8,152 research outputs found
Estimation under group actions: recovering orbits from invariants
Motivated by geometric problems in signal processing, computer vision, and
structural biology, we study a class of orbit recovery problems where we
observe very noisy copies of an unknown signal, each acted upon by a random
element of some group (such as Z/p or SO(3)). The goal is to recover the orbit
of the signal under the group action in the high-noise regime. This generalizes
problems of interest such as multi-reference alignment (MRA) and the
reconstruction problem in cryo-electron microscopy (cryo-EM). We obtain
matching lower and upper bounds on the sample complexity of these problems in
high generality, showing that the statistical difficulty is intricately
determined by the invariant theory of the underlying symmetry group.
In particular, we determine that for cryo-EM with noise variance
and uniform viewing directions, the number of samples required scales as
. We match this bound with a novel algorithm for ab initio
reconstruction in cryo-EM, based on invariant features of degree at most 3. We
further discuss how to recover multiple molecular structures from heterogeneous
cryo-EM samples.Comment: 54 pages. This version contains a number of new result
Generating Series for Interconnected Nonlinear Systems and the Formal Laplace-Borel Transform
Formal power series methods provide effective tools for nonlinear system analysis. For a broad range of analytic nonlinear systems, their input-output mapping can be described by a Fliess operator associated with a formal power series. In this dissertation, the inter connection of two Fliess operators is characterized by the generating series of the composite system. In addition, the formal Laplace-Borel transform of a Fliess operator is defined and its fundamental properties are presented. The formal Laplace-Borel transform produces an elegant description of system interconnections in a purely algebraic context.
Specifically, four basic interconnections of Fliess operators are addressed: the parallel, product, cascade and feedback connections. For each interconnection, the generating series of the overall system is given, and a growth condition is developed, which guarantees the convergence property of the output of the corresponding Fliess operator.
Motivated by the relationship between operations on formal power series and system interconnections, and following the idea of the classical integral Laplace-Borel transform, a new formal Laplace-Borel transform of a Fliess operator is proposed. The properties of this Laplace-Borel transform are provided, and in particular, a fundamental semigroup isomorphism is identified between the set of all locally convergent power series and the set of all well-defined Fliess operators.
A software package was produced in Maple based on the ACE package developed by the ACE group in Université de Marne-la Vallée led by Sébastien Veigneau. The ACE package provided the binary operations of addition, concatenation and shuffle product on the free monoid of formal polynomials. In this dissertation, the operations of composition, modified composition, chronological products and the evaluation of Fliess operators are implemented in software. The package was used to demonstrate various aspects of the new interconnection theory
Waterfilling Theorems for Linear Time-Varying Channels and Related Nonstationary Sources
The capacity of the linear time-varying (LTV) channel, a continuous-time LTV
filter with additive white Gaussian noise, is characterized by waterfilling in
the time-frequency plane. Similarly, the rate distortion function for a related
nonstationary source is characterized by reverse waterfilling in the
time-frequency plane. Constraints on the average energy or on the squared-error
distortion, respectively, are used. The source is formed by the white Gaussian
noise response of the same LTV filter as before. The proofs of both
waterfilling theorems rely on a Szego theorem for a class of operators
associated with the filter. A self-contained proof of the Szego theorem is
given. The waterfilling theorems compare well with the classical results of
Gallager and Berger. In the case of a nonstationary source, it is observed that
the part of the classical power spectral density is taken by the Wigner-Ville
spectrum. The present approach is based on the spread Weyl symbol of the LTV
filter, and is asymptotic in nature. For the spreading factor, a lower bound is
suggested by means of an uncertainty inequality.Comment: 13 pages, 5 figures; channel model in Section III now restricted to
LTV filters with real-valued kerne
Sampling of operators
Sampling and reconstruction of functions is a central tool in science. A key
result is given by the sampling theorem for bandlimited functions attributed to
Whittaker, Shannon, Nyquist, and Kotelnikov. We develop an analogous sampling
theory for operators which we call bandlimited if their Kohn-Nirenberg symbols
are bandlimited. We prove sampling theorems for such operators and show that
they are extensions of the classical sampling theorem
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