44,513 research outputs found
Linearly-Recurrent Autoencoder Networks for Learning Dynamics
This paper describes a method for learning low-dimensional approximations of
nonlinear dynamical systems, based on neural-network approximations of the
underlying Koopman operator. Extended Dynamic Mode Decomposition (EDMD)
provides a useful data-driven approximation of the Koopman operator for
analyzing dynamical systems. This paper addresses a fundamental problem
associated with EDMD: a trade-off between representational capacity of the
dictionary and over-fitting due to insufficient data. A new neural network
architecture combining an autoencoder with linear recurrent dynamics in the
encoded state is used to learn a low-dimensional and highly informative
Koopman-invariant subspace of observables. A method is also presented for
balanced model reduction of over-specified EDMD systems in feature space.
Nonlinear reconstruction using partially linear multi-kernel regression aims to
improve reconstruction accuracy from the low-dimensional state when the data
has complex but intrinsically low-dimensional structure. The techniques
demonstrate the ability to identify Koopman eigenfunctions of the unforced
Duffing equation, create accurate low-dimensional models of an unstable
cylinder wake flow, and make short-time predictions of the chaotic
Kuramoto-Sivashinsky equation.Comment: 37 pages, 16 figure
Semi-Supervised Learning with Ladder Networks
We combine supervised learning with unsupervised learning in deep neural
networks. The proposed model is trained to simultaneously minimize the sum of
supervised and unsupervised cost functions by backpropagation, avoiding the
need for layer-wise pre-training. Our work builds on the Ladder network
proposed by Valpola (2015), which we extend by combining the model with
supervision. We show that the resulting model reaches state-of-the-art
performance in semi-supervised MNIST and CIFAR-10 classification, in addition
to permutation-invariant MNIST classification with all labels.Comment: Revised denoising function, updated results, fixed typo
Statistical inference for dynamical systems: a review
The topic of statistical inference for dynamical systems has been studied
extensively across several fields. In this survey we focus on the problem of
parameter estimation for non-linear dynamical systems. Our objective is to
place results across distinct disciplines in a common setting and highlight
opportunities for further research.Comment: Some minor typos correcte
Computing Reduced Order Models via Inner-Outer Krylov Recycling in Diffuse Optical Tomography
In nonlinear imaging problems whose forward model is described by a partial
differential equation (PDE), the main computational bottleneck in solving the
inverse problem is the need to solve many large-scale discretized PDEs at each
step of the optimization process. In the context of absorption imaging in
diffuse optical tomography, one approach to addressing this bottleneck proposed
recently (de Sturler, et al, 2015) reformulates the viewing of the forward
problem as a differential algebraic system, and then employs model order
reduction (MOR). However, the construction of the reduced model requires the
solution of several full order problems (i.e. the full discretized PDE for
multiple right-hand sides) to generate a candidate global basis. This step is
then followed by a rank-revealing factorization of the matrix containing the
candidate basis in order to compress the basis to a size suitable for
constructing the reduced transfer function. The present paper addresses the
costs associated with the global basis approximation in two ways. First, we use
the structure of the matrix to rewrite the full order transfer function, and
corresponding derivatives, such that the full order systems to be solved are
symmetric (positive definite in the zero frequency case). Then we apply MOR to
the new formulation of the problem. Second, we give an approach to computing
the global basis approximation dynamically as the full order systems are
solved. In this phase, only the incrementally new, relevant information is
added to the existing global basis, and redundant information is not computed.
This new approach is achieved by an inner-outer Krylov recycling approach which
has potential use in other applications as well. We show the value of the new
approach to approximate global basis computation on two DOT absorption image
reconstruction problems
Reachable Space Characterization of Markov Decision Processes with Time Variability
We propose a solution to a time-varying variant of Markov Decision Processes
which can be used to address decision-theoretic planning problems for
autonomous systems operating in unstructured outdoor environments. We explore
the time variability property of the planning stochasticity and investigate the
state reachability, based on which we then develop an efficient iterative
method that offers a good trade-off between solution optimality and time
complexity. The reachability space is constructed by analyzing the means and
variances of states' reaching time in the future. We validate our algorithm
through extensive simulations using ocean data, and the results show that our
method achieves a great performance in terms of both solution quality and
computing time.Comment: 10 pages, 9 figures, 1 table, accepted by RSS 201
Technical Report: Observability of a Linear System under Sparsity Constraints
Consider an n-dimensional linear system where it is known that there are at
most k<n non-zero components in the initial state. The observability problem,
that is the recovery of the initial state, for such a system is considered. We
obtain sufficient conditions on the number of the available observations to be
able to recover the initial state exactly for such a system. Both deterministic
and stochastic setups are considered for system dynamics. In the former
setting, the system matrices are known deterministically, whereas in the latter
setting, all of the matrices are picked from a randomized class of matrices.
