A recovering of violated metric in machine learning Dvoenko, Sergey D.; Pshenichny, Denis O.
Proceedings of the Seventh Symposium on Information and Communication Technology,
12/2016
Conference Proceeding
Experimental results in machine learning, data analysis and data mining often appear as comparisons between elements from a limited set. If a matrix of pairwise similarities is positively definite, ...then the set of elements is considered to be immersed in some metric space, e.g. Euclidean one, with dimensionality no more than the rank of the matrix. But it can be the non-positively definite matrix, because measurement results usually are not scalar products. It is necessary to recover metric for correct use of it in clustering or machine learning problems. Violations arise not only in the triangle inequality, but also relative to positions of more than three elements. In general, all similarity submatrices for triples of elements are positively definite, but simultaneously the whole matrix is negatively definite. We discuss here the approach to recover violated metric based on the idea of appropriate corrections of normalized similarity matrix and develop it for non-normalized similarity and dissimilarity ones.
A generalization of Sourour's theorem Ellers, Erich W.; Gordeev, Nikolai
Linear algebra and its applications,
1999, 1999-01-00, Volume:
286, Issue:
1
Journal Article
Peer reviewed
Open access
Let
X be an invertible
n ×
n matrix,
n > 1, with entries in some field
K. Assume
X ≠ diag(
a, …
a) for any
a ε K. Then for every sequence (
a
1, …,
a
n−1
),
a
i
ε
K, there is a matrix
Y with entries ...in K and det
Y = 1 such that the
n − 1 principal minors of
YXY
−1 have the values
a
1,…
a
n−1
), respectively. This generalizes Sourour' theorem, where
a
i
≠ 0 is assumed for all
i.
Inverses of Unipathic M-Matrices McDonald, J. J.; Neumann, M.; Schneider, H. ...
SIAM journal on matrix analysis and applications,
10/1996, Volume:
17, Issue:
4
Journal Article
Peer reviewed
Open access
In this paper we characterize all nonnegative matrices whose inverses are M-matrices with unipathic digraphs. A digraph is called unipathic if there is at most one simple path from any vertex $j$ to ...any other vertex $k$. The set of unipathic digraphs on $n$ vertices includes the simple $n$-cycle and all digraphs whose underlying undirected graphs are trees (or forests). Our results facilitate the construction of nonnegative matrices whose inverses are M-matrices with unipathic digraphs. We highlight this procedure for inverses of tridiagonal M-matrices and of M-matrices whose digraphs are simple $n$-cycles with loops.
The question of which ratios of products of principal minors are bounded over all matrices in a given class has been of interest historically. This question is settled herein for the class of ...tridiagonal sign-symmetric P matrices, which essentially lies in each of the classes: positive definite, in vertible totally nonnegative and M-matrices. It happens that all bounded ratios are bounded by one.
Let $a$, $b$ and $c$ be fixed complex numbers. Let $M_n(a,b,c)$ be the $n\times n$ Toeplitz matrix all of whose entries above the diagonal are $a$, all of whose entries below the diagonal are $b$, ...and all of whose entries on the diagonal are $c$. For $1\leq k\leq n$, each $k\times k$ principal minor of $M_n(a,b,c)$ has the same value. We find explicit and recursive formulae for the principal minors and the characteristic polynomial of $M_n(a,b,c)$. We also show that all complex polynomials in $M_n(a,b,c)$ are Toeplitz matrices. In particular, the inverse of $M_n(a,b,c)$ is a Toeplitz matrix when it exists.
A content-based image retrieval system with query image classification prior to retrieving procedure is proposed. Query image is compared to representative patterns of image classes, not to all ...images from database, accelerating thus initial retrieving step. Such procedure is possible when images from database are grouped into classes with similar content. Classification is performed using minor component (MC) analysis. Since it is expectable that MCs mainly depend on image details, not on an image background, this approach seems to be more efficient than classic CBIR. Minor components may be calculated by using single-layer neural network. The efficiency of proposed system is tested over images from Corel dataset
Jeffrey L. Stuart.
Obsahuje seznam literatury
Given a sequence of real or complex numbers, we construct a sequence of nested, symmetric matrices. We determine the LU- and QR-factorizations, the ...determinant and the principal minors for such a matrix. When the sequence is real, positive and strictly increasing, the matrices are strictly positive, inverse M-matrices with symmetric, irreducible, tridiagonal inverses.
A characterization of a class of totally nonnegative matrices whose inverses are M-matrices is given. It is then shown that if A is nonnegative of order n and A-1is an M-matrix, then the almost ...principal minors of A of all orders are nonnegative.
Suppose $A$ is an $n$-square matrix over the real numbers such that all principal minors are nonzero. If $A$ is nonnegative, then necessary and sufficient conditions are determined for $A$ to be ...factored into a product $L \cdot U$, where $L$ is a lower triangular nonnegative matrix and $U$ is an upper triangular nonnegative matrix with $u_{ii} = 1$. These conditions are given in terms of the nonnegativity of certain almost-principal minors of $A$.
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