Showing papers on "Disjoint sets published in 2021"

Dynamic Type Matching

Abstract: Problem definition: We consider an intermediary’s problem of dynamically matching demand and supply of heterogeneous types in a periodic-review fashion. Specifically, there are two disjoint sets of...

Fractal geometry of Airy_2 processes coupled via the Airy sheet

Abstract: In last passage percolation models lying in the Kardar–Parisi–Zhang universality class, maximizing paths that travel over distances of order $n$ accrue energy that fluctuates on scale $n^{1/3}$; and these paths deviate from the linear interpolation of their endpoints on scale $n^{2/3}$. These maximizing paths and their energies may be viewed via a coordinate system that respects these scalings. What emerges by doing so is a system indexed by $x,y\in \mathbb{R}$ and $s,t\in \mathbb{R}$ with $s

Flexible Body Partition-Based Adversarial Learning for Visible Infrared Person Re-Identification.

Abstract: Person re-identification (Re-ID) aims to retrieve images of the same person across disjoint camera views. Most Re-ID studies focus on pedestrian images captured by visible cameras, without considering the infrared images obtained in the dark scenarios. Person retrieval between visible and infrared modalities is of great significance to public security. Current methods usually train a model to extract global feature descriptors and obtain discriminative representations for visible infrared person Re-ID (VI-REID). Nevertheless, they ignore the detailed information of heterogeneous pedestrian images, which affects the performance of Re-ID. In this article, we propose a flexible body partition (FBP) model-based adversarial learning method (FBP-AL) for VI-REID. To learn more fine-grained information, FBP model is exploited to automatically distinguish part representations according to the feature maps of pedestrian images. Specially, we design a modality classifier and introduce adversarial learning which attempts to discriminate features between visible and infrared modality. Adaptive weighting-based representation learning and threefold triplet loss-based metric learning compete with modality classification to obtain more effective modality-sharable features, thus shrinking the cross-modality gap and enhancing the feature discriminability. Extensive experimental results on two cross-modality person Re-ID data sets, i.e., SYSU-MM01 and RegDB, exhibit the superiority of the proposed method compared with the state-of-the-art solutions.

A new principal component analysis by particle swarm optimization with an environmental application for data science

Abstract: In this paper, we propose a new method for disjoint principal component analysis based on an intelligent search. The method consists of a principal component analysis with constraints, allowing us to determine components that are linear combinations of disjoint subsets of the original variables. The effectiveness of the proposed method contributes to solve one of the crucial problems of multivariate analysis, that is, the interpretation of the vectorial subspaces in the reduction of the dimensionality. The method selects the variables that contribute the most to each of the principal components in a clear and direct way. Numerical results are provided to confirm the quality of the solutions attained by the proposed method. This method avoids a local optimum and obtains a high success rate when reaching the best solution, which occurs in all the cases of our simulation study. An illustration with environmental real data shows the good performance of the method and its potential applications.

Optimal Scale Combination Selection Integrating Three-Way Decision With Hasse Diagram.

Abstract: Multi-scale decision system (MDS) is an effective tool to describe hierarchical data in machine learning. Optimal scale combination (OSC) selection and attribute reduction are two key issues related to knowledge discovery in MDSs. However, searching for all OSCs may result in a combinatorial explosion, and the existing approaches typically incur excessive time consumption. In this study, searching for all OSCs is considered as an optimization problem with the scale space as the search space. Accordingly, a sequential three-way decision model of the scale space is established to reduce the search space by integrating three-way decision with the Hasse diagram. First, a novel scale combination is proposed to perform scale selection and attribute reduction simultaneously, and then an extended stepwise optimal scale selection (ESOSS) method is introduced to quickly search for a single local OSC on a subset of the scale space. Second, based on the obtained local OSCs, a sequential three-way decision model of the scale space is established to divide the search space into three pair-wise disjoint regions, namely the positive, negative, and boundary regions. The boundary region is regarded as a new search space, and it can be proved that a local OSC on the boundary region is also a global OSC. Therefore, all OSCs of a given MDS can be obtained by searching for the local OSCs on the boundary regions in a step-by-step manner. Finally, according to the properties of the Hasse diagram, a formula for calculating the maximal elements of a given boundary region is provided to alleviate space complexity. Accordingly, an efficient OSC selection algorithm is proposed to improve the efficiency of searching for all OSCs by reducing the search space. The experimental results demonstrate that the proposed method can significantly reduce computational time.

