Arbitrary Partition Research Articles

Improved PTAS for the constrained k-means problem

The k-means problem has been paid lots of attention in many fields, and each cluster of the k-means problem always satisfies locality property. In this paper, we study the constrained k-means problem, where the clusters do not satisfy locality property and can be an arbitrary partition of the set of points. Ding and Xu presented a unified framework with running time $$O(2^{poly (k/\epsilon )} (\log n)^{k+1} nd)$$ by applying uniform sampling and simplex lemma techniques such that a collection of size $$O(2^{poly (k/\epsilon )} (\log n)^{k+1})$$ of candidate sets containing approximate centers is obtained. Then, the collection is enumerated to get the one that can induce a $$(1+\epsilon )$$ -approximation solution for the constrained k-means problem. By applying $$D^2$$ -sampling technique, Bhattacharya, Jaiswal, and Kumar presented an algorithm with running time $$O(2^{{\tilde{O}}(k/\epsilon )}nd)$$ , which is bounded by $$O(2^k( \frac{2123ek}{\epsilon ^3})^{64k/\epsilon }knd)$$ . The algorithm outputs a collection of size $$O(2^k( \frac{2123ek}{\epsilon ^3})^{64k/\epsilon })$$ of candidate sets containing approximate centers. In this paper, we present an algorithm with running time $$O((\frac{1891ek}{\epsilon ^2})^{8k/\epsilon }nd)$$ such that a collection of size $$O((\frac{1891ek}{\epsilon ^2})^{8k/\epsilon }n)$$ of candidate sets containing approximate centers can be obtained.

An iterative algorithm to find maximum spanning sets and minimum linearly independent sets which partition a finite generator

ABSTRACTMotivated by existence problems of the maximum number of spanning sets (MS) and the minimum number of Linearly independent sets (ML) which partition a generator in a finite dimensional vector space, we introduce a real time iterative algorithm with polynomial complexity. The algorithm starts with an arbitrary partition and improves it to obtain the final partition. In addition this algorithm gives the conclusion of the Rado–Horn theorem, and facilitates using of this theorem.

Nilpotent matrices having a given Jordan type as maximum commuting nilpotent orbit

The Jordan type of a nilpotent matrix is the partition giving the sizes of its Jordan blocks. We study pairs of partitions (P,Q), where Q=Q(P) is the Jordan type of a generic nilpotent matrix A commuting with a nilpotent matrix B of Jordan type P. T. Košir and P. Oblak have shown that Q has parts that differ pairwise by at least two. Such partitions, which are also known as “super distinct” or “Rogers–Ramanujan”, are exactly those that are stable or “self-large” in the sense that Q(Q)=Q.In 2012 P. Oblak formulated a conjecture concerning the cardinality of Q−1(Q) when Q has two parts, and proved some special cases. R. Zhao refined this to posit that the partitions in Q−1(Q) for Q=(u,u−r) with u>r>1 could be arranged in an (r−1)×(u−r) table T(Q) where the entry in the k-th row and ℓ-th column has k+ℓ parts. We prove this Table Theorem, and then generalize the statement to propose a Box Conjecture for the set of partitions Q−1(Q) for an arbitrary partition Q whose parts differ pairwise by at least two.

Faster Algorithms for the Constrained k-means Problem

The classical center based clustering problems such as k-means/median/center assume that the optimal clusters satisfy the locality property that the points in the same cluster are close to each other. A number of clustering problems arise in machine learning where the optimal clusters do not follow such a locality property. For instance, consider the r -gather clustering problem where there is an additional constraint that each of the clusters should have at least r points or the capacitated clustering problem where there is an upper bound on the cluster sizes. Consider a variant of the k-means problem that may be regarded as a general version of such problems. Here, the optimal clusters O 1, ..., O k are an arbitrary partition of the dataset and the goal is to output k-centers c 1, ..., c k such that the objective function ${\sum }_{i = 1}^{k} {\sum }_{x \in O_{i}} ||x - c_{i}||^{2}$ is minimized. It is not difficult to argue that any algorithm (without knowing the optimal clusters) that outputs a single set of k centers, will not behave well as far as optimizing the above objective function is concerned. However, this does not rule out the existence of algorithms that output a list of such k centers such that at least one of these k centers behaves well. Given an error parameter e > 0, let l denote the size of the smallest list of k-centers such that at least one of the k-centers gives a (1 + e) approximation w.r.t. the objective function above. In this paper, we show an upper bound on l by giving a randomized algorithm that outputs a list of $2^{\tilde {O}(k/\varepsilon )}$ k-centers. We also give a closely matching lower bound of $2^{\tilde {\Omega }(k/\sqrt {\varepsilon })}$ . Moreover, our algorithm runs in time $O \left (n d \cdot 2^{\tilde {O}(k/\varepsilon )} \right )$ . This is a significant improvement over the previous result of Ding and Xu (2015) who gave an algorithm with running time O(n d ⋅ (log n) k ⋅ 2 p o l y(k/e)) and output a list of size O((log n) k ⋅ 2 p o l y(k/e)). Our techniques generalize for the k-median problem and for many other settings where non-Euclidean distance measures are involved.

