symmetric positive definite manifold

Top PDF symmetric positive definite manifold:

Emotion Recognition by Body Movement Representation on the Manifold of Symmetric Positive Definite Matrices

Several works used the special Riemannian manifold of SPD matrices. One typical case for which such matrices arise in practice is when covariance de- scriptors are used to model image sets or temporal frame sequences in videos. Covariance features were first introduced by Tuzel et al. [18] for texture matching and classification. Several studies have extended the use of covariance descriptors to the temporal dimension, with application to human action and gesture recog- nition. Sanin et al. [16], proposed an action and gesture recognition method from videos based on spatio-temporal covariance descriptors. Prior to classification, points on the manifold were mapped to an Euclidean space, through Riemannian Locality Preserving Projection [7]. Bhattacharya et al. [4] constructed covariance matrices, which capture joint statistics of both low-level motion and appearance features extracted from a video. To facilitate the classification task, matrices were mapped to an equivalent vector space obtained by the matrix logarithm operation, which approximates the tangent space of the original SPSD space of covariance matrices. Then, human action recognition was formulated as a sparse linear approximation problem, in which these mapped features are used to construct an overcomplete dictionary of the covariance based descriptors built from labeled training samples. In [5], Faraki et al. noted that when covariance descriptors are used to represent image sets, the result is often rank-deficient. Most of the existing methods solve this problem by accepting small perturba- tions to avoid null eigenvalues and thus, employ standard inference tools. What they proposed, instead, were novel similarity measures specifically designed for the particular case where symmetric matrices are not full-rank (i.e., Symmetric Positive Semi-Definite matrices, SPSD).
En savoir plus

Riemannian Framework for estimating Symmetric Positive Definite 4th Order Diffusion Tensors

5. Liu, C., Bammer, R., Moseley, M.E.: Generalized diffusion tensor imaging (gdti): A method for characterizing and imaging diffusion anisotropy caused by non-gaussian diffusion. Israel Journal of Chemistry 43 (2003) 145154 6. Lenglet, C., Rousson, M., Deriche, R., Faugeras, O.: Statistics on the manifold of multivariate normal distributions: Theory and application to diffusion tensor MRI processing. Journal of Mathematical Imaging and Vision 25(3) (2006) 423–444 7. Arsigny, V., Fillard, P., Pennec, X., Ayache, N.: Log-Euclidean metrics for fast

Exploration of Balanced Metrics on Symmetric Positive Definite Matrices

1 Introduction Symmetric Positive Definite (SPD) matrices are used in many applications: for example, they represent covariance matrices in signal or image processing [1–3] and they are diffusion tensors in diffusion tensor imaging [4–6]. Many Rieman- nian structures have been introduced on the manifold of SPD matrices depending on the problem and showing significantly different results from one another on statistical procedures such as the computation of barycenters or the principal component analysis. Non exhaustively, we can cite Euclidean metrics, power- Euclidean metrics [7], log-Euclidean metrics [8], which are flat; affine-invariant metrics [5, 6, 9] which are negatively curved; the Bogoliubov-Kubo-Mori metric [10] whose curvature has a quite complex expression.
En savoir plus

Kernel density estimation on spaces of Gaussian distributions and symmetric positive definite matrices

depend on the geometry of the space: if the space is not homogeneous and the isotropy group is empty, these indifference principles have no meaning. The convergence of the different estimation techniques is widely studied. Results were first obtained in the Euclidean case, and are gradually ex- tended to the probability densities on manifold, see [11, 19, 13, 4]. The last relevant aspect, is computational. Each estimation technique has its own computational framework, which presents pro and cons given the different applications. For instance, the estimation by orthogonal series presents an initial pre-processing, but provides a fast evaluation of the estimated density in compact manifolds.
En savoir plus

