287 research outputs found
Empirical Risk Minimization over Artificial Neural Networks Overcomes the Curse of Dimensionality in the Numerical Approximation of Linear Kolmogorov Partial Differential Equations with Unbounded Initial Functions
Deep learning algorithms have been successfully applied to numerically solve
linear Kolmogorov partial differential equations (PDEs). A recent research
shows that the empirical risk minimization~(ERM) over deep artificial neural
networks overcomes the curse of dimensionality in the numerical approximation
of linear Kolmogorov PDEs with bounded initial functions. However, the initial
functions may be unbounded in many applications such as the Black Scholes PDEs
in pricing call options. In this paper, we extend this result to the cases
involving unbounded initial functions. We prove that for -dimensional linear
Kolmogorov PDEs with unbounded initial functions, under suitable assumptions,
the number of training data and the size of the artificial neural network
required to achieve an accuracy for the ERM grow polynomially in
both and . Moreover, we verify that the required
assumptions hold for Black-Scholes PDEs and heat equations which are two
important cases of linear Kolmogorov PDEs
Space-Invariant Projection in Streaming Network Embedding
Newly arriving nodes in dynamics networks would gradually make the node
embedding space drifted and the retraining of node embedding and downstream
models indispensable. An exact threshold size of these new nodes, below which
the node embedding space will be predicatively maintained, however, is rarely
considered in either theory or experiment. From the view of matrix perturbation
theory, a threshold of the maximum number of new nodes that keep the node
embedding space approximately equivalent is analytically provided and
empirically validated. It is therefore theoretically guaranteed that as the
size of newly arriving nodes is below this threshold, embeddings of these new
nodes can be quickly derived from embeddings of original nodes. A generation
framework, Space-Invariant Projection (SIP), is accordingly proposed to enables
arbitrary static MF-based embedding schemes to embed new nodes in dynamics
networks fast. The time complexity of SIP is linear with the network size. By
combining SIP with four state-of-the-art MF-based schemes, we show that SIP
exhibits not only wide adaptability but also strong empirical performance in
terms of efficiency and efficacy on the node classification task in three real
datasets
Analysis of the Generalization Error of deep learning based on Randomized Quasi-Monte Carlo for Solving Linear Kolmogorov PDEs
Deep learning algorithms have been widely used to solve linear Kolmogorov
partial differential equations~(PDEs) in high dimensions, where the loss
function is defined as a mathematical expectation. We propose to use the
randomized quasi-Monte Carlo (RQMC) method instead of the Monte Carlo (MC)
method for computing the loss function. In theory, we decompose the error from
empirical risk minimization~(ERM) into the generalization error and the
approximation error. Notably, the approximation error is independent of the
sampling methods. We prove that the convergence order of the mean
generalization error for the RQMC method is for
arbitrarily small , while for the MC method it is
for arbitrarily small . Consequently, we
find that the overall error for the RQMC method is asymptotically smaller than
that for the MC method as increases. Our numerical experiments show that
the algorithm based on the RQMC method consistently achieves smaller relative
error than that based on the MC method
Real estate rental market: a 10-year bibliometricbased review
The real estate rental market (RERM) is considered to have an
important role in the entire real estate market. It refers to a property composed of land and its buildings, including the natural
resources that can be rented or leased. Previous researches show
that most developed countries have experienced the historical
process of passively renting, actively buying, and actively renting.
Moreover, academic interest in the impact of different sectors of
the RERM has been reviewed increasingly over the past decade.
However, previous studies provide limited insights into a comprehensive review of the RERM. Based on a 10-year database of 790
articles collected from the Web of Science, a comprehensive literature review is presented to discover the knowledge structure
of RERM using CiteSpace software. First, this study recognizes the
cluster of the articles, and discusses six major clusters in detail.
Next, this study has identified four research trends that emerged
during the past decade. To reveal the differences between the
studies in the United States (US), China and the United Kingdom
(UK), this study compares their publication scales and co-word
networks. Finally, this study suggests six meaningful future
research directions
Coherent structure analysis of spatiotemporal chaos
We introduce a measure to quantify spatiotemporal turbulence in extended systems. It is based on the statistical analysis of a coherent structure decomposition of the evolving system. Applied to a cellular excitable medium and a reaction-diffusion model describing the oxidation of CO on Pt(100), it reveals power-law scaling of the size distribution of coherent space-time structures for the state of spiral turbulence. The coherent structure decomposition is also used to define an entropy measure, which sharply increases in these systems at the transition to turbulence
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