Predictive SIRT dosimetry based on a territorial model

BackgroundIn the planning of selective internal radiation therapy (SIRT) for liver cancer treatment, one major aspect is to determine the prescribed activity and to estimate the resulting absorbed dose inside normal liver and tumor tissue. An optimized partition model for SIRT dosimetry based on arterial liver territories is proposed. This model is dedicated to characterize the variability of dose within the whole liver. For an arbitrary partition, the generalized absorbed dose is derived from the classical partition model. This enables to consider normal liver partitions for each arterial perfusion supply area and one partition for each tumor for activity and dose calculation. The proposed method excludes a margin of 11 mm emitting range around tumor volumes from normal liver to investigate the impact on activity calculation. Activity and dose calculation was performed for five patients using the body-surface-area (BSA) method, the classical and territorial partition model.ResultsThe territorial model reaches smaller normal liver doses and significant higher tumor doses compared to the classical partition model. The exclusion of a small region around tumors has a significant impact on mean liver dose. Determined tumor activities for the proposed method are higher in all patients when limited by normal liver dose. Activity calculation based on BSA achieves in all cases the lowest amount.ConclusionsThe territorial model provides a more local and patient-individual dose distribution in normal liver taking into account arterial supply areas. This proposed arterial liver territory-based partition model may be used for SPECT-independent activity calculation and dose prediction under the condition of an artery-based simulation for particle distribution.

Bernstein-Bézier techniques for divergence of polynomial spline vector fields in ℝ n

Bernstein-Bezier techniques for analyzing polynomial spline fields in n variables and their divergence are developed. Dimension and a minimal determining set for continuous piecewise divergence-free spline fields on the Alfeld split of a simplex in ℝ n are obtained using the new techniques, as well as the dimension formula for continuous piecewise divergence-free splines on the Alfeld refinement of an arbitrary simplicial partition in ℝ n .

Multi-boundary entanglement in Chern-Simons theory and link invariants

We consider Chern-Simons theory for gauge group G at level k on 3-manifolds Mn with boundary consisting of n topologically linked tori. The Euclidean path integral on Mn defines a quantum state on the boundary, in the n-fold tensor product of the torus Hilbert space. We focus on the case where Mn is the link-complement of some n-component link inside the three-sphere S3. The entanglement entropies of the resulting states define framing-independent link invariants which are sensitive to the topology of the chosen link. For the Abelian theory at level k (G = U(1)k) we give a general formula for the entanglement entropy associated to an arbitrary (m|n − m) partition of a generic n-component link into sub-links. The formula involves the number of solutions to certain Diophantine equations with coefficients related to the Gauss linking numbers (mod k) between the two sublinks. This formula connects simple concepts in quantum information theory, knot theory, and number theory, and shows that entanglement entropy between sublinks vanishes if and only if they have zero Gauss linking (mod k). For G = SU(2)k, we study various two and three component links. We show that the 2-component Hopf link is maximally entangled, and hence analogous to a Bell pair, and that the Whitehead link, which has zero Gauss linking, nevertheless has entanglement entropy. Finally, we show that the Borromean rings have a “W-like” entanglement structure (i.e., tracing out one torus does not lead to a separable state), and give examples of other 3-component links which have “GHZ-like” entanglement (i.e., tracing out one torus does lead to a separable state).

Generalized Kapchinskij-Vladimirskij Distribution and Beam Matrix for Phase-Space Manipulations of High-Intensity Beams.

