101,248 research outputs found
Operator-Based Detecting, Learning, and Stabilizing Unstable Periodic Orbits of Chaotic Attractors
This paper examines the use of operator-theoretic approaches to the analysis
of chaotic systems through the lens of their unstable periodic orbits (UPOs).
Our approach involves three data-driven steps for detecting, identifying, and
stabilizing UPOs. We demonstrate the use of kernel integral operators within
delay coordinates as an innovative method for UPO detection. For identifying
the dynamic behavior associated with each individual UPO, we utilize the
Koopman operator to present the dynamics as linear equations in the space of
Koopman eigenfunctions. This allows for characterizing the chaotic attractor by
investigating its principal dynamical modes across varying UPOs. We extend this
methodology into an interpretable machine learning framework aimed at
stabilizing strange attractors on their UPOs. To illustrate the efficacy of our
approach, we apply it to the Lorenz attractor as a case study.Comment: arXiv admin note: text overlap with arXiv:2304.0783
A Bode Sensitivity Integral for Linear Time-Periodic Systems
Bode's sensitivity integral is a well-known formula that quantifies some of the limitations in feedback control for linear time-invariant systems. In this note, we show that there is a similar formula for linear time-periodic systems. The harmonic transfer function is used to prove the result. We use the notion of roll-off 2, which means that the first time-varying Markov parameter is equal to zero. It then follows that the harmonic transfer function is an analytic operator and a trace class operator. These facts are used to prove the result
A theoretical framework for supervised learning from regions
Supervised learning is investigated, when the data are represented not only by labeled points but also labeled regions of the input space. In the limit case, such
regions degenerate to single points and the proposed approach changes back to the classical learning context. The adopted framework entails the minimization
of a functional obtained by introducing a loss function that involves such regions. An additive regularization term is expressed via differential operators that model
the smoothness properties of the desired input/output relationship. Representer
theorems are given, proving that the optimization problem associated to learning
from labeled regions has a unique solution, which takes on the form of a linear
combination of kernel functions determined by the differential operators together
with the regions themselves. As a relevant situation, the case of regions given
by multi-dimensional intervals (i.e., “boxes”) is investigated, which models prior
knowledge expressed by logical propositions
Group Invariance, Stability to Deformations, and Complexity of Deep Convolutional Representations
The success of deep convolutional architectures is often attributed in part
to their ability to learn multiscale and invariant representations of natural
signals. However, a precise study of these properties and how they affect
learning guarantees is still missing. In this paper, we consider deep
convolutional representations of signals; we study their invariance to
translations and to more general groups of transformations, their stability to
the action of diffeomorphisms, and their ability to preserve signal
information. This analysis is carried by introducing a multilayer kernel based
on convolutional kernel networks and by studying the geometry induced by the
kernel mapping. We then characterize the corresponding reproducing kernel
Hilbert space (RKHS), showing that it contains a large class of convolutional
neural networks with homogeneous activation functions. This analysis allows us
to separate data representation from learning, and to provide a canonical
measure of model complexity, the RKHS norm, which controls both stability and
generalization of any learned model. In addition to models in the constructed
RKHS, our stability analysis also applies to convolutional networks with
generic activations such as rectified linear units, and we discuss its
relationship with recent generalization bounds based on spectral norms
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