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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
Computing Algebraic Matroids
An affine variety induces the structure of an algebraic matroid on the set of
coordinates of the ambient space. The matroid has two natural decorations: a
circuit polynomial attached to each circuit, and the degree of the projection
map to each base, called the base degree. Decorated algebraic matroids can be
computed via symbolic computation using Groebner bases, or through linear
algebra in the space of differentials (with decorations calculated using
numerical algebraic geometry). Both algorithms are developed here. Failure of
the second algorithm occurs on a subvariety called the non-matroidal or NM-
locus. Decorated algebraic matroids have widespread relevance anywhere that
coordinates have combinatorial significance. Examples are computed from applied
algebra, in algebraic statistics and chemical reaction network theory, as well
as more theoretical examples from algebraic geometry and matroid theory.Comment: 15 pages; added link to references, note on page 1, and small
formatting fixe
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
The Maximum Likelihood Threshold of a Graph
The maximum likelihood threshold of a graph is the smallest number of data
points that guarantees that maximum likelihood estimates exist almost surely in
the Gaussian graphical model associated to the graph. We show that this graph
parameter is connected to the theory of combinatorial rigidity. In particular,
if the edge set of a graph is an independent set in the -dimensional
generic rigidity matroid, then the maximum likelihood threshold of is less
than or equal to . This connection allows us to prove many results about the
maximum likelihood threshold.Comment: Added Section 6 and Section
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
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