Submodularity and Randomized rounding techniques for Optimal Experimental Design

Bouhtou, Mustapha; Gaubert, Stéphane; Sagnol, Guillaume

doi:10.1016/j.endm.2010.05.086

Cited by 22 publications

(20 citation statements)

References 11 publications

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“…It is not also the concave closure of F (see definition in §5.1); therefore maximizing g(w) with respect to w ∈ [0, 1] p is not equivalent to maximizing F (A) with respect to A ⊆ V . However, it readily leads to a convex relaxation of the problem of maximizing G, which is common in experimental design (see, e.g., [180,29]). …”

Section: Entropiesmentioning

confidence: 99%

Learning with Submodular Functions: A Convex Optimization Perspective

Bach

2013

FNT in Machine Learning

214

226

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Section: Entropiesmentioning

confidence: 99%

Learning with Submodular Functions: A Convex Optimization Perspective

Bach

2013

FNT in Machine Learning

214

226

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“…Identical with these classical pivotingbased matrix decomposition algorithms, the time and space computational complexity of the main algorithm in Algorithm 3.1 are thus O L 3 and O n 2 , respectively, where n is the total number of candidate points and L is the desired number of parameters. Note that these complexities are both polynomial and comparable with those in the computer science literature [15,58,86,1,2]. This numerical linear algebraic perspective motivates us to investigate variants of Algorithm 3.1 based on other numerical linear algebraic algorithms with pivoting in future work.…”

Section: Numericalmentioning

confidence: 77%

“…For computer scientists, the interesting problem is to find polynomial algorithms that efficiently find (1 + O ( ))approximations of the optimal solution to the exact problem, where > 0 is expected to be as small as possible but depends on the size of the problem and the pre-fixed budget for the number of design points; oftentimes these approximation results also require certain constraints on the relative sizes of the dimension of the ambient space, the number of design points, and the total number of candidate points. Different from those approaches, our theoretical contribution assumes no relations between these quantities, and the convergence rate is with respect to the increasing number of landmark points (as opposed to a pre-fixed budget); nevertheless, similar to results obtained in [15,7,58,86,1,2], our proposed algorithm has polynomial complexity and is thus computationally tractable; see Subsection 3.2 for more details. We refer interested readers to [62] for more exhaustive discussions of the optimality criteria used in experimental design.…”

Section: Our Contribution Andmentioning

confidence: 88%

“…We point out that, though experimental design is a classical problem in the statistical literature [34,20,62], it is only very recently that interests in computationally efficient experimental design algorithms began to arise in the computer science community [15,7,58,86,1,2]. Most experimental design algorithms, based on various types of optimality criteria including but not limited to A(verage)-, D(eterminant)-, E(igen)-, V(ariance)-, G-optimality and Bayesian alphabetical optimality, are NP-hard computational in their exact form [23,19], with the only exception of T(race)-optimality which is trivial to solve.…”

Section: Our Contribution Andmentioning

confidence: 99%

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Gaussian Process Landmarking on Manifolds

Gao¹,

Kovalsky²,

Daubechies³

2019

SIAM Journal on Mathematics of Data Science

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As a means of improving analysis of biological shapes, we propose an algorithm for sampling a Riemannian manifold by sequentially selecting points with maximum uncertainty under a Gaussian process model. This greedy strategy is known to be near-optimal in the experimental design literature, and appears to outperform the use of user-placed landmarks in representing the geometry of biological objects in our application. In the noiseless regime, we establish an upper bound for the mean squared prediction error (MSPE) in terms of the number of samples and geometric quantities of the manifold, demonstrating that the MSPE for our proposed sequential design decays at a rate comparable to the oracle rate achievable by any sequential or non-sequential optimal design; to our knowledge this is the first result of this type for sequential experimental design. The key is to link the greedy algorithm to reduced basis methods in the context of model reduction for partial differential equations. We expect this approach will find additional applications in other fields of research.

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“…Bouhou et al [9] give a n k 1 m -approximation algorithm. Wang et al [32] building on [6] give a (1 + )-approximation if k ≥ m 2 .…”

Section: Our Results and Contributionsmentioning

confidence: 99%

Approximate Positive Correlated Distributions and Approximation Algorithms for D-optimal Design

Singh

Xie

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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Experimental design is a classical area in statistics [21] and has also found new applications in machine learning [2]. In the combinatorial experimental design problem, the aim is to estimate an unknown m-dimensional vector x from linear measurements where a Gaussian noise is introduced in each measurement. The goal is to pick k out of the given n experiments so as to make the most accurate estimate of the unknown parameter x. Given a set S of chosen experiments, the most likelihood estimate x can be obtained by a least squares computation. One of the robust measures of error estimation is the D-optimality criterion [27] which aims to minimize the generalized variance of the estimator. This corresponds to minimizing the volume of the standard confidence ellipsoid for the estimation error x − x . The problem gives rise to two natural variants depending on whether repetitions of experiments is allowed or not. The latter variant, while being more general, has also found applications in geographical location of sensors [19].We show a close connection between approximation algorithms for the D-optimal design problem and constructions of approximately m-wise positively correlated distributions.This connection allows us to obtain a 1 e -approximation for the D-optimal design problem with and without repetitions giving the first constant factor approximation for the problem. We then consider the case when the number of experiments chosen is much larger than the dimension m and show one can obtain (1 − )-approximation if k ≥ 2m when repetitions are allowed andwhen no repetitions are allowed improving on previous work.

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Submodularity and Randomized rounding techniques for Optimal Experimental Design

Cited by 22 publications

References 11 publications

Learning with Submodular Functions: A Convex Optimization Perspective

Learning with Submodular Functions: A Convex Optimization Perspective

Gaussian Process Landmarking on Manifolds

Approximate Positive Correlated Distributions and Approximation Algorithms for D-optimal Design

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