8,161 research outputs found
Theoretical Properties of Projection Based Multilayer Perceptrons with Functional Inputs
Many real world data are sampled functions. As shown by Functional Data
Analysis (FDA) methods, spectra, time series, images, gesture recognition data,
etc. can be processed more efficiently if their functional nature is taken into
account during the data analysis process. This is done by extending standard
data analysis methods so that they can apply to functional inputs. A general
way to achieve this goal is to compute projections of the functional data onto
a finite dimensional sub-space of the functional space. The coordinates of the
data on a basis of this sub-space provide standard vector representations of
the functions. The obtained vectors can be processed by any standard method. In
our previous work, this general approach has been used to define projection
based Multilayer Perceptrons (MLPs) with functional inputs. We study in this
paper important theoretical properties of the proposed model. We show in
particular that MLPs with functional inputs are universal approximators: they
can approximate to arbitrary accuracy any continuous mapping from a compact
sub-space of a functional space to R. Moreover, we provide a consistency result
that shows that any mapping from a functional space to R can be learned thanks
to examples by a projection based MLP: the generalization mean square error of
the MLP decreases to the smallest possible mean square error on the data when
the number of examples goes to infinity
Nonparametric regression using deep neural networks with ReLU activation function
Consider the multivariate nonparametric regression model. It is shown that
estimators based on sparsely connected deep neural networks with ReLU
activation function and properly chosen network architecture achieve the
minimax rates of convergence (up to -factors) under a general
composition assumption on the regression function. The framework includes many
well-studied structural constraints such as (generalized) additive models.
While there is a lot of flexibility in the network architecture, the tuning
parameter is the sparsity of the network. Specifically, we consider large
networks with number of potential network parameters exceeding the sample size.
The analysis gives some insights into why multilayer feedforward neural
networks perform well in practice. Interestingly, for ReLU activation function
the depth (number of layers) of the neural network architectures plays an
important role and our theory suggests that for nonparametric regression,
scaling the network depth with the sample size is natural. It is also shown
that under the composition assumption wavelet estimators can only achieve
suboptimal rates.Comment: article, rejoinder and supplementary materia
A survey on modern trainable activation functions
In neural networks literature, there is a strong interest in identifying and
defining activation functions which can improve neural network performance. In
recent years there has been a renovated interest of the scientific community in
investigating activation functions which can be trained during the learning
process, usually referred to as "trainable", "learnable" or "adaptable"
activation functions. They appear to lead to better network performance.
Diverse and heterogeneous models of trainable activation function have been
proposed in the literature. In this paper, we present a survey of these models.
Starting from a discussion on the use of the term "activation function" in
literature, we propose a taxonomy of trainable activation functions, highlight
common and distinctive proprieties of recent and past models, and discuss main
advantages and limitations of this type of approach. We show that many of the
proposed approaches are equivalent to adding neuron layers which use fixed
(non-trainable) activation functions and some simple local rule that
constraints the corresponding weight layers.Comment: Published in "Neural Networks" journal (Elsevier
Multilayer Networks
In most natural and engineered systems, a set of entities interact with each
other in complicated patterns that can encompass multiple types of
relationships, change in time, and include other types of complications. Such
systems include multiple subsystems and layers of connectivity, and it is
important to take such "multilayer" features into account to try to improve our
understanding of complex systems. Consequently, it is necessary to generalize
"traditional" network theory by developing (and validating) a framework and
associated tools to study multilayer systems in a comprehensive fashion. The
origins of such efforts date back several decades and arose in multiple
disciplines, and now the study of multilayer networks has become one of the
most important directions in network science. In this paper, we discuss the
history of multilayer networks (and related concepts) and review the exploding
body of work on such networks. To unify the disparate terminology in the large
body of recent work, we discuss a general framework for multilayer networks,
construct a dictionary of terminology to relate the numerous existing concepts
to each other, and provide a thorough discussion that compares, contrasts, and
translates between related notions such as multilayer networks, multiplex
networks, interdependent networks, networks of networks, and many others. We
also survey and discuss existing data sets that can be represented as
multilayer networks. We review attempts to generalize single-layer-network
diagnostics to multilayer networks. We also discuss the rapidly expanding
research on multilayer-network models and notions like community structure,
connected components, tensor decompositions, and various types of dynamical
processes on multilayer networks. We conclude with a summary and an outlook.Comment: Working paper; 59 pages, 8 figure
Post-processing partitions to identify domains of modularity optimization
We introduce the Convex Hull of Admissible Modularity Partitions (CHAMP)
algorithm to prune and prioritize different network community structures
identified across multiple runs of possibly various computational heuristics.
Given a set of partitions, CHAMP identifies the domain of modularity
optimization for each partition ---i.e., the parameter-space domain where it
has the largest modularity relative to the input set---discarding partitions
with empty domains to obtain the subset of partitions that are "admissible"
candidate community structures that remain potentially optimal over indicated
parameter domains. Importantly, CHAMP can be used for multi-dimensional
parameter spaces, such as those for multilayer networks where one includes a
resolution parameter and interlayer coupling. Using the results from CHAMP, a
user can more appropriately select robust community structures by observing the
sizes of domains of optimization and the pairwise comparisons between
partitions in the admissible subset. We demonstrate the utility of CHAMP with
several example networks. In these examples, CHAMP focuses attention onto
pruned subsets of admissible partitions that are 20-to-1785 times smaller than
the sets of unique partitions obtained by community detection heuristics that
were input into CHAMP.Comment: http://www.mdpi.com/1999-4893/10/3/9
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