# (1,1)-geodesic maps into grassmann manifolds

Grassmann manifolds and the Grassmann image of submanifolds
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We saw one Metrics via Geodesic Distances: The task of measuring dis- example of a mapping, namely the inverse exponential map or the tances on a manifold is accomplished using a Riemannian metric. Using properties that make it attractive for indexing and searching.

However, one Tangent-plane embedding: If M is a Riemannian manifold needs to know the error incurred under this embedding. The table shows the highly non-linear and sometimes computationally intensive distance functions on the manifolds.

## (1,1)-geodesic maps into grassmann manifolds

Depending on data, the dimension of the manifold can be quite high. Thus which shall be used to define the embedding. However, unlike Rn there is when the manifold is very high dimensional, we can leverage the no point that naturally serves as the pole for manifolds. Given a fact that i. Gaussian vectors in high dimensions are nearly or- dataset of points on a manifold, the intrinsic mean of the dataset or thogonal, also known as concentration inequalities for projections the Karcher mean [21] is a natural choice for the pole.

We shall in high dimensions [4]. Hence, in most practical applications it later discuss how to compute it in a fast and efficient manner. In general, experiments. For Pole and Centroid Computation: Selecting the pole as the points that are close to the pole, i. We refer the reader This is a non-iterative procedure, and requires only a single pass to [14] for an example of the exact relation between these quantities over the dataset.

Further it is a recursive procedure and does not for the special case of a sphere. For points that are far away from the pole, in general the approx- imation error can be large.

### Kodai Mathematical Journal

In certain cases, as we show next, even 4. Then, by the triangle inequality developed for Euclidean spaces. We note that since the embedding we have provides approximate distances, any search algorithm using this embedding will necessarily be approximate. Any other algorithm can be similarly deployed. Since, we chose the pole as the centroid A good introduction and survey of LSH can be found in [11].

LSH solves this problem by ing. We randomly sample such function. For randomly chosen direction whose entries are chosen independently each of these matrices, we find the nearest neighbor in the train- from a stable distribution, and b is a random number chosen be- ing dataset and assign the image-class of the corresponding nearest tween [0, w]. In this case, the hash function takes on integer val- neighbor.

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Then, from the such labels obtained, we use a major- ues. Each image is rep- Hk. Then, L hash tables are constructed by randomly choosing resented as a set of covariance matrices. This gives rise to a H1 , H2. All the training examples are hashed into total of 22, covariance matrices in the training set. The testing the L hash tables. For a query point q, an exhaustive search is car- scheme as described above would require us to find nearest neigh- ried out among the examples in the union of the L hash-buckets bors in this large set.

We show in table 2 how the proposed hashing indexed by q. To extend this to manifolds, the basic idea is to first pick a pole p such that the approximation in equation 10 is valid. Method Geodesic Computa- Accuracy Once this pole is chosen, we embed all points in the tangent space tions per test image at p.

For a query point q, an exhaustive geodesic distance based search is carried out among the examples in the For comparison, we show in table 3 the state-of-the art recog- union of the L hash-buckets indexed by q. This is illustrated in nition accuracies on the Brodatz database using several methods figure 2. Note that even though we use the same covari- ance features and testing methodology of [41], we obtained an ac- 5.

This can be attributed In this section we demonstrate the utility of the proposed frame- to the variation due to the random choice of s windows in the im- work to enable fast nearest neighbor searches in non-Euclidean ages over which covariance is computed.

### mathematics and statistics online

However, the goal is not manifolds. Covariance features as region descriptors were introduced in [41] and have been suc- Method Performance cessfully applied to human detection [40], object tracking [31] and Oriented Filters To perform hashing, we need the exponen- tial and logarithmic maps.

