5,656 research outputs found
Simultaneously Structured Models with Application to Sparse and Low-rank Matrices
The topic of recovery of a structured model given a small number of linear
observations has been well-studied in recent years. Examples include recovering
sparse or group-sparse vectors, low-rank matrices, and the sum of sparse and
low-rank matrices, among others. In various applications in signal processing
and machine learning, the model of interest is known to be structured in
several ways at the same time, for example, a matrix that is simultaneously
sparse and low-rank.
Often norms that promote each individual structure are known, and allow for
recovery using an order-wise optimal number of measurements (e.g.,
norm for sparsity, nuclear norm for matrix rank). Hence, it is reasonable to
minimize a combination of such norms. We show that, surprisingly, if we use
multi-objective optimization with these norms, then we can do no better,
order-wise, than an algorithm that exploits only one of the present structures.
This result suggests that to fully exploit the multiple structures, we need an
entirely new convex relaxation, i.e. not one that is a function of the convex
relaxations used for each structure. We then specialize our results to the case
of sparse and low-rank matrices. We show that a nonconvex formulation of the
problem can recover the model from very few measurements, which is on the order
of the degrees of freedom of the matrix, whereas the convex problem obtained
from a combination of the and nuclear norms requires many more
measurements. This proves an order-wise gap between the performance of the
convex and nonconvex recovery problems in this case. Our framework applies to
arbitrary structure-inducing norms as well as to a wide range of measurement
ensembles. This allows us to give performance bounds for problems such as
sparse phase retrieval and low-rank tensor completion.Comment: 38 pages, 9 figure
Node harvest
When choosing a suitable technique for regression and classification with
multivariate predictor variables, one is often faced with a tradeoff between
interpretability and high predictive accuracy. To give a classical example,
classification and regression trees are easy to understand and interpret. Tree
ensembles like Random Forests provide usually more accurate predictions. Yet
tree ensembles are also more difficult to analyze than single trees and are
often criticized, perhaps unfairly, as `black box' predictors. Node harvest is
trying to reconcile the two aims of interpretability and predictive accuracy by
combining positive aspects of trees and tree ensembles. Results are very sparse
and interpretable and predictive accuracy is extremely competitive, especially
for low signal-to-noise data. The procedure is simple: an initial set of a few
thousand nodes is generated randomly. If a new observation falls into just a
single node, its prediction is the mean response of all training observation
within this node, identical to a tree-like prediction. A new observation falls
typically into several nodes and its prediction is then the weighted average of
the mean responses across all these nodes. The only role of node harvest is to
`pick' the right nodes from the initial large ensemble of nodes by choosing
node weights, which amounts in the proposed algorithm to a quadratic
programming problem with linear inequality constraints. The solution is sparse
in the sense that only very few nodes are selected with a nonzero weight. This
sparsity is not explicitly enforced. Maybe surprisingly, it is not necessary to
select a tuning parameter for optimal predictive accuracy. Node harvest can
handle mixed data and missing values and is shown to be simple to interpret and
competitive in predictive accuracy on a variety of data sets.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS367 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Variational Data Assimilation via Sparse Regularization
This paper studies the role of sparse regularization in a properly chosen
basis for variational data assimilation (VDA) problems. Specifically, it
focuses on data assimilation of noisy and down-sampled observations while the
state variable of interest exhibits sparsity in the real or transformed domain.
We show that in the presence of sparsity, the -norm regularization
produces more accurate and stable solutions than the classic data assimilation
methods. To motivate further developments of the proposed methodology,
assimilation experiments are conducted in the wavelet and spectral domain using
the linear advection-diffusion equation
Fitting Prediction Rule Ensembles with R Package pre
Prediction rule ensembles (PREs) are sparse collections of rules, offering
highly interpretable regression and classification models. This paper presents
the R package pre, which derives PREs through the methodology of Friedman and
Popescu (2008). The implementation and functionality of package pre is
described and illustrated through application on a dataset on the prediction of
depression. Furthermore, accuracy and sparsity of PREs is compared with that of
single trees, random forest and lasso regression in four benchmark datasets.
Results indicate that pre derives ensembles with predictive accuracy comparable
to that of random forests, while using a smaller number of variables for
prediction
Knowledge mining sensory evaluation data: genetic programming, statistical techniques, and swarm optimization
Knowledge mining sensory evaluation data is a challenging process due to extreme sparsity of the data, and a large variation in responses from different members (called assessors) of the panel. The main goals of knowledge mining in sensory sciences are understanding the dependency of the perceived liking score on the concentration levels of flavors’ ingredients, identifying ingredients that drive liking, segmenting the panel into groups with similar liking preferences and optimizing flavors to maximize liking per group. Our approach employs (1) Genetic programming (symbolic regression) and ensemble methods to generate multiple diverse explanations of assessor liking preferences with confidence information; (2) statistical techniques to extrapolate using the produced ensembles to unobserved regions of the flavor space, and segment the assessors into groups which either have the same propensity to like flavors, or are driven by the same ingredients; and (3) two-objective swarm optimization to identify flavors which are well and consistently liked by a selected segment of assessors
Modeling Financial Time Series with Artificial Neural Networks
Financial time series convey the decisions and actions of a population of human actors over time. Econometric and regressive models have been developed in the past decades for analyzing these time series. More recently, biologically inspired artificial neural network models have been shown to overcome some of the main challenges of traditional techniques by better exploiting the non-linear, non-stationary, and oscillatory nature of noisy, chaotic human interactions. This review paper explores the options, benefits, and weaknesses of the various forms of artificial neural networks as compared with regression techniques in the field of financial time series analysis.CELEST, a National Science Foundation Science of Learning Center (SBE-0354378); SyNAPSE program of the Defense Advanced Research Project Agency (HR001109-03-0001
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