2,614 research outputs found
Matroid Regression
We propose an algebraic combinatorial method for solving large sparse linear
systems of equations locally - that is, a method which can compute single
evaluations of the signal without computing the whole signal. The method scales
only in the sparsity of the system and not in its size, and allows to provide
error estimates for any solution method. At the heart of our approach is the
so-called regression matroid, a combinatorial object associated to sparsity
patterns, which allows to replace inversion of the large matrix with the
inversion of a kernel matrix that is constant size. We show that our method
provides the best linear unbiased estimator (BLUE) for this setting and the
minimum variance unbiased estimator (MVUE) under Gaussian noise assumptions,
and furthermore we show that the size of the kernel matrix which is to be
inverted can be traded off with accuracy
Dual-to-kernel learning with ideals
In this paper, we propose a theory which unifies kernel learning and symbolic
algebraic methods. We show that both worlds are inherently dual to each other,
and we use this duality to combine the structure-awareness of algebraic methods
with the efficiency and generality of kernels. The main idea lies in relating
polynomial rings to feature space, and ideals to manifolds, then exploiting
this generative-discriminative duality on kernel matrices. We illustrate this
by proposing two algorithms, IPCA and AVICA, for simultaneous manifold and
feature learning, and test their accuracy on synthetic and real world data.Comment: 15 pages, 1 figur
Algebraic matroids with graph symmetry
This paper studies the properties of two kinds of matroids: (a) algebraic
matroids and (b) finite and infinite matroids whose ground set have some
canonical symmetry, for example row and column symmetry and transposition
symmetry.
For (a) algebraic matroids, we expose cryptomorphisms making them accessible
to techniques from commutative algebra. This allows us to introduce for each
circuit in an algebraic matroid an invariant called circuit polynomial,
generalizing the minimal poly- nomial in classical Galois theory, and studying
the matroid structure with multivariate methods.
For (b) matroids with symmetries we introduce combinatorial invariants
capturing structural properties of the rank function and its limit behavior,
and obtain proofs which are purely combinatorial and do not assume algebraicity
of the matroid; these imply and generalize known results in some specific cases
where the matroid is also algebraic. These results are motivated by, and
readily applicable to framework rigidity, low-rank matrix completion and
determinantal varieties, which lie in the intersection of (a) and (b) where
additional results can be derived. We study the corresponding matroids and
their associated invariants, and for selected cases, we characterize the
matroidal structure and the circuit polynomials completely
Obtaining error-minimizing estimates and universal entry-wise error bounds for low-rank matrix completion
We propose a general framework for reconstructing and denoising single entries
of incomplete and noisy entries. We describe: effective algorithms for deciding if and
entry can be reconstructed and, if so, for reconstructing and denoising it; and a priori
bounds on the error of each entry, individually. In the noiseless case our algorithm is
exact. For rank-one matrices, the new algorithm is fast, admits a highly-parallel implementation,
and produces an error minimizing estimate that is qualitatively close
to our theoretical and the state-of-the-art Nuclear Norm and OptSpace methods
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Algebraic matroids with graph symmetry
This paper studies the properties of two kinds of matroids: (a) algebraic matroids
and (b) finite and infinite matroids whose ground set have some canonical symmetry,
for example row and column symmetry and transposition symmetry.
For (a) algebraic matroids, we expose cryptomorphisms making them accessible to
techniques from commutative algebra. This allows us to introduce for each circuit in an
algebraic matroid an invariant called circuit polynomial, generalizing the minimal polynomial
in classical Galois theory, and studying the matroid structure with multivariate
methods.
For (b) matroids with symmetries we introduce combinatorial invariants capturing
structural properties of the rank function and its limit behavior, and obtain proofs which
are purely combinatorial and do not assume algebraicity of the matroid; these imply
and generalize known results in some specific cases where the matroid is also algebraic.
These results are motivated by, and readily applicable to framework rigidity, low-rank
matrix completion and determinantal varieties, which lie in the intersection of (a) and (b)
where additional results can be derived. We study the corresponding matroids and their
associated invariants, and for selected cases, we characterize the matroidal structure
and the circuit polynomials completely
Disease spread through animal movements: a static and temporal network analysis of pig trade in Germany
Background: Animal trade plays an important role for the spread of infectious
diseases in livestock populations. As a case study, we consider pig trade in
Germany, where trade actors (agricultural premises) form a complex network. The
central question is how infectious diseases can potentially spread within the
system of trade contacts. We address this question by analyzing the underlying
network of animal movements.
Methodology/Findings: The considered pig trade dataset spans several years
and is analyzed with respect to its potential to spread infectious diseases.
Focusing on measurements of network-topological properties, we avoid the usage
of external parameters, since these properties are independent of specific
pathogens. They are on the contrary of great importance for understanding any
general spreading process on this particular network. We analyze the system
using different network models, which include varying amounts of information:
(i) static network, (ii) network as a time series of uncorrelated snapshots,
(iii) temporal network, where causality is explicitly taken into account.
Findings: Our approach provides a general framework for a
topological-temporal characterization of livestock trade networks. We find that
a static network view captures many relevant aspects of the trade system, and
premises can be classified into two clearly defined risk classes. Moreover, our
results allow for an efficient allocation strategy for intervention measures
using centrality measures. Data on trade volume does barely alter the results
and is therefore of secondary importance. Although a static network description
yields useful results, the temporal resolution of data plays an outstanding
role for an in-depth understanding of spreading processes. This applies in
particular for an accurate calculation of the maximum outbreak size.Comment: main text 33 pages, 17 figures, supporting information 7 pages, 7
figure
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The algebraic combinatorial approach for low-rank matrix completion
We propose an algebraic combinatorial framework for the problem of completing
partially observed low-rank matrices. We show that the intrinsic properties of
the problem, including which entries can be reconstructed, and the degrees of freedom
in the reconstruction, do not depend on the values of the observed entries, but
only on their position. We associate combinatorial and algebraic objects, differentials
and matroids, which are descriptors of the particular reconstruction task, to the
set of observed entries, and apply them to obtain reconstruction bounds. We show
how similar techniques can be used to obtain reconstruction bounds on general compressed
sensing problems with algebraic compression constraints. Using the new
theory, we develop several algorithms for low-rank matrix completion, which allow
to determine which set of entries can be potentially reconstructed and which not,
and how, and we present algorithms which apply algebraic combinatorial methods
in order to reconstruct the missing entries
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