In an uncoupled linear lattice system, the Kapchinskij-Vladimirskij (KV) distribution formulated on the basis of the single-particle Courant-Snyder invariants has served as a fundamental theoretical basis for the analyses of the equilibrium, stability, and transport properties of high-intensity beams for the past several decades. Recent applications of high-intensity beams, however, require beam phase-space manipulations by intentionally introducing strong coupling. In this Letter, we report the full generalization of the KV model by including all of the linear (both external and space-charge) coupling forces, beam energy variations, and arbitrary emittance partition, which all form essential elements for phase-space manipulations. The new generalized KV model yields spatially uniform density profiles and corresponding linear self-field forces as desired. The corresponding matrix envelope equations and beam matrix for the generalized KV model provide important new theoretical tools for the detailed design and analysis of high-intensity beam manipulations, for which previous theoretical models are not easily applicable.

The granular partition lattice of an information table

In this paper we study the lattice of all indiscernibility partitions induced from attribute subsets of a knowledge representation system (information table in the finite case). This lattice, that we here call granular partition lattice, is a very well studied order structure in granular computing and data base theory and it provides a complete hierarchical classification of the knowledge obtained from all possible choices of attribute subsets. We show that it has a lattice structure also in the infinite case and we provide several isomorphic characterizations for this lattice. We discuss the potentiality of this order structure from both a micro-granular and a macro-granular perspective. Furthermore, the sub-poset of all the indiscernibility closures needed to determine when an arbitrary partition is an indiscernibility one is studied. Finally, we show the monotonic behaviour of the granular partition lattice with respect to entropy of partitions and attribute dependency in decision tables.

Approximation of separable numerical range using simulated annealing

In this work we provide a method for approximating the separable numerical range of a matrix. We also recall the connection between restricted numerical range and entanglement of a quantum state. We show the possibility to establish state separability using computed restricted numerical range. In particular we present a method to obtain separability criteria for arbitrary system partition with use of the separable numerical range.

The rank of the semigroup of transformations stabilising a partition of a finite set

AbstractLet $\mathcal{P}$ be a partition of a finite set X. We say that a transformation f : X → X preserves (or stabilises) the partition $\mathcal{P}$ if for all P ∈ $\mathcal{P}$ there exists Q ∈ $\mathcal{P}$ such that Pf ⊆ Q. Let T(X, $\mathcal{P}$) denote the semigroup of all full transformations of X that preserve the partition $\mathcal{P}$.In 2005 Pei Huisheng found an upper bound for the minimum size of the generating sets of T(X, $\mathcal{P}$), when $\mathcal{P}$ is a partition in which all of its parts have the same size. In addition, Pei Huisheng conjectured that his bound was exact. In 2009 the first and last authors used representation theory to solve Pei Huisheng's conjecture.The aim of this paper is to solve the more complex problem of finding the minimum size of the generating sets of T(X, $\mathcal{P}$), when $\mathcal{P}$ is an arbitrary partition. Again we use representation theory to find the minimum number of elements needed to generate the wreath product of finitely many symmetric groups, and then use this result to solve the problem.The paper ends with a number of problems for experts in group and semigroup theories.

Perturbative string thermodynamics near black hole horizons

We provide further computations and ideas to the problem of near-Hagedorn string thermodynamics near (uncharged) black hole horizons, building upon our earlier work JHEP 1403 (2014) 086. The relevance of long strings to one-loop black hole thermodynamics is emphasized. We then provide an argument in favor of the absence of $\alpha'$-corrections for the (quadratic) heterotic thermal scalar action in Rindler space. We also compute the large $k$ limit of the cigar orbifold partition functions (for both bosonic and type II superstrings) which allows a better comparison between the flat cones and the cigar cones. A discussion is made on the general McClain-Roth-O'Brien-Tan theorem and on the fact that different torus embeddings lead to different aspects of string thermodynamics. The black hole/string correspondence principle for the 2d black hole is discussed in terms of the thermal scalar. Finally, we present an argument to deal with arbitrary higher genus partition functions, suggesting the breakdown of string perturbation theory (in $g_s$) to compute thermodynamical quantities in black hole spacetimes.

Beamforming properties and design of the phased arrays in terms of irregular subarrays

Large phased arrays are often organised by subarrays given the current state of technology. The irregular subarray architecture, which can effectively mitigate the quantisation lobes, has received a renewed interest in recent years. Recent researches present some innovative particular irregular subarrays to provide practical means to eliminate the quantisation lobes using several (even only one) kinds of subarray. However, there is no single, neat theory of the irregularly partitioned array antennas in the literature. This study presents some derivations of the beamforming properties for arrays with arbitrary subarray partition to fill this gap. Derivations have led us to introduce a novel approach to assess the radiation properties which is useful for the systematic design of phased arrays. Numerical experiments are carried out to validate the authors’ derivations. Additionally, an elaborate example illustrates a design discipline for a limited-field-of-view array with polyomino-shaped subarrays.