The main message is that, one does not need to obtain full n observations to be
able to uniquely identify the initial state of the linear system, even when the
observations are picked randomly, when the initial condition is known to be
sparse
On Optimal Zero-Delay Coding of Vector Markov Sources
Optimal zero-delay coding (quantization) of a vector-valued Markov source
driven by a noise process is considered. Using a stochastic control problem
formulation, the existence and structure of optimal quantization policies are
studied. For a finite-horizon problem with bounded per-stage distortion
measure, the existence of an optimal zero-delay quantization policy is shown
provided that the quantizers allowed are ones with convex codecells. The
bounded distortion assumption is relaxed to cover cases that include the linear
quadratic Gaussian problem. For the infinite horizon problem and a stationary
Markov source the optimality of deterministic Markov coding policies is shown.
The existence of optimal stationary Markov quantization policies is also shown
provided randomization that is shared by the encoder and the decoder is
allowed.Comment: IEEE Transactions on Information Theory, accepted for publicatio
Time-Delay Observables for Koopman: Theory and Applications
Nonlinear dynamical systems are ubiquitous in science and engineering, yet
analysis and prediction of these systems remains a challenge. Koopman operator
theory circumvents some of these issues by considering the dynamics in the
space of observable functions on the state, in which the dynamics are
intrinsically linear and thus amenable to standard techniques from numerical
analysis and linear algebra. However, practical issues remain with this
approach, as the space of observables is infinite-dimensional and selecting a
subspace of functions in which to accurately represent the system is a
nontrivial task. In this work we consider time-delay observables to represent
nonlinear dynamics in the Koopman operator framework. We prove the surprising
result that Koopman operators for different systems admit universal
(system-independent) representations in these coordinates, and give analytic
expressions for these representations. In addition, we show that for certain
systems a restricted class of these observables form an optimal
finite-dimensional basis for representing the Koopman operator, and that the
analytic representation of the Koopman operator in these coordinates coincides
with results computed by the dynamic mode decomposition. We provide numerical
examples to complement our results. In addition to being theoretically
interesting, these results have implications for a number of linearization
algorithms for dynamical systems.Comment: 28 pages, 6 figure
Deconstructing the Ladder Network Architecture
The Manual labeling of data is and will remain a costly endeavor. For this
reason, semi-supervised learning remains a topic of practical importance. The
recently proposed Ladder Network is one such approach that has proven to be
very successful. In addition to the supervised objective, the Ladder Network
also adds an unsupervised objective corresponding to the reconstruction costs
of a stack of denoising autoencoders. Although the empirical results are
impressive, the Ladder Network has many components intertwined, whose
contributions are not obvious in such a complex architecture. In order to help
elucidate and disentangle the different ingredients in the Ladder Network
recipe, this paper presents an extensive experimental investigation of variants
of the Ladder Network in which we replace or remove individual components to
gain more insight into their relative importance. We find that all of the
components are necessary for achieving optimal performance, but they do not
contribute equally. For semi-supervised tasks, we conclude that the most
important contribution is made by the lateral connection, followed by the
application of noise, and finally the choice of what we refer to as the
`combinator function' in the decoder path. We also find that as the number of
labeled training examples increases, the lateral connections and reconstruction
criterion become less important, with most of the improvement in generalization
being due to the injection of noise in each layer. Furthermore, we present a
new type of combinator function that outperforms the original design in both
fully- and semi-supervised tasks, reducing record test error rates on
Permutation-Invariant MNIST to 0.57% for the supervised setting, and to 0.97%
and 1.0% for semi-supervised settings with 1000 and 100 labeled examples
respectively.Comment: Proceedings of the 33 rd International Conference on Machine
Learning, New York, NY, USA, 201
Self-taught learning of a deep invariant representation for visual tracking via temporal slowness principle
Visual representation is crucial for a visual tracking method's performances.
Conventionally, visual representations adopted in visual tracking rely on
hand-crafted computer vision descriptors. These descriptors were developed
generically without considering tracking-specific information. In this paper,
we propose to learn complex-valued invariant representations from tracked
sequential image patches, via strong temporal slowness constraint and stacked
convolutional autoencoders. The deep slow local representations are learned
offline on unlabeled data and transferred to the observational model of our
proposed tracker. The proposed observational model retains old training samples
to alleviate drift, and collect negative samples which are coherent with
target's motion pattern for better discriminative tracking. With the learned
representation and online training samples, a logistic regression classifier is
adopted to distinguish target from background, and retrained online to adapt to
appearance changes. Subsequently, the observational model is integrated into a
particle filter framework to peform visual tracking. Experimental results on
various challenging benchmark sequences demonstrate that the proposed tracker
performs favourably against several state-of-the-art trackers.Comment: Pattern Recognition (Elsevier), 201
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