mixing Markov chain models. Based on classical analysis of proximal point algorithms (PPAs) and recent developments on semismooth analysis of nonseparable spectral operators, we propose a semismooth Newton-CG based dual PPA for solving the matrix norm approximation problems. Furthermore, when the primal constraint nondegeneracy condition holds for the subproblems, our semismooth Newton-CG method is proven to have at least a superlinear convergence rate. We also design efficient implementations for our proposed algorithm to solve a variety of instances and compare its performance with the nowadays popular first order alternating direction method of multipliers (ADMM). The results show that our algorithm, which is warm-started with an initial point obtained from the ADMM, substantially outperforms the pure ADMM, especially for the constrained cases and it is able to solve the problems robustly and efficiently to a relatively high accuracy.

we present a method based on a locally affineinvariant constraint for volumetric registration of 3D solid shapes. The core idea of this method is that an affine combination of the given point in 3D solid shapes that are directly connected to the given point, and the corresponding weight of each neighboring point can be obtained by the method of generalized least square. The input of our method is a pair of 3D solid shapes that are represented by a tetrahedral mesh and a voxelized object consisting of a set of voxel cells segmented from Digital Imaging and Communications in Medicine(DICOM) scans. To achieve the registration between the two input DICOM images, we firstly need to do some preprocessing to segment the bones out from the DICOM images and represent the segmented healthy bone and lesion bone using a generic template tetrahedral mesh and a set of voxel cells, respectively. Secondly we apply the standard Iterative Closest Point (ICP) method to briefly align the tetrahedral mesh and the voxelized object. Thirdly we execute a novel registration process that uses as much volumetric information and local geometry information as possible while deforming the generic template tetrahedral mesh of a healthy human bone towards the undelying geometry of the voxel cells. Compared with the previous methods that are based on point or surface, our method requires less auxiliary variables and can better capture the volumetric information of the 3D solid shapes, such as the thickness of the bones. Besides that, using a tetrahedral mesh to represent a solid shape can make the precision of registration greatly improved. Our experimental results demonstrate that the proposed method is robust and is of high registration accuracy.

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.

We present a novel method, called Simplex Assembly , to compute inversion-free mappings with low or bounded distortion on simplicial meshes. Our method involves two steps: simplex disassembly and simplex assembly. Given a simplicial mesh and its initial piecewise affine mapping, we project the affine transformation associated with each simplex into the inversion-free and distortion-bounded space. The projection disassembles the input mesh into disjoint simplices. The disjoint simplices are then assembled to recover the original connectivity by minimizing the mapping distortion and the difference of the disjoint vertices with respect to the piecewise affine transformations, while the piecewise affine mapping is restricted inside the feasible space. Due to the use of affine transformations as variables, our method explicitly guarantees that no inverted simplex occurs, and that the mapping distortion is below the bound during the optimization. Compared with existing methods, our method is robust to an initialization with many inverted elements and positional constraints. We demonstrate the efficiency and robustness of our method through a variety of geometric processing tasks.

Correlation filter achieves promising performance with high speed in visual tracking. However, conventional correlation filter based trackers cannot tackle affine transformation issues such as scale variation, rotation and skew. To address this problem, in this paper, we propose a part-based representation tracker via kernelized correlation filter (KCF) for visual tracking. A Spatial-Temporal Angle Matrix (STAM), severed as confidence metric, is proposed to select reliable patches from parts via multiple correlation filters. These stable patches are used to estimate a 2D affine transformation matrix of the target in a geometric method. Specially, the whole combination scheme for these stable patches is proposed to exploit sampling space in order to obtain numerous affine matrices and their corresponding candidates. The diversiform candidates would help to seek for the optimal candidate to represent the object's accurate affine transformation in a higher probability. Both qualitative and quantitative evaluations on VOT2014 challenge and Object Tracking Benchmark (OTB) show that the proposed tracking method achieves favorable performance compared with other state-of-the-art methods.

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 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.