Parallel algorithms for CFD–DEM modeling of dense particulate flows

Most of the CFD codes are parallelized under distributed-memory parallel computing environments. For this reason, many attempts have been made to develop parallel DPM/DEM algorithms for dense particulate flows under the same parallel architecture. For such a development, it is very difficult to achieve efficient load balancing of processors due to the heterogeneous particle spatial distribution that often characterizes dense particulate systems. In this work, parallel CFD–DEM algorithms have been developed under the distributed memory environment with a fluid flow solver based on finite volume method and arbitrary 3D unstructured meshes. An implicit two-phase coupling scheme was proposed to enhance numerical stability for complex dense particulate flow problems. Parallelization-generated numerical difficulties such as void fraction calculation, two-phase momentum exchange, and contact force calculations for particles at irregular and arbitrary partition boundaries were efficiently addressed. The load-balancing difficulty due to heterogeneous particle distribution was partly overcome by the introduction of multi-threading. An efficient algorithm is proposed to handle data-exclusive access of the shared-memory by multi-threads on a compute node. The developed parallel DPM model has been successfully used to simulate many important applications such as bubbling fluidized bed, granular Rayleigh–Taylor instability, and particle swarm dynamics.

Projection-Based Model Reduction of Multi-Agent Systems Using Graph Partitions

In this paper, we establish a projection-based model reduction method for multiagent systems defined on a graph. Reduced order models are obtained by clustering the vertices (agents) of the underlying communication graph by means of suitable graph partitions. In the reduction process, the spatial structure of the network is preserved and the reduced order models can again be realized as multiagent systems defined on a graph. The agents are assumed to have single-integrator dynamics and the communication graph of the original system is weighted and undirected. The proposed model reduction technique reduces the number of vertices of the graph (which is equal to the dynamic order of the original multi-agent system) and yields a reduced order multiagent system defined on a new graph with a reduced number of vertices. This new graph is a weighted symmetric directed graph. It is shown that if the original multiagent system reaches consensus, then so does the reduced order model. For the case that the clusters are chosen using an almost equitable partition (AEP) of the graph, we obtain an explicit formula for the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -norm of the error system obtained by comparing the input-output behaviors of the original model and the reduced order model. We also prove that the error obtained by taking an arbitrary partition of the graph is bounded from below by the error obtained using the largest AEP finer than the given partition. The proposed results are illustrated by means of a running example.

Representing Partitions on Trees

In evolutionary biology, biologists often face the problem of constructing a phylogenetic tree on a set $X$ of species from a multiset $\Pi$ of partitions corresponding to various attributes of these species. One approach that is used to solve this problem is to try instead to associate a tree (or even a network) to the multiset $\Sigma_{\Pi}$ consisting of all those bipartitions $\{A,X-A\}$ with $A$ a part of some partition in $\Pi$. The rationale behind this approach is that a phylogenetic tree with leaf set $X$ can be uniquely represented by the set of bipartitions of $X$ induced by its edges. Motivated by these considerations, given a multiset $\Sigma$ of bipartitions corresponding to a phylogenetic tree on $X$, in this paper we introduce and study the set $\mathbb{P}(\Sigma)$ consisting of those multisets of partitions $\Pi$ of $X$ with $\Sigma_{\Pi}=\Sigma$. More specifically, we characterize when $\mathbb{P}(\Sigma)$ is nonempty and also identify some partitions in $\mathbb{P}(\Sigma)$ that are of maximum and minimum size. We also show that it is NP-complete to decide when $\mathbb{P}(\Sigma)$ is nonempty in the case when $\Sigma$ is an arbitrary multiset of bipartitions of $X$. Ultimately, we hope that by gaining a better understanding of the mapping that takes an arbitrary partition system $\Pi$ to the multiset $\Sigma_{\Pi}$, we will obtain new insights into the use of median networks and, more generally, split networks, to visualize sets of partitions.

An Optimal Decision Population Code that Accounts for Correlated Variability Unambiguously Predicts a Subject’s Choice

Decisions emerge from the concerted activity of neuronal populations distributed across brain circuits. However, the analytical tools best suited to decode decision signals from neuronal populations remain unknown. Here we show that knowledge of correlated variability between pairs of cortical neurons allows perfect decoding of decisions from population firing rates. We recorded pairs of neurons from secondary somatosensory (S2) and premotor (PM) cortices while monkeys reported the presence or absence of a tactile stimulus. We found that while populations of S2 and sensory-like PM neurons are only partially correlated with behavior, those PM neurons active during a delay period preceding the motor report predict unequivocally the animal's decision report. Thus, a population rate code that optimally reveals a subject's perceptual decisions can be implemented just by knowing the correlations of PM neurons representing decision variables.

