36 research outputs found
Data Filtering for Cluster Analysis by -Norm Regularization
A data filtering method for cluster analysis is proposed, based on minimizing
a least squares function with a weighted -norm penalty. To overcome the
discontinuity of the objective function, smooth non-convex functions are
employed to approximate the -norm. The convergence of the global
minimum points of the approximating problems towards global minimum points of
the original problem is stated. The proposed method also exploits a suitable
technique to choose the penalty parameter. Numerical results on synthetic and
real data sets are finally provided, showing how some existing clustering
methods can take advantages from the proposed filtering strategy.Comment: Optimization Letters (2017
An almost cyclic 2-coordinate descent method for singly linearly constrained problems
A block decomposition method is proposed for minimizing a (possibly
non-convex) continuously differentiable function subject to one linear equality
constraint and simple bounds on the variables. The proposed method iteratively
selects a pair of coordinates according to an almost cyclic strategy that does
not use first-order information, allowing us not to compute the whole gradient
of the objective function during the algorithm. Using first-order search
directions to update each pair of coordinates, global convergence to stationary
points is established for different choices of the stepsize under an
appropriate assumption on the level set. In particular, both inexact and exact
line search strategies are analyzed. Further, linear convergence rate is proved
under standard additional assumptions. Numerical results are finally provided
to show the effectiveness of the proposed method.Comment: Computational Optimization and Application
A decomposition method for lasso problems with zero-sum constraint
In this paper, we consider lasso problems with zero-sum constraint, commonly
required for the analysis of compositional data in high-dimensional spaces. A
novel algorithm is proposed to solve these problems, combining a tailored
active-set technique, to identify the zero variables in the optimal solution,
with a 2-coordinate descent scheme. At every iteration, the algorithm chooses
between two different strategies: the first one requires to compute the whole
gradient of the smooth term of the objective function and is more accurate in
the active-set estimate, while the second one only uses partial derivatives and
is computationally more efficient. Global convergence to optimal solutions is
proved and numerical results are provided on synthetic and real datasets,
showing the effectiveness of the proposed method. The software is publicly
available
Active-set identification with complexity guarantees of an almost cyclic 2-coordinate descent method with Armijo line search
In this paper, it is established finite active-set identification of an
almost cyclic 2-coordinate descent method for problems with one linear coupling
constraint and simple bounds. First, general active-set identification results
are stated for non-convex objective functions. Then, under convexity and a
quadratic growth condition (satisfied by any strongly convex function),
complexity results on the number of iterations required to identify the active
set are given. In our analysis, a simple Armijo line search is used to compute
the stepsize, thus not requiring exact minimizations or additional information
On global minimizers of quadratic functions with cubic regularization
In this paper, we analyze some theoretical properties of the problem of
minimizing a quadratic function with a cubic regularization term, arising in
many methods for unconstrained and constrained optimization that have been
proposed in the last years. First we show that, given any stationary point that
is not a global solution, it is possible to compute, in closed form, a new
point with a smaller objective function value. Then, we prove that a global
minimizer can be obtained by computing a finite number of stationary points.
