532 research outputs found
An Optimized Dynamic Mode Decomposition Model Robust to Multiplicative Noise
Dynamic mode decomposition (DMD) is an efficient tool for decomposing
spatio-temporal data into a set of low-dimensional modes, yielding the
oscillation frequencies and the growth rates of physically significant modes.
In this paper, we propose a novel DMD model that can be used for dynamical
systems affected by multiplicative noise. We first derive a maximum a
posteriori (MAP) estimator for the data-based model decomposition of a linear
dynamical system corrupted by certain multiplicative noise. Applying penalty
relaxation to the MAP estimator, we obtain the proposed DMD model whose
epigraphical limits are the MAP estimator and the conventional optimized DMD
model. We also propose an efficient alternating gradient descent method for
solving the proposed DMD model, and analyze its convergence behavior. The
proposed model is demonstrated on both the synthetic data and the numerically
generated one-dimensional combustor data, and is shown to have superior
reconstruction properties compared to state-of-the-art DMD models. Considering
that multiplicative noise is ubiquitous in numerous dynamical systems, the
proposed DMD model opens up new possibilities for accurate data-based modal
decomposition.Comment: 35 pages, 10 figure
Pop2Piano : Pop Audio-based Piano Cover Generation
The piano cover of pop music is widely enjoyed by people. However, the
generation task of the pop piano cover is still understudied. This is partly
due to the lack of synchronized {Pop, Piano Cover} data pairs, which made it
challenging to apply the latest data-intensive deep learning-based methods. To
leverage the power of the data-driven approach, we make a large amount of
paired and synchronized {pop, piano cover} data using an automated pipeline. In
this paper, we present Pop2Piano, a Transformer network that generates piano
covers given waveforms of pop music. To the best of our knowledge, this is the
first model to directly generate a piano cover from pop audio without melody
and chord extraction modules. We show that Pop2Piano trained with our dataset
can generate plausible piano covers
On the linear convergence of additive Schwarz methods for the -Laplacian
We consider additive Schwarz methods for boundary value problems involving
the -Laplacian. While the existing theoretical estimates for the convergence
rate of additive Schwarz methods for the -Laplacian are sublinear, the
actual convergence rate observed by numerical experiments is linear. In this
paper, we bridge the gap between these theoretical and numerical results by
analyzing the linear convergence rate of additive Schwarz methods for the
-Laplacian. In order to estimate the linear convergence rate of the methods,
we present two essential components. Firstly, we present a new abstract
convergence theory of additive Schwarz methods written in terms of a
quasi-norm. This quasi-norm exhibits behavior similar to the Bregman distance
of the convex energy functional associated to the problem. Secondly, we provide
a quasi-norm version of the Poincar'{e}--Friedrichs inequality, which is
essential for deriving a quasi-norm stable decomposition for a two-level domain
decomposition setting.Comment: 23 pages, 2 figure
Balanced Group Convolution: An Improved Group Convolution Based on Approximability Estimates
The performance of neural networks has been significantly improved by
increasing the number of channels in convolutional layers. However, this
increase in performance comes with a higher computational cost, resulting in
numerous studies focused on reducing it. One promising approach to address this
issue is group convolution, which effectively reduces the computational cost by
grouping channels. However, to the best of our knowledge, there has been no
theoretical analysis on how well the group convolution approximates the
standard convolution. In this paper, we mathematically analyze the
approximation of the group convolution to the standard convolution with respect
to the number of groups. Furthermore, we propose a novel variant of the group
convolution called balanced group convolution, which shows a higher
approximation with a small additional computational cost. We provide
experimental results that validate our theoretical findings and demonstrate the
superior performance of the balanced group convolution over other variants of
group convolution.Comment: 26pages, 2 figure
Parareal Neural Networks Emulating a Parallel-in-time Algorithm
As deep neural networks (DNNs) become deeper, the training time increases. In
this perspective, multi-GPU parallel computing has become a key tool in
accelerating the training of DNNs. In this paper, we introduce a novel
methodology to construct a parallel neural network that can utilize multiple
GPUs simultaneously from a given DNN. We observe that layers of DNN can be
interpreted as the time step of a time-dependent problem and can be
parallelized by emulating a parallel-in-time algorithm called parareal. The
parareal algorithm consists of fine structures which can be implemented in
parallel and a coarse structure which gives suitable approximations to the fine
structures. By emulating it, the layers of DNN are torn to form a parallel
structure which is connected using a suitable coarse network. We report
accelerated and accuracy-preserved results of the proposed methodology applied
to VGG-16 and ResNet-1001 on several datasets
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