Sentence examples for matrix decomposition problem from inspiring English sources

A first step in the solution of the Hermitean matrix decomposition problem is the introduction of the matricial Hermitean Téodorescu transform: (4.1).

Lemma 1 (uniqueness of matrix decomposition with Vandermonde structure) [24]: Consider a matrix decomposition problem of X = A B T, in which A ∈ C I × F is a Vandermonde matrix and B ∈ C J × F is a non-singular square matrix.

As mentioned-above, NCA requires three criteria to be satisfied in advance to ensure unique solutions for the matrix decomposition problem [ 16, 24].

Both matrices are therefore obtained by the least-square objective as expressed in the form: Three criteria for the original NCA must be satisfied [ 16, 24] to ensure unique solutions to the matrix decomposition problem.

All subspace methods can be formulated as a matrix decomposition problem: X ~ S P T, S ∈ ℜ n × k, P ∈ ℜ p × k where different methods construct different basis matrices S and different feature matrices P according to different termination conditions.

Specifically speaking, we set L = u1 T, where u ∈ ℂ m and 1 denotes the vector in ℝ n whose entries are all 1, which leads to the following rank-one and transformed sparse matrix decomposition problem: (5) m i n   E u 1 T + S − d 2 s. t.   T S 0 ≤ s, and its unconstrained version (6) min ⁡ E u 1 T + S − d 2 + λ T S 0, where λ > 0 is a regularized parameter.

> GPGPUs can be used in matrix decomposition problems [23,24].

It transforms the problem of polynomial matrix decomposition to the problem of, pointwise in frequency, constant matrix decomposition.

Then, the same idea is in turn applied to extend the solution of a reduced matrix eigen-decomposition problem to approximate the eigenvectors of an SPSD matrix.

Observing that the group sparsity also implies a low rank structure, we reformulate the problem using matrix decomposition, which can handle large scale training samples by reducing the memory requirement at each iteration from O k2) to O(k) where k is the number of samples.

To address the above problem, we adopted the matrix decomposition methodology "low-rank and sparse decomposition" (LRSDec) to decompose EMAP data matrix into low-rank part and sparse part.