135 research outputs found
Statistical and Computational Tradeoff in Genetic Algorithm-Based Estimation
When a Genetic Algorithm (GA), or a stochastic algorithm in general, is
employed in a statistical problem, the obtained result is affected by both
variability due to sampling, that refers to the fact that only a sample is
observed, and variability due to the stochastic elements of the algorithm. This
topic can be easily set in a framework of statistical and computational
tradeoff question, crucial in recent problems, for which statisticians must
carefully set statistical and computational part of the analysis, taking
account of some resource or time constraints. In the present work we analyze
estimation problems tackled by GAs, for which variability of estimates can be
decomposed in the two sources of variability, considering some constraints in
the form of cost functions, related to both data acquisition and runtime of the
algorithm. Simulation studies will be presented to discuss the statistical and
computational tradeoff question.Comment: 17 pages, 5 figure
High-dimensional change-point estimation: Combining filtering with convex optimization
We consider change-point estimation in a sequence of high-dimensional signals given noisy observations. Classical approaches to this problem such as the filtered derivative method are useful for sequences of scalar-valued signals, but they have undesirable scaling behavior in the high-dimensional setting. However, many high-dimensional signals encountered in practice frequently possess latent low-dimensional structure. Motivated by this observation, we propose a technique for high-dimensional change-point estimation that combines the filtered derivative approach from previous work with convex optimization methods based on atomic norm regularization, which are useful for exploiting structure in high-dimensional data. Our algorithm is applicable in online settings as it operates on small portions of the sequence of observations at a time, and it is well-suited to the high-dimensional setting both in terms of computational scalability and of statistical efficiency. The main result of this paper shows that our method performs change-point estimation reliably as long as the product of the smallest-sized change (the Euclidean-norm-squared of the difference between signals at a change-point) and the smallest distance between change-points (number of time instances) is larger than a Gaussian width parameter that characterizes the low-dimensional complexity of the underlying signal sequence. A full version of this paper is available online [1]
High-dimensional change-point estimation: Combining filtering with convex optimization
We consider change-point estimation in a sequence of high-dimensional signals given noisy observations. Classical approaches to this problem such as the filtered derivative method are useful for sequences of scalar-valued signals, but they have undesirable scaling behavior in the high-dimensional setting. However, many high-dimensional signals encountered in practice frequently possess latent low-dimensional structure. Motivated by this observation, we propose a technique for high-dimensional change-point estimation that combines the filtered derivative approach from previous work with convex optimization methods based on atomic norm regularization, which are useful for exploiting structure in high-dimensional data. Our algorithm is applicable in online settings as it operates on small portions of the sequence of observations at a time, and it is well-suited to the high-dimensional setting both in terms of computational scalability and of statistical efficiency. The main result of this paper shows that our method performs change-point estimation reliably as long as the product of the smallest-sized change (the Euclidean-norm-squared of the difference between signals at a change-point) and the smallest distance between change-points (number of time instances) is larger than a Gaussian width parameter that characterizes the low-dimensional complexity of the underlying signal sequence
Statistical Estimation of Correlated Genome Associations to a Quantitative Trait Network
Many complex disease syndromes, such as asthma, consist of a large number of highly related, rather than independent, clinical or molecular phenotypes. This raises a new technical challenge in identifying genetic variations associated simultaneously with correlated traits. In this study, we propose a new statistical framework called graph-guided fused lasso (GFlasso) to directly and effectively incorporate the correlation structure of multiple quantitative traits such as clinical metrics and gene expressions in association analysis. Our approach represents correlation information explicitly among the quantitative traits as a quantitative trait network (QTN) and then leverages this network to encode structured regularization functions in a multivariate regression model over the genotypes and traits. The result is that the genetic markers that jointly influence subgroups of highly correlated traits can be detected jointly with high sensitivity and specificity. While most of the traditional methods examined each phenotype independently and combined the results afterwards, our approach analyzes all of the traits jointly in a single statistical framework. This allows our method to borrow information across correlated phenotypes to discover the genetic markers that perturb a subset of the correlated traits synergistically. Using simulated datasets based on the HapMap consortium and an asthma dataset, we compared the performance of our method with other methods based on single-marker analysis and regression-based methods that do not use any of the relational information in the traits. We found that our method showed an increased power in detecting causal variants affecting correlated traits. Our results showed that, when correlation patterns among traits in a QTN are considered explicitly and directly during a structured multivariate genome association analysis using our proposed methods, the power of detecting true causal SNPs with possibly pleiotropic effects increased significantly without compromising performance on non-pleiotropic SNPs
Divide and Conquer Kernel Ridge Regression: A Distributed Algorithm with Minimax Optimal Rates
We establish optimal convergence rates for a decomposition-based scalable
approach to kernel ridge regression. The method is simple to describe: it
randomly partitions a dataset of size N into m subsets of equal size, computes
an independent kernel ridge regression estimator for each subset, then averages
the local solutions into a global predictor. This partitioning leads to a
substantial reduction in computation time versus the standard approach of
performing kernel ridge regression on all N samples. Our two main theorems
establish that despite the computational speed-up, statistical optimality is
retained: as long as m is not too large, the partition-based estimator achieves
the statistical minimax rate over all estimators using the set of N samples. As
concrete examples, our theory guarantees that the number of processors m may
grow nearly linearly for finite-rank kernels and Gaussian kernels and
polynomially in N for Sobolev spaces, which in turn allows for substantial
reductions in computational cost. We conclude with experiments on both
simulated data and a music-prediction task that complement our theoretical
results, exhibiting the computational and statistical benefits of our approach
Time–Data Tradeoffs by Aggressive Smoothing
This paper proposes a tradeoff between sample complexity and computation time that applies to statistical estimators based on convex optimization. As the amount of data increases, we can smooth optimization problems more and more aggressively to achieve accurate estimates more quickly. This work provides theoretical and experimental evidence of this tradeoff for a class of regularized linear inverse problems
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