### mathematics and statistics online

Unfortunately tree-based indexing gent plane. Papapetrou, and G. An overview of various manifolds that frequently appear in vi- sion literature, and their associated non-linear distance functions are described in table 1. Standard embeddings of Grassmann manifolds in Euclidean space 4. Thus which shall be used to define the embedding. Click here to close this overlay, or press the "Escape" key on your keyboard. Jermyn, and S.

Given a set of points on a manifold, we first compute a suitable pole as discussed in section 4. Then, we shoot K random geodesics from the pole. Projections onto these geodesics are computed as described in section 4. From the projections, a hash function is computed such as in equation 14 or 15 which are then appended to form a K-bit hash function.

Spencer Dowdall - Geometry and dynamics of free-by-cyclic groups I

Then, L hash-tables are created by choosing L different K-bit hash functions 5. We see that we can get comparable accuracy to exhaustive In this experiment, we consider the problem of face recognition nearest neighbors with significantly fewer geodesic computations. By facial geometry we refer to the location of 2D facial landmarks on images.

gatsbyinteriors.co.uk/8502-citas-en-linea.php In several face recognition tasks, Method Geodesic Computa- Accuracy tions per test face the locations of the landmarks have been shown to be extremely Median Mean Max informative [45, 32]. A shape is represented by a set of landmark Exhaustive 1-NN Then, all affine trans- a21 a22 6. Note that, multiplication by a full-rank matrix on the searching problem on non-Euclidean manifolds for various vision right preserves the column-space of the matrix Lbase. Thus, the applications. We discussed that the complexity of computing dis- 2D subspace of Rn spanned by the columns of the matrix Lbase is tance functions and centroids sets this problem apart from usual an affine-invariant representation of the shape.

Then, we provided a formal framework is invariant to affine transforms of the shape.

Subspaces such as for addressing the problem. From the geometric framework, we these can be identified as points on a Grassmann manifold. The derived an approximate method of searching via approximate em- Grassmann manifold Gn,d is the space whose points are d-planes beddings using the logarithmic map. We studied the error incurred or d-dimensional hyperplanes containing the origin in Rn. Experiments demonstrate that it is possible to signif- tient of the special orthogonal group SO n.

Using this it tive nearest-neighbors. The ex- [2] O. Arandjelovic, G. Shakhnarovich, J. Fisher, R. Cipolla, and BT 0 T. Face recognition with image sets using manifold pression for inverse exponential map is not available analytically density divergence. The details of and Pattern Recognition, pages —, June We use the publicly available FG-Net dataset [1], which con- [3] V.

Athitsos, M. Potamias, P. Papapetrou, and G. For this dataset, 68 Nearest neighbor retrieval using distance-based hashing. In fiducial points are available with each face. There are a total of on Data Engineering, pages —, We perform a leave-one-out test to quantity the face recog- [4] R. Baraniuk, M. Davenport, R. DeVore, and M. This is a challenging dataset as there are many Wakin. A simple proof of the restricted isometry property for sources of appearance variations for each person such as age, fa- random matrices.

Constructive Approximation, cial hair, expression, pose and illumination. We show the results 28 3 —, Dec. Basri, T. Hassner, and L. Zelnik Manor. Samples shown correspond to an individual from the FG-net dataset. Pattern Recognition, pages 1—8, Indyk and R. Approximate nearest neighbors: [6] E. Begelfor and M. Affine invariance revisited.

In Proc. Recognition, Riemannian center of mass and mollifier [7] J. On lipschitz embedding of finite metric spaces smoothing. Communications on Pure and Applied in hilbert space.

### Kodai Mathematical Journal

[OU] Y. Ohnita and S. Udagawa,Complex-analyticity of pluriharmonic maps and their constructions, Springer Lecture Notes in Mathematics (). [en] Nontrivial (1,1)-geodesic immersions in complex and real Grassmannians, even for the limit dimension m=(p-1)(q-1)+1 in the case q=2, are constructed.

Israel Journal of Mathematics, 52 1 —52, Mathematics, 30 5 —, March Shape manifolds, Procrustean metrics and [8] R.