Gram matrix

We consider a 2 x 2 real symmetric Gram matrix M associated with an underlying matrix R = (Ri R2) of real vectors R, whose scalar products are its elements ... 

The matrix = [ (xj)] is called the Gram matrix, when the matrix is non-singular (has an inverse). If the matrix is non-singular, the equation I>a = f has a unique solution however, according to Golberg [21], this fails to be true even in very simple cases [21]. 

We want to note finally that, as pointed out by Blumstein and Wheeler and by Magnus, the results of this section can also be obtained within the original procedure of Gautshi, starting from the knowledge of the Gram matrix corresponding to the sequence (Pj ( ) with density fimction /i( ). 

It can be demonstrated that the linear independence of the elements d guarantees that matrix Fj is nonsingular, which means that the solution to (B.4) aj, i = 1,2,. n always exists for any do and is unique. The Gram matrix can be calculated as follows ... 

The advantage of having the orthonormal set 61,62, ,6 is that for this set the Gram matrix is reduced to the Kronecker matrix ... 

From the last expression we can see that x, z) are the data kernels for this problem. The Gram matrix takes the form... 

Kriging uses the same strategy of maximising the likelihood it finds the parameters Oh and ph (h = 1, 2,. .., d) in the so-called Gram matrix R,... 

Unfortunately, EFT does not exist for all the combinations of d, N. In [15], it can be shown that, by alternating projection, we can find a matrix that is predicted to be close to ETF. The aim of parametric dictionary (PD) is to find a set of parameters that minimizes the distance between dictionary and the Gram matrix. 

For all columns of Gg, we have Lg = 1 + (d — V)g. It means that with this invariant feature of Gram matrix, we can eliminate the iteration method that has been proposed in [12]. In order to minimize this objective function, we proposed the Genetic Algorithm (GA) approach instead. 

Fig. 3 Distance with Gram Matrix for different redundancy(result of our algorithm is plotted with o)...

In this paper, we studied the problems of parametric dictionary (PD) design for sparse representations. We can see that by minimizing the distance between C norm of column of designed dictionary and Gram matrix, the resultant dictionary will have the minimum mutual coherence compared to initial dictionary. By the mean of constant characteristic of gram matrix, we eliminate the iteration method that was used in previous methods, enabling better results. 

The first item of business is to show how any set of multivectors in the geometric algebra of three dimensions can be characterized, up to rotation, by a system of scalar-valued expressions in these fundamental invariants. Any multivector can always be separated into its scalar, vector, bivector, and trivector parts. The scalar part is ready to go, while the trivector part can be converted to a scalar simply by multiplying it by the unit pseudo-scalar. We next observe that any set of vectors is determined, up to rotation, by their Gram matrix of inner products. This is easily seen by taking any maximal linearly... 

Then if Y, is the /th column of the scaled coordinate matrix Y, we have BY, = X, Y, for i = 1,..., 3. It follows that these columns are proportional to eigenvectors of the scaled estimated Gram matrix B, while the moments of inertia Xi, X2, X.3 are the corresponding eigenvalues. Since the eigenvectors have unit norm, the diagonal form of the inertial tensor implies that the constant of proportionality is VXJ. 

Optimal Multivariate Interpolation 399 The entries of the symmetric Gram matrix B of size N + 2) are given... 

Both PCA and Classical MDS give rise to the same low-dimensional embedding and the Gram matrix (Eq. 2.4) has the same rank and eigenvalues up to a constant factor as the feature (covariance) matrix of PCA [5]. 

The solution to the maximum variance unfolding problem is found by constructing a Gram matrix, F, whose top eigenvectors give rise to the low-dimensional representation of the data. MVU seeks to maximise llyi yj II with yij e Y, subject to the following constraints... 

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