Quadratic Assignment Problem

Pitsoulis, Leonidas; Pardalos, Pãnos M.

doi:10.1007/0-306-48332-7_405

Cited by 3 publications

(1 citation statement)

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“…The worst-case risk behavior of the LSE over the set M perm (L n,d ) ∩ B ∞ (1) was already discussed in Corollary 1(a): It attains the minimax lower bound (5) up to a poly-logarithmic factor. However, computing such an estimator is NP-hard in the worst-case even when d = 2, since the notoriously difficult max-clique instance can be straightforwardly reduced to the corresponding quadratic assignment optimization problem (see, e.g., Pitsoulis and Pardalos (2001) for reductions of this type).…”

Section: Adaptation Properties Of Existing Estimatorsmentioning

confidence: 99%

Isotonic regression with unknown permutations: Statistics, computation, and adaptation

Pananjady¹,

Samworth²

2020

Preprint

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Motivated by models for multiway comparison data, we consider the problem of estimating a coordinate-wise isotonic function on the domain [0, 1] d from noisy observations collected on a uniform lattice, but where the design points have been permuted along each dimension. While the univariate and bivariate versions of this problem have received significant attention, our focus is on the multivariate case d ≥ 3. We study both the minimax risk of estimation (in empirical L 2 loss) and the fundamental limits of adaptation (quantified by the adaptivity index) to a family of piecewise constant functions. We provide a computationally efficient Mirsky partition estimator that is minimax optimal while also achieving the smallest adaptivity index possible for polynomial time procedures. Thus, from a worst-case perspective and in sharp contrast to the bivariate case, the latent permutations in the model do not introduce significant computational difficulties over and above vanilla isotonic regression. On the other hand, the fundamental limits of adaptation are significantly different with and without unknown permutations: Assuming a hardness conjecture from average-case complexity theory, a statistical-computational gap manifests in the former case. In a complementary direction, we show that natural modifications of existing estimators fail to satisfy at least one of the desiderata of optimal worst-case statistical performance, computational efficiency, and fast adaptation. Along the way to showing our results, we improve adaptation results in the special case d = 2 and establish some properties of estimators for vanilla isotonic regression, both of which may be of independent interest.

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Section: Adaptation Properties Of Existing Estimatorsmentioning

confidence: 99%

Isotonic regression with unknown permutations: Statistics, computation, and adaptation

Pananjady¹,

Samworth²

2020

Preprint

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Quadratic Optimization Models and Convex Extensions on Permutation Matrix Set

Pichugina

Yakovlev

2019

Advances in Intelligent Systems and Computing IV

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Isotonic regression with unknown permutations: Statistics, computation and adaptation

Pananjady¹,

Samworth²

2022

Ann. Statist.

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It suffices to prove the upper bound, since the minimax lower bound was already shown by Han et al. (2019) for isotonic regression without unknown permutations. We also focus on the case d ≥ 3 since the result is already available for d = 2 (Shah et al., 2017;Mao et al., 2020). Our proof proceeds in two parts. First, we show that the bounded least squares estimator over isotonic tensors (without unknown permutations) enjoys the claimed risk bound. We then use our proof of this result to prove the upper bound (7). The proof of the first result is also useful in establishing part (b) of Proposition 2.Bounded LSE over isotonic tensors. For a tensor A ∈ R d,n and x 1 , . . . , x d−2 ∈ [n 1 ], let A x1,...,xd−2 denote the matrix formed by fixing the first d − 2 dimensions (variables) of A to x 1 , . . . , x d−2 , i.e., entry (i, j) of this matrix is given by A(x 1 , . . . , x d−2 , i, j). Recall that M(L 2,n1,n1 ) denotes the set of all bivariate isotonic n 1 × n 1 matrices.For convenience, let M(L d,n | r) and M(L 2,n1,n1 | r) denote the intersection of the respective sets with the ∞ ball of radius r. Letting A − B denote the Minkowski difference between the sets A and B, defineWe observe that there is a bijection between M full (r) and theNote that through this notation, we are indexing the components of this Cartesian product by the elements of L d−2,n1,...,n1 , so that a generic element A of this set has (x 1 , . . . ,Note that by construction, we have ensured, for each r ≥ 0, the inclusions ∆∈M full (1)−M full (1) ∆ 2≤ ∆ 2 , ∆ . For convenience, define for each t ≥ 0 the random variable ξ(t) : = sup ∆∈M full (1)−M full (1) ∆ 2≤t, ∆ .

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Quadratic Assignment Problem

Cited by 3 publications

References 95 publications

Isotonic regression with unknown permutations: Statistics, computation, and adaptation

Isotonic regression with unknown permutations: Statistics, computation, and adaptation

Quadratic Optimization Models and Convex Extensions on Permutation Matrix Set

Isotonic regression with unknown permutations: Statistics, computation and adaptation

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