Probability Theory of Classical Euclidean Optimization Problems

Yukich, J. E.

doi:10.1007/bfb0093472

Cited by 197 publications

(283 citation statements)

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“…Essentially all of the Euclidean functionals discussed in Steele [60] or Yukich [65] may be represented as H ξ (X ) for an appropriate choice of ξ(x; X ).…”

Section: Sums Of Local Contributionsmentioning

confidence: 99%

The Objective Method: Probabilistic Combinatorial Optimization and Local Weak Convergence

Aldous¹,

Steele²

2004

Encyclopaedia of Mathematical Sciences

308

684

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“…Essentially all of the Euclidean functionals discussed in Steele [60] or Yukich [65] may be represented as H ξ (X ) for an appropriate choice of ξ(x; X ).…”

Section: Sums Of Local Contributionsmentioning

confidence: 99%

The Objective Method: Probabilistic Combinatorial Optimization and Local Weak Convergence

Aldous¹,

Steele²

2004

Encyclopaedia of Mathematical Sciences

308

684

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“…Many aspects of the large-sample asymptotic theory for such graphs, which are locally determined in a certain sense, are by now quite well understood. See for example [10,13,14,16,17,23,26].…”

Section: Introductionmentioning

confidence: 99%

Limit theory for the random on‐line nearest‐neighbor graph

Penrose

Wade

2007

Random Struct Algorithms

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In the on-line nearest-neighbour graph (ONG), each point after the first in a sequence of points in R d is joined by an edge to its nearest-neighbour amongst those points that precede it in the sequence. We study the large-sample asymptotic behaviour of the total power-weighted length of the ONG on uniform random points in (0, 1) d . In particular, for d = 1 and weight exponent α > 1/2, the limiting distribution of the centred total weight is characterized by a distributional fixed-point equation. As an ancillary result, we give exact expressions for the expectation and variance of the standard nearest-neighbour (directed) graph on uniform random points in the unit interval.

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“…Entropy estimation (similar to [3]) based on k-nearest neighbors [6,7]: asymptotically unbiased and strongly consistent [6]. Basic idea:…”

Section: Multi-dimensional Entropy Estimation By the K-nearest Neighbmentioning

confidence: 99%

Cross-Entropy Optimization for Independent Process Analysis

Szabó

Póczos

Lörincz

2006

Independent Component Analysis and Blind Signal Separation

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We treat the problem of searching for hidden multidimensional independent auto-regressive processes. First, we transform the problem to Independent Subspace Analysis (ISA) Here: the unknown mixing matrix A ∈ R D×D , the hidden components s m ∈ R d , andGoal of IPA: estimate s(t) and A (or W := A −1 : separation matrix) by using observations z(t) only. Specially:when ∀F m = 0 and d = 1. AssumptionsFor an AR process, the innovation is identical to the noise that drives the process ⇒ IPA models(t + 1) = Fs(t) + e(t), (6) z(t) = AFA −1 z(t − 1) + Ae(t − 1), (7) z(t) = Ae(t − 1) = As(t). Assuming that W ICA (z) is unique (up to permutation and sign of the components), then it is W ISA (z) (up to permutation and sign of the components). In other wordswhere P ∈ R D×D is a permutation matrix to be determined. (Proof in [5], e.g., for elliptically symmetric sources) ⇒ IPA estimation steps: 1. observe z(t) and estimate the AR model, 2. whiten the innovation of the AR process and perform ICA on it, 3. solve the combinatorial problem: search for the permutation of the ICA sources that minimizes the cost J. Thus IPA needs only two (more) steps: (i)Ĥ, and (ii) optimization of J in S D (permutations of length D). Assistants 3

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Probability Theory of Classical Euclidean Optimization Problems

Cited by 197 publications

References 0 publications

The Objective Method: Probabilistic Combinatorial Optimization and Local Weak Convergence

The Objective Method: Probabilistic Combinatorial Optimization and Local Weak Convergence

Limit theory for the random on‐line nearest‐neighbor graph

Cross-Entropy Optimization for Independent Process Analysis

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