A flexible subhalo abundance matching model for galaxy clustering in redshift space

Abstract: We develop an extension of subhalo abundance matching (SHAM) capable of accurately reproducing the real and redshift-space clustering of galaxies in a state-of-the-art hydrodynamical simulation. Our method uses a low-resolution gravity-only simulation and it includes orphan and tidal disruption prescriptions for satellite galaxies, and a flexible amount of galaxy assembly bias. Furthermore, it includes recipes for star formation rate (SFR) based on the dark matter accretion rate. We test the accuracy of our model against catalogues of stellar-mass- and SFR-selected galaxies in the TNG300 hydrodynamic simulation. By fitting a small number of free parameters, our extended SHAM reproduces the projected correlation function and redshift-space multipoles for number densities $10^{-3} - 10^{-2}\, h^{3}{\rm Mpc}^{-3}$, at $z=1$ and $z=0$, and for scales $r \in [0.3 - 20] h^{-1}{\rm Mpc}$. Simultaneously, the SHAM results also retrieve the correct halo occupation distribution, the level of galaxy assembly bias, and higher-order statistics present in the TNG300 galaxy catalogues. As an application, we show that our model simultaneously fits the projected correlation function of the SDSS in 3 disjoint stellar mass bins, with an accuracy similar to that of TNG300 galaxies. This SHAM extension can be used to get accurate clustering prediction even when using low and moderate-resolution simulations.

Sidorenko’s conjecture for blow-ups

Abstract: Sidorenko's conjecture for blow-ups, Discrete Analysis 2021:2, 13 pp Let $G$ be a bipartite graph with finite vertex sets $X$ and $Y$ If $G$ has density $\alpha$, then the average degree of the vertices in $X$ is $\alpha|Y|$, so the mean-square degree is at least $\alpha^2|Y|^2$ This is easily seen to be equivalent to the statement that if two vertices $y_1,y_2$ are selected independently and uniformly at random from $Y$, then the average number of neighbours they have in common in $X$ is at least $\alpha^2|X|$ It follows that the mean-square number of common neighbours is at least $\alpha^4|X|^2$, and this translates into the statement that if $(x_1,x_2,y_1,y_2)$ is a random element of $X^2\times Y^2$, then the probability that all four pairs $x_iy_j$ are edges of $G$ is at least $\alpha^4$ This can be interpreted as saying that the 4-cycle density of a graph $G$ is always at least as large as it would be for a typical random graph of the same density The above argument can be straightforwardly generalized from 4-cycles to general complete bipartite graphs $K_{r,s}$ The famous Sidorenko conjecture (also asked by Erdős and Simonovits) states that the conclusion holds for _all_ bipartite graphs That is, if $H$ is a bipartite graph with vertex sets $U$ and $V$ of size $r$ and $s$, $G$ is as above, and $\phi:U\to X$ and $\psi:V\to Y$ are random functions, then the probability that $\phi(u)\psi(v)$ is an edge of $G$ for every $uv\in E(H)$ is at least $\alpha^{|E(H)|}$ One might think that this conjecture would either have a simple counterexample or would follow fairly easily from known inequalities, but this appears not to be the case A good way to get a feel for the difficulty is to try to prove it for paths of length 3 It is known to be true in this case, but a certain amount of ingenuity is required to prove it In fact, it is known to be true for quite a large class of bipartite graphs, with results typically taking the form that a bipartite graph satisfies the conjecture if it can be built up in a certain way from graphs that belong to a particularly simple class The achievement of this paper is that it proves the conjecture for a class of graphs that _cannot_ be built up in that way In other words, it obtains a proof that is not just a refinement of existing methods but a genuine extension of the available techniques A particular class of graphs for which they obtain a positive result is that of _blow-ups_ These are bipartite graphs obtained as follows: let $H$ be a bipartite graph with vertex sets $A$ and $B$ and let $p$ be a positive integer, take $p$ disjoint copies $H_1,\dots,H_p$ of $H$ and identify corresponding vertices that belong to $A$ (For example, if $H$ is a single edge, then the blow-up is the complete bipartite graph $K_{1,p}$ -- that is, a star with $p$ edges) The authors show that for every bipartite graph $H$ there is some $p$ such that the blow-up satisfies Sidorenko's conjecture The rough idea behind the proof is that blow-ups have a property that allows one to replace the graph by a simpler graph to which known techniques apply Perhaps the simplest case for which Sidorenko's conjecture is still unknown is the graph obtained from the complete bipartite graph $K_{5,5}$ by removing a 10-cycle While the authors do not resolve this question, their result does imply that the "square" of this graph (that is, the blow-up when $p=2$) satisfies the conjecture Another consequence of their results is that for every bipartite graph $H$ there is a bipartite graph $H'$ such that the disjoint union of $H$ and $H'$ satisfies the conjecture So is Sidorenko's conjecture true? The experts on the problem are reluctant to take a strong view either way The results proved so far place significant restrictions on what a counterexample could be like (though these leave open relatively small possibilities such as the $K_{5,5}\setminus C_{10}$ example), which perhaps points in the positive direction On the other hand, a natural analogue of the conjecture for 3-uniform hypergraphs, which places a restