2,602 research outputs found
On the existence of identifiable reparametrizations for linear compartment models
The parameters of a linear compartment model are usually estimated from
experimental input-output data. A problem arises when infinitely many parameter
values can yield the same result; such a model is called unidentifiable. In
this case, one can search for an identifiable reparametrization of the model: a
map which reduces the number of parameters, such that the reduced model is
identifiable. We study a specific class of models which are known to be
unidentifiable. Using algebraic geometry and graph theory, we translate a
criterion given by Meshkat and Sullivant for the existence of an identifiable
scaling reparametrization to a new criterion based on the rank of a weighted
adjacency matrix of a certain bipartite graph. This allows us to derive several
new constructions to obtain graphs with an identifiable scaling
reparametrization. Using these constructions, a large subclass of such graphs
is obtained. Finally, we present a procedure of subdividing or deleting edges
to ensure that a model has an identifiable scaling reparametrization
Identifiability and Unmixing of Latent Parse Trees
This paper explores unsupervised learning of parsing models along two
directions. First, which models are identifiable from infinite data? We use a
general technique for numerically checking identifiability based on the rank of
a Jacobian matrix, and apply it to several standard constituency and dependency
parsing models. Second, for identifiable models, how do we estimate the
parameters efficiently? EM suffers from local optima, while recent work using
spectral methods cannot be directly applied since the topology of the parse
tree varies across sentences. We develop a strategy, unmixing, which deals with
this additional complexity for restricted classes of parsing models
Identifiability of Large Phylogenetic Mixture Models
Phylogenetic mixture models are statistical models of character evolution
allowing for heterogeneity. Each of the classes in some unknown partition of
the characters may evolve by different processes, or even along different
trees. The fundamental question of whether parameters of such a model are
identifiable is difficult to address, due to the complexity of the
parameterization. We analyze mixture models on large trees, with many mixture
components, showing that both numerical and tree parameters are indeed
identifiable in these models when all trees are the same. We also explore the
extent to which our algebraic techniques can be employed to extend the result
to mixtures on different trees.Comment: 15 page
Parameter identifiability in a class of random graph mixture models
We prove identifiability of parameters for a broad class of random graph
mixture models. These models are characterized by a partition of the set of
graph nodes into latent (unobservable) groups. The connectivities between nodes
are independent random variables when conditioned on the groups of the nodes
being connected. In the binary random graph case, in which edges are either
present or absent, these models are known as stochastic blockmodels and have
been widely used in the social sciences and, more recently, in biology. Their
generalizations to weighted random graphs, either in parametric or
non-parametric form, are also of interest in many areas. Despite a broad range
of applications, the parameter identifiability issue for such models is
involved, and previously has only been touched upon in the literature. We give
here a thorough investigation of this problem. Our work also has consequences
for parameter estimation. In particular, the estimation procedure proposed by
Frank and Harary for binary affiliation models is revisited in this article
Complex Random Vectors and ICA Models: Identifiability, Uniqueness and Separability
In this paper the conditions for identifiability, separability and uniqueness
of linear complex valued independent component analysis (ICA) models are
established. These results extend the well-known conditions for solving
real-valued ICA problems to complex-valued models. Relevant properties of
complex random vectors are described in order to extend the Darmois-Skitovich
theorem for complex-valued models. This theorem is used to construct a proof of
a theorem for each of the above ICA model concepts. Both circular and
noncircular complex random vectors are covered. Examples clarifying the above
concepts are presented.Comment: To appear in IEEE TR-IT March 200
Effective criteria for specific identifiability of tensors and forms
In applications where the tensor rank decomposition arises, one often relies
on its identifiability properties for interpreting the individual rank-
terms appearing in the decomposition. Several criteria for identifiability have
been proposed in the literature, however few results exist on how frequently
they are satisfied. We propose to call a criterion effective if it is satisfied
on a dense, open subset of the smallest semi-algebraic set enclosing the set of
rank- tensors. We analyze the effectiveness of Kruskal's criterion when it
is combined with reshaping. It is proved that this criterion is effective for
both real and complex tensors in its entire range of applicability, which is
usually much smaller than the smallest typical rank. Our proof explains when
reshaping-based algorithms for computing tensor rank decompositions may be
expected to recover the decomposition. Specializing the analysis to symmetric
tensors or forms reveals that the reshaped Kruskal criterion may even be
effective up to the smallest typical rank for some third, fourth and sixth
order symmetric tensors of small dimension as well as for binary forms of
degree at least three. We extended this result to symmetric tensors by analyzing the Hilbert function, resulting in a
criterion for symmetric identifiability that is effective up to symmetric rank
, which is optimal.Comment: 31 pages, 2 Macaulay2 code
The average condition number of most tensor rank decomposition problems is infinite
The tensor rank decomposition, or canonical polyadic decomposition, is the
decomposition of a tensor into a sum of rank-1 tensors. The condition number of
the tensor rank decomposition measures the sensitivity of the rank-1 summands
with respect to structured perturbations. Those are perturbations preserving
the rank of the tensor that is decomposed. On the other hand, the angular
condition number measures the perturbations of the rank-1 summands up to
scaling.
We show for random rank-2 tensors with Gaussian density that the expected
value of the condition number is infinite. Under some mild additional
assumption, we show that the same is true for most higher ranks as
well. In fact, as the dimensions of the tensor tend to infinity, asymptotically
all ranks are covered by our analysis. On the contrary, we show that rank-2
Gaussian tensors have finite expected angular condition number.
Our results underline the high computational complexity of computing tensor
rank decompositions. We discuss consequences of our results for algorithm
design and for testing algorithms that compute the CPD. Finally, we supply
numerical experiments
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