Finally, we extend these results to the case where stationary conditions are
approximately satisfied, discussing some possible algorithmic applications.Comment: Optimization Letters (2018
An Active-Set Algorithmic Framework for Non-Convex Optimization Problems over the Simplex
In this paper, we describe a new active-set algorithmic framework for
minimizing a non-convex function over the unit simplex. At each iteration, the
method makes use of a rule for identifying active variables (i.e., variables
that are zero at a stationary point) and specific directions (that we name
active-set gradient related directions) satisfying a new "nonorthogonality"
type of condition. We prove global convergence to stationary points when using
an Armijo line search in the given framework. We further describe three
different examples of active-set gradient related directions that guarantee
linear convergence rate (under suitable assumptions). Finally, we report
numerical experiments showing the effectiveness of the approach.Comment: 29 pages, 3 figure
Total variation based community detection using a nonlinear optimization approach
Maximizing the modularity of a network is a successful tool to identify an
important community of nodes. However, this combinatorial optimization problem
is known to be NP-complete. Inspired by recent nonlinear modularity eigenvector
approaches, we introduce the modularity total variation and show that
its box-constrained global maximum coincides with the maximum of the original
discrete modularity function. Thus we describe a new nonlinear optimization
approach to solve the equivalent problem leading to a community detection
strategy based on . The proposed approach relies on the use of a fast
first-order method that embeds a tailored active-set strategy. We report
extensive numerical comparisons with standard matrix-based approaches and the
Generalized RatioDCA approach for nonlinear modularity eigenvectors, showing
that our new method compares favourably with state-of-the-art alternatives
Louvain-like Methods for Community Detection in Multi-Layer Networks
In many complex systems, entities interact with each other through
complicated patterns that embed different relationships, thus generating
networks with multiple levels and/or multiple types of edges. When trying to
improve our understanding of those complex networks, it is of paramount
importance to explicitly take the multiple layers of connectivity into account
in the analysis. In this paper, we focus on detecting community structures in
multi-layer networks, i.e., detecting groups of well-connected nodes shared
among the layers, a very popular task that poses a lot of interesting questions
and challenges. Most of the available algorithms in this context either reduce
multi-layer networks to a single-layer network or try to extend algorithms for
single-layer networks by using consensus clustering. Those approaches have
anyway been criticized lately. They indeed ignore the connections among the
different layers, hence giving low accuracy. To overcome these issues, we
propose new community detection methods based on tailored Louvain-like
strategies that simultaneously handle the multiple layers. We consider the
informative case, where all layers show a community structure, and the noisy
case, where some layers only add noise to the system. We report experiments on
both artificial and real-world networks showing the effectiveness of the
proposed strategies.Comment: 16 pages, 4 figure
Learning the Right Layers: a Data-Driven Layer-Aggregation Strategy for Semi-Supervised Learning on Multilayer Graphs
Clustering (or community detection) on multilayer graphs poses several
additional complications with respect to standard graphs as different layers
may be characterized by different structures and types of information. One of
the major challenges is to establish the extent to which each layer contributes
to the cluster assignment in order to effectively take advantage of the
multilayer structure and improve upon the classification obtained using the
individual layers or their union. However, making an informed a-priori
assessment about the clustering information content of the layers can be very
complicated. In this work, we assume a semi-supervised learning setting, where
the class of a small percentage of nodes is initially provided, and we propose
a parameter-free Laplacian-regularized model that learns an optimal nonlinear
combination of the different layers from the available input labels. The
learning algorithm is based on a Frank-Wolfe optimization scheme with inexact
gradient, combined with a modified Label Propagation iteration. We provide a
detailed convergence analysis of the algorithm and extensive experiments on
synthetic and real-world datasets, showing that the proposed method compares
favourably with a variety of baselines and outperforms each individual layer
when used in isolation
Machine Learning-Based Classification to Disentangle EEG Responses to TMS and Auditory Input
The combination of transcranial magnetic stimulation (TMS) and electroencephalography (EEG) offers an unparalleled opportunity to study cortical physiology by characterizing brain electrical responses to external perturbation, called transcranial-evoked potentials (TEPs). Although these reflect cortical post-synaptic potentials, they can be contaminated by auditory evoked potentials (AEPs) due to the TMS click, which partly show a similar spatial and temporal scalp distribution. Therefore, TEPs and AEPs can be difficult to disentangle by common statistical methods, especially in conditions of suboptimal AEP suppression. In this work, we explored the ability of machine learning
algorithms to distinguish TEPs recorded with masking of the TMS click, AEPs and non-masked TEPs in a sample of healthy subjects. Overall, our classifier provided reliable results at the single subject level, even for signals where differences were not shown in previous works. Classification accuracy (CA) was lower at the group level, when different subjects were used for training and test phases, and when three stimulation conditions instead of two were compared. Lastly, CA was higher when average, rather than single-trial TEPs, were used. In conclusion, this proof-of-concept study proposes machine learning as a promising tool to separate pure TEPs from those contaminated by sensory input