370 research outputs found
Support-based lower bounds for the positive semidefinite rank of a nonnegative matrix
The positive semidefinite rank of a nonnegative -matrix~ is
the minimum number~ such that there exist positive semidefinite -matrices , such that S(k,\ell) =
\mbox{tr}(A_k^* B_\ell).
The most important, lower bound technique for nonnegative rank is solely
based on the support of the matrix S, i.e., its zero/non-zero pattern. In this
paper, we characterize the power of lower bounds on positive semidefinite rank
based on solely on the support.Comment: 9 page
Uniqueness of Nonnegative Tensor Approximations
We show that for a nonnegative tensor, a best nonnegative rank-r
approximation is almost always unique, its best rank-one approximation may
always be chosen to be a best nonnegative rank-one approximation, and that the
set of nonnegative tensors with non-unique best rank-one approximations form an
algebraic hypersurface. We show that the last part holds true more generally
for real tensors and thereby determine a polynomial equation so that a real or
nonnegative tensor which does not satisfy this equation is guaranteed to have a
unique best rank-one approximation. We also establish an analogue for real or
nonnegative symmetric tensors. In addition, we prove a singular vector variant
of the Perron--Frobenius Theorem for positive tensors and apply it to show that
a best nonnegative rank-r approximation of a positive tensor can never be
obtained by deflation. As an aside, we verify that the Euclidean distance (ED)
discriminants of the Segre variety and the Veronese variety are hypersurfaces
and give defining equations of these ED discriminants
Low Rank Approximation of Binary Matrices: Column Subset Selection and Generalizations
Low rank matrix approximation is an important tool in machine learning. Given
a data matrix, low rank approximation helps to find factors, patterns and
provides concise representations for the data. Research on low rank
approximation usually focus on real matrices. However, in many applications
data are binary (categorical) rather than continuous. This leads to the problem
of low rank approximation of binary matrix. Here we are given a
binary matrix and a small integer . The goal is to find two binary
matrices and of sizes and respectively, so
that the Frobenius norm of is minimized. There are two models of this
problem, depending on the definition of the dot product of binary vectors: The
model and the Boolean semiring model. Unlike low rank
approximation of real matrix which can be efficiently solved by Singular Value
Decomposition, approximation of binary matrix is -hard even for .
In this paper, we consider the problem of Column Subset Selection (CSS), in
which one low rank matrix must be formed by columns of the data matrix. We
characterize the approximation ratio of CSS for binary matrices. For
model, we show the approximation ratio of CSS is bounded by
and this bound is asymptotically tight. For
Boolean model, it turns out that CSS is no longer sufficient to obtain a bound.
We then develop a Generalized CSS (GCSS) procedure in which the columns of one
low rank matrix are generated from Boolean formulas operating bitwise on
columns of the data matrix. We show the approximation ratio of GCSS is bounded
by , and the exponential dependency on is inherent.Comment: 38 page
Algorithms for Approximate Subtropical Matrix Factorization
Matrix factorization methods are important tools in data mining and analysis.
They can be used for many tasks, ranging from dimensionality reduction to
visualization. In this paper we concentrate on the use of matrix factorizations
for finding patterns from the data. Rather than using the standard algebra --
and the summation of the rank-1 components to build the approximation of the
original matrix -- we use the subtropical algebra, which is an algebra over the
nonnegative real values with the summation replaced by the maximum operator.
Subtropical matrix factorizations allow "winner-takes-it-all" interpretations
of the rank-1 components, revealing different structure than the normal
(nonnegative) factorizations. We study the complexity and sparsity of the
factorizations, and present a framework for finding low-rank subtropical
factorizations. We present two specific algorithms, called Capricorn and
Cancer, that are part of our framework. They can be used with data that has
been corrupted with different types of noise, and with different error metrics,
including the sum-of-absolute differences, Frobenius norm, and Jensen--Shannon
divergence. Our experiments show that the algorithms perform well on data that
has subtropical structure, and that they can find factorizations that are both
sparse and easy to interpret.Comment: 40 pages, 9 figures. For the associated source code, see
http://people.mpi-inf.mpg.de/~pmiettin/tropical
A module-theoretic approach to matroids
Speyer recognized that matroids encode the same data as a special class of
tropical linear spaces and Shaw interpreted tropically certain basic matroid
constructions; additionally, Frenk developed the perspective of tropical linear
spaces as modules over an idempotent semifield. All together, this provides
bridges between the combinatorics of matroids, the algebra of idempotent
modules, and the geometry of tropical linear spaces. The goal of this paper is
to strengthen and expand these bridges by systematically developing the
idempotent module theory of matroids. Applications include a geometric
interpretation of strong matroid maps and the factorization theorem; a
generalized notion of strong matroid maps, via an embedding of the category of
matroids into a category of module homomorphisms; a monotonicity property for
the stable sum and stable intersection of tropical linear spaces; a novel
perspective of fundamental transversal matroids; and a tropical analogue of
reduced row echelon form.Comment: 22 pages; v3 minor corrections/clarifications; to appear in JPA
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