On the derivative of the $\alpha$-Farey-Minkowski function

In this paper we study the family of $\alpha$-Farey-Minkowski functions $\theta_\alpha$, for an arbitrary countable partition $\alpha$ of the unit interval with atoms which accumulate only at the origin, which are the conjugating homeomorphisms between each of the $\alpha$-Farey systems and the tent map. We first show that each function $\theta_\alpha$ is singular with respect to the Lebesgue measure and then demonstrate that the unit interval can be written as the disjoint union of the following three sets: $\Theta_0 : = \{x \in [0,1] : \theta_\alpha'(x)=0\}, \Theta_{\infty} : = \{ x \in [0,1] : \theta_\alpha'(x)=\infty \} $ and $\Theta_\sim : = \{ x \in [0,1] : \theta_\alpha'(x) does not exist \} $. The main result is that \[ \dim_{\mathrm{H}}(\Theta_\infty)=\dim_{\mathrm{H}}(\Theta_\sim)=\sigma_\alpha(\log2)<\dim_{\mathrm{H}}(\Theta_0)=1, \] where $\sigma_\alpha(\log2)$ denotes the Hausdorff dimension of the level set $\{x\in [0,1]:\Lambda(F_\alpha, x)=\log2\}$ and $\Lambda(F_\alpha, x)$ is the Lyapunov exponent of the map $F_\alpha$ at the point $x$. The proof of the theorem employs the multifractal formalism for $\alpha$-Farey systems.

Quantifying single nucleotide variant detection sensitivity in exome sequencing

BackgroundThe targeted capture and sequencing of genomic regions has rapidly demonstrated its utility in genetic studies. Inherent in this technology is considerable heterogeneity of target coverage and this is expected to systematically impact our sensitivity to detect genuine polymorphisms. To fully interpret the polymorphisms identified in a genetic study it is often essential to both detect polymorphisms and to understand where and with what probability real polymorphisms may have been missed.ResultsUsing down-sampling of 30 deeply sequenced exomes and a set of gold-standard single nucleotide variant (SNV) genotype calls for each sample, we developed an empirical model relating the read depth at a polymorphic site to the probability of calling the correct genotype at that site. We find that measured sensitivity in SNV detection is substantially worse than that predicted from the naive expectation of sampling from a binomial. This calibrated model allows us to produce single nucleotide resolution SNV sensitivity estimates which can be merged to give summary sensitivity measures for any arbitrary partition of the target sequences (nucleotide, exon, gene, pathway, exome). These metrics are directly comparable between platforms and can be combined between samples to give “power estimates” for an entire study. We estimate a local read depth of 13X is required to detect the alleles and genotype of a heterozygous SNV 95% of the time, but only 3X for a homozygous SNV. At a mean on-target read depth of 20X, commonly used for rare disease exome sequencing studies, we predict 5–15% of heterozygous and 1–4% of homozygous SNVs in the targeted regions will be missed.ConclusionsNon-reference alleles in the heterozygote state have a high chance of being missed when commonly applied read coverage thresholds are used despite the widely held assumption that there is good polymorphism detection at these coverage levels. Such alleles are likely to be of functional importance in population based studies of rare diseases, somatic mutations in cancer and explaining the “missing heritability” of quantitative traits.

A characterization of the optimal sets for self-similar measures with respect to the geometric mean error

Let μ be a self-similar measure on ℝq associated with a set of contractive similitudes \({\{S_{i}\}}_{i=1}^{N}\) and a probability vector \({(p_{i})}_{i=1}^{N}\). Assume that \({\{S_{i}\}}_{i=1}^{N}\) satisfies the strong separation condition. In terms of the n-optimal sets for a finite number n∈ℕ, we give a characterization for the n-optimal sets for all n∈ℕ in the quantization for μ with respect to the geometric mean error. As an application, we determine the convergence order for the logarithmic difference of the asymptotic geometric mean error. This characterization also allows us to show that the μ-measure of every element of an arbitrary Voronoi partition with respect to an n-optimal set is uniformly comparable with n−1, provided that μ vanishes on every hyperplane in ℝq.