Recent subspace clustering algorithms, which use sparse or low-rank representations, conduct clustering by considering the errors and noises into their objective functions. Then, the similarity matrix is solved via alternating direction method of multipliers. However, these approaches are subject to the restriction that the characteristic of errors and outliers in sample points should be known as the prior information. Furthermore, these algorithms are time-consuming during the iterative process. Motivated by this observation, this paper proposes a new subspace clustering algorithm: an affine subspace clustering algorithm based on ridge regression. The method introduces ridge regression as objective function which applies affine criteria into subspace clustering. An analytic solution to the problem has been determined for the coefficient matrix. Experimental results obtained on face datasets demonstrate that the proposed method not only improves the accuracy of the clustering results, but also enhances the robustness. Furthermore, the proposed method reduces the computational complexity.

The linear semidefinite programming problem is considered. The dual affine scaling method in which all current iterations belong to the feasible set is proposed for its solution. Moreover, the boundaries of the feasible set may be reached. This method is a generalization of a version of the affine scaling method that was earlier developed for linear programs to the case of semidefinite programming.

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

Since unity step size could guarantee the fastest convergence and more detailed analysis for the affine projection (AP) algorithm, a statistical tracking behavior of AP algorithm is discussed in this paper. Deterministic recursive equations are derived for the mean weight error and mean-square error. All the possible correlations between the adaptive filtering coefficients and the past measurement noise are considered as well.

We investigate an algorithm to find a point of an affine subspace in the positive orthant such that it is the closest one to the original point with respect to the Kullback---Leibler distance. This problem is solved by means of the classical Darroch---Ratcliff algorithm (see [1]), while we use ideas of the information geometry founded by Chentsov (see [2]) and Csiszar (see [3]). The main theorem of the present work proves the convergence of that algorithm (the method of the proof is different from previous ones). The proposed algorithm can be applied, e.g., to find the maximum likelihood estimates in an exponential family (see the last section of the paper).

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.

Using the classical Elie Cartan method, we construct a canonical frame for a smooth curve in multidimensional affine space with canonical and non-canonical parameters.

Quantum circuits of the generalized affine transform are devised based on the novel enhanced quantum representation of digital images. A novel quantum image encryption algorithm combining the generalized affine transform with logistic map is suggested. The gray-level information of the quantum image is encrypted by the XOR operation with a key generator controlled by the logistic map, while the position information of the quantum image is encoded by the generalized affine transform. The encryption keys include the independent control parameters used in the generalized affine transform and the logistic map. Thus, the key space is large enough to frustrate the possible brute-force attack. Numerical simulations and analyses indicate that the proposed algorithm is realizable, robust and has a better performance than its classical counterpart in terms of computational complexity.

We prove some general stability results for a family of equations, which generalizes the equation of p-Wright affine functions. In this way we obtain some hyperstability properties for those equations, as well. We also provide some applications of those outcomes in proving inequalities characterizing the inner product spaces and stability of *-homomorphisms of C*-algebras. The main tool in the proofs is a fixed point result in Brzd?k, Chudziak, Pales (2011).

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.

We study projections onto a subspace and reflections with respect to a subspace in an arbitrary vector space with an inner product. We give necessary and sufficient conditions for two such transformations to commute. We then generalize the result to affine subspaces and transformations.

In this paper, we discuss the digitization and applications of smooth affine shear tight frames, a recently developed new class of directional multiscale representation systems. An affine wavelet tight frame is generated by isotropic dilations and translations of directional wavelet generators, while an affine shear tight frame is generated by anisotropic dilations, shears, and translations of shearlet generators. These two tight frames are actually connected in the sense that an affine shear tight frame can be obtained from an affine wavelet tight frame through subsampling. Consequently, an affine shear tight frame has an underlying filter bank from the MRA structure of its associated affine wavelet tight frame. We discuss the digitization of digital affine shear filter banks associated with the affine shear tight frames. Moreover, we provide the detailed algorithmic steps for both the forward and backward digital affine shear transforms. Analysis of the redundancy rate and computational complexity shows...

Piecewise affine models often provide a good approximation to describe continuous systems, but may involve a high degree of simplification. To compare solutions of the continuous and piecewise affine models, it is important to quantify the differences between solutions in each region of the state space. As an approach, we will use enveloping "bands" to characterize continuous activation or inhibition functions, and then describe the differences between continuous and piecewise affine solutions in terms of the width δ of these bands. As a case study, we will consider the negative feedback loop, a classical motif in two dimensions which results in oscillating behaviour. For this example, it is shown that the two types of models may differ only on a compact invariant set (the interior of a limit cycle), whose diameter is a function of the band width δ.

