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).

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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

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.

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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.

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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].

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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.

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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

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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.

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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,

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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

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]

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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

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 ).

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.

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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 ].

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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

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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.

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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