406 research outputs found
Estimation of high-dimensional low-rank matrices
Suppose that we observe entries or, more generally, linear combinations of
entries of an unknown -matrix corrupted by noise. We are
particularly interested in the high-dimensional setting where the number
of unknown entries can be much larger than the sample size . Motivated by
several applications, we consider estimation of matrix under the assumption
that it has small rank. This can be viewed as dimension reduction or sparsity
assumption. In order to shrink toward a low-rank representation, we investigate
penalized least squares estimators with a Schatten- quasi-norm penalty term,
. We study these estimators under two possible assumptions---a modified
version of the restricted isometry condition and a uniform bound on the ratio
"empirical norm induced by the sampling operator/Frobenius norm." The main
results are stated as nonasymptotic upper bounds on the prediction risk and on
the Schatten- risk of the estimators, where . The rates that we
obtain for the prediction risk are of the form (for ), up to
logarithmic factors, where is the rank of . The particular examples of
multi-task learning and matrix completion are worked out in detail. The proofs
are based on tools from the theory of empirical processes. As a by-product, we
derive bounds for the th entropy numbers of the quasi-convex Schatten class
embeddings , , which are of independent
interest.Comment: Published in at http://dx.doi.org/10.1214/10-AOS860 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Quadruple Neutrosophic Theory And Applications Volume I
Neutrosophic set has been derived from a new branch of philosophy, namely Neutrosophy. Neutrosophic set is capable of dealing with uncertainty, indeterminacy and inconsistent information. Neutrosophic set approaches are suitable to modeling problems with uncertainty, indeterminacy and inconsistent information in which human knowledge is necessary, and human evaluation is needed. Neutrosophic set theory firstly proposed in 1998 by Florentin Smarandache, who also developed the concept of single valued neutrosophic set, oriented towards real world scientific and engineering applications. Since then, the single valued neutrosophic set theory has been extensively studied in books and monographs introducing neutrosophic sets and its applications, by many authors around the world. Also, an international journal - Neutrosophic Sets and Systems started its journey in 2013. Smarandache introduce for the first time the neutrosophic quadruple numbers (of the form + + + ) and the refined neutrosophic quadruple numbers
Multivariate Shortfall Risk Allocation and Systemic Risk
The ongoing concern about systemic risk since the outburst of the global
financial crisis has highlighted the need for risk measures at the level of
sets of interconnected financial components, such as portfolios, institutions
or members of clearing houses. The two main issues in systemic risk measurement
are the computation of an overall reserve level and its allocation to the
different components according to their systemic relevance. We develop here a
pragmatic approach to systemic risk measurement and allocation based on
multivariate shortfall risk measures, where acceptable allocations are first
computed and then aggregated so as to minimize costs. We analyze the
sensitivity of the risk allocations to various factors and highlight its
relevance as an indicator of systemic risk. In particular, we study the
interplay between the loss function and the dependence structure of the
components. Moreover, we address the computational aspects of risk allocation.
Finally, we apply this methodology to the allocation of the default fund of a
CCP on real data.Comment: Code, results and figures can also be consulted at
https://github.com/yarmenti/MSR
Elastic-Net Regularization in Learning Theory
Within the framework of statistical learning theory we analyze in detail the
so-called elastic-net regularization scheme proposed by Zou and Hastie for the
selection of groups of correlated variables. To investigate on the statistical
properties of this scheme and in particular on its consistency properties, we
set up a suitable mathematical framework. Our setting is random-design
regression where we allow the response variable to be vector-valued and we
consider prediction functions which are linear combination of elements ({\em
features}) in an infinite-dimensional dictionary. Under the assumption that the
regression function admits a sparse representation on the dictionary, we prove
that there exists a particular ``{\em elastic-net representation}'' of the
regression function such that, if the number of data increases, the elastic-net
estimator is consistent not only for prediction but also for variable/feature
selection. Our results include finite-sample bounds and an adaptive scheme to
select the regularization parameter. Moreover, using convex analysis tools, we
derive an iterative thresholding algorithm for computing the elastic-net
solution which is different from the optimization procedure originally proposed
by Zou and HastieComment: 32 pages, 3 figure
Learning weights in the generalized OWA operators
This paper discusses identification of parameters of generalized ordered weighted averaging (GOWA) operators from empirical data. Similarly to ordinary OWA operators, GOWA are characterized by a vector of weights, as well as the power to which the arguments are raised. We develop optimization techniques which allow one to fit such operators to the observed data. We also generalize these methods for functional defined GOWA and generalized Choquet integral based aggregation operators.<br /
Machine Learning (ML) module
Lectures notes of the machine learning content of the course TOML (Topics on Optimization and Machine Learning) at Master in Innovation and Research in Informatics (MIRI) at FIB, UPC.2023/202
Double Backpropagation with Applications to Robustness and Saliency Map Interpretability
This thesis is concerned with works in connection to double backpropagation, which is a phenomenon that arises when first-order optimization methods are applied to a neural network's loss function, if this contains derivatives. Its connection to robustness and saliency map interpretability is explained
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