Capon [Proc. IEEE (1969)] designed the minimum variance distortionless response (MVDR) beamformer to obtain spectral estimates with better resolution than the conventional averaged-periodogram estimator. The MVDR spectrum is a function of the inverse of the sample covariance matrix (SCM), which often must be regularized prior to inversion. To address conditioning problems, Abraham and Owsley [IEEE Oceans (1990)] developed a modified MVDR approach called dominant mode rejection (DMR). The DMR beamformer defines its weights using a structured covariance consisting of a low-rank interference subspace plus an orthogonal noise subspace. It assumes the rank of the interference is known. Recently, Buck and Singer [IEEE SAM (2018)] proposed the blended DMR beamformer that eliminates the need for rank estimation by defining a weight vector that is an affine combination of fixed-rank DMR beamformers. This talk investigates frequency-wavenumber estimation using blended DMR, focusing particularly on efficient implementations for large linear arrays. Adapting the approach described by Therrien [Prentice Hall (1992)] for MVDR, the blended DMR spectrum for an equally spaced array can be computed using fast Fourier transforms of the sample eigenvectors. Results will be illustrated using experimental data from underwater vertical arrays. [Work supported by ONR.]

The Maxwell extension of the affine algebra in four dimensions with additional tensor generator is given. Using the methods of nonlinear realizations, we find the transformation rules for the group parameters and the corresponding generators. Gauging the Maxwell-affine algebra we present two possible invariant actions for gravity: one is first order and the other one is second order in the affine curvature. We notice that equations of motion for the action, second order in the affine curvature, lead to the generalized Bianchi identities on the choice of appropriate coefficients for a particular solution of the constraint equation.

We formulate an affine-invariant implementation of the accelerated first-order algorithm in [Y. Nesterov, Dokl. Math., 27 (1983), pp. 372--376]. Its complexity bound is proportional to an affine-invariant regularity constant defined with respect to the Minkowski gauge of the feasible set. We extend these results to more general problems, optimizing Holder smooth functions using $p$-uniformly convex prox terms, and derive an algorithm whose complexity better fits the geometry of the feasible set and adapts to both the best Holder smoothness parameter and the best gradient Lipschitz constant. Finally, we detail matching complexity lower bounds when the feasible set is an $\ell_p$ ball. In this setting, our upper bounds on iteration complexity for the algorithm in [Y. Nesterov, Dokl. Math., 27 (1983), pp. 372--376] are thus optimal in terms of target precision, smoothness, and problem dimension.

Affine λ-terms are λ-terms in which each bound variable occurs at most once, and linear λ-terms are λ-terms in which each bound variable occurs once and only once. In this article, we count the number of affine closed λ-terms of size n , linear closed λ-terms of size n , affine closed β-normal forms of size n , and linear closed β-normal forms of size n , for several measures of the size of λ-terms. From these formulas, we show how we can derive programs for generating all the terms of size n for each class. The foundation of all of this is a specific data structure, made of contexts in which one counts all the holes at each level of abstractions by λ’s.

International audience; We construct algebraic families of exotic affine 3-spheres, that is, smooth affine threefolds diffeomorphic to a non-degenerate smooth complex affine quadric of dimension 3 but non-algebraically isomorphic to it. We show in particular that for every smooth topologically contractible affine surface $S$ with trivial automorphism group, there exists a canonical smooth family of pairwise non-isomorphic exotic affine 3-spheres parametrized by the closed points of $S$.

The behavior of affine processes, which are ubiquitous in a wide range of applications, depends crucially on the choice of state space. We study the case where the state space is compact, and prove in particular that (i) no diffusion is possible; (ii) jumps are possible and enforce a grid-like structure of the state space; (iii) jump components can feed into drift components, but not vice versa. Using our main structural theorem, we classify all bivariate affine processes with compact state space. Unlike the classical case, the characteristic function of an affine process with compact state space may vanish, even in very simple cases., Electronic Journal of Probability, 23, ISSN:1083-6489