Symmetric tensor decomposition

Arithmetic complexity is also an important field where the understanding of tensor decompositions has made a lot of progress, especially third order tensors, which represent bilinear maps [35] [3] [50] [37]. Another important application field is Data Analysis. For instance, Independent Component Analysis, originally introduced for symmetric tensors whose rank did not exceed dimension [12] [6]. Now, it has become possible to estimate more factors than the dimension [23] [32]. In some applications, tensors may be symmetric only in some modes [14], or may not be symmetric nor have equal dimensions [10] [49]. Further numerous applications of tensor decompositions may be found in [10] [49].
En savoir plus

Noncommutative symmetric functions

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignemen[r]

Manifold embedding for curve registration

classification procedure : the classical k-nearest neighbours. In Figure 4 , we observe that the CAM average oversmoothes the peaks of activity at times 12 and 22 to make them almost nonexistent. This is a clear defect since, according to the experts of landscape remote sensing, these peaks of activity are representative of the nature of landscape. Indeed, these peaks convey essential informations which determines, among other things, the type of landscape. On the other hand, these changes are very well rendered by the pattern obtained by Manifold Warping. The same con- clusions can be drawn in Figure 5 for an other landscape. In this application domain, extracting a curve by Manifold Warping is best able to report data as reflecting their structure and thus to obtain a better representative.
En savoir plus

On positive functions with positive fourier transforms

The present note attempts to give general positivity criteria, in a con- structive way, by taking advantage of a representation under which the FT is essentially “transparent”. Our method combines the advantages of self- duality properties with those allowed by an algebra of polynomials, where positivity means absence of real roots, hence reasonably simple conditions for the polynomial coefficients. For this sake, in the 1-d case, we select a ba- sis made of convenient eigenstates of the FT, the Hermite–Fourier functions, i.e. the harmonic oscillator eigenstates. The method extends to the 2-d case, or Fourier–Bessel transform, by replacing Hermite by Laguerre polynomials. There are general theorems about the characterization of Fourier trans- forms of positive functions [9]. Let us quote in the first place the Bochner theorem and its generalizations [10] which state that the Fourier transform of a positive function is positive-definite. But positive definiteness in the sense of such theorems does not imply plain positivity 2
En savoir plus

Symmetric tensor decomposition

ABSTRACT We present an algorithm for decomposing a symmetric tensor of dimension n and order d as a sum of of rank-1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for symmetric tensors of dimension 2. We exploit the known fact that every symmetric tensor is equivalently represented by a homogeneous polyno- mial in n variables of total degree d. Thus the decom- position corresponds to a sum of powers of linear forms. The impact of this contribution is two-fold. First it permits an efficient computation of the decomposi- tion of any tensor of sub-generic rank, as opposed to widely used iterative algorithms with unproved conver- gence (e.g. Alternate Least Squares or gradient de- scents). Second, it gives tools for understanding unique- ness conditions, and for detecting the tensor rank.
En savoir plus

Computation of Euclidean minima in totally definite quaternion fields

1. Introduction If K is a number field, we denote by n its degree, by Z K its ring of integers, by Z × K its unit group, and by N K/Q : K → Q the norm form. Throughout this paper F will be a totally definite quaternion field over a number field K. Let us recall the relevant definitions, the reader may refer to [9, 13, 15] for a complete theory. Let F be a quaternion algebra over a number field K, i.e. a 4-dimensional algebra over K with basis (1, i, j, k) such that i 2 = a, j 2 = b and k = ij = −ji,

Proof methods of declarative properties of definite programs

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignemen[r]

Orderable 3-manifold groups

5. Left-orderability and foliations. In this paragraph we shall focus on a diﬀerent class of objects, namely on codimension 1 foliations. Such foliations will play an important role in our analysis of the left-orderability of the fundamental groups of Seifert ﬁbred manifolds. Their connection to orderabilty comes through the induced action of the fundamental group of the ambient manifold on the leaf space of the induced foliation on the universal cover. Under certain natural hypotheses this leaf space can be shown to be homeomorphic to

Definite integration using the generalized hypergeometric functions.

Chapter 2 next demonstrates the policy we adopted for each of the approximately fifty Special Functions so that our goal, viewing each Special Function as a Genera[r]

Definite quaternion orders of class number one

0(d(A)) :!~ 12. Thus d(A) E {8,16,24}. But d(A) = 8 is impossible, since then d(A’) = 1, which can not happen for an order in a definite algebra. If d(A) = 24, then d(A’) = 3, which gives e3 (A) = - 1 (and, of course, e2 (A) = 0). Using the formula of [B2], (4.5), an easy computation shows

Exact Random Generation of Symmetric and Quasi-symmetric Alternating-sign Matrices

tribution is very similar to the classical proofs for CFTP, and is omitted due to space constraints. 2.4 A note on coalescence detection The stopping criterion for the symmetric-CFTP algorithm, as suggested by Theorem 1, is that the coupled chains started at the minimum and maximum states (or at the lower and higher bounds Elow and Ehigh as defined earlier) have coalesced to the same state, which, by monotonicity, implies that the composed update function is constant on the whole of J(P ).

Synthesis Method for Manifold-Coupled Multiplexers

Keywords — multiplexers, matching filters synthesis, manifold. I. I NTRODUCTION In general, multiplexer synthesis involves dealing with manifolds peaks (spikes) appearing inside the passbands. Those are narrow transmission zeros that appear due to phase recombinations happening inside the manifold. In the design of multiplexers, we often face the problem of avoiding these peaks and there are many contributions to this problem in the literature [1], [2], most of them providing heuristic procedures to solve or simplify the problem. Nevertheless, these procedures usually apply after the design is completed and an EM simulation reveals the presence of the peaks.
En savoir plus

Estimating the Reach of a Manifold

In this paper we study the problem of estimating reach. To do so, we first provide new geometric results on the reach. We also give the first bounds on the minimax rate for estimating reach. As a first attempt to study reach es- timation in the literature, we will mainly work in a framework where a point cloud is observed jointly with tangent spaces, before relaxing this constraint in Section 6 . Such an oracle framework has direct applications in digital imag- ing [ 32 , 28 ], where a very high resolution image or 3D-scan, represented as a manifold, enables to determine precisely tangent spaces for arbitrary finite set of points [ 28 ].
En savoir plus

Minimax adaptive estimation in manifold inference

our method largely stems from the Penalized Comparison to Overfitting method introduced in [ LMR17 ] in the setting of kernel density estimation. Outline of the paper The framework of minimax adaptive estimation as well as preliminary results on manifold esti- mation are detailed in Section 2 . In Section 3 , we define the t-convex hull of a set, and show that the estimator Conv d (t; X n ) is minimax for some choice of t. In Section 4 , we introduce the convexity defect function of a set, originally defined in [ ALS13 ], and study in details the behavior of the convexity defect of the observation set X n . This study is then used to select a
En savoir plus

Lexicographic optimal chains and manifold triangulations

Second, we consider OHCP in the particular case where K is a (d + 1)-pseudo-manifold, for example when it triangulates a (d + 1)-sphere. In this case, there is a O(n log n) algorithm which can be seen, by duality, as a lexicographic minimum cut in the dual graph of K. Third, we introduce a total order on n-simplices for which, when the points lie in the n- Euclidean space, the support of the lexicographic-minimal chain with the convex hull boundary as boundary constraint is precisely the n-dimensional Delaunay triangulation, or in a more general setting, the regular triangulation of a set of weighted points.
En savoir plus

Homoclinic orbit to a center manifold

Among the appli ations let us give the example of the sti elasti spatial pendulum. This is a pendulum where the bar has been repla ed by a sti spring whi h has v ariable length but remains always straight, see Figure 1. The enter manifold here is the set of os illations of the spring in unstable equilibrium. W e obtain an orbit homo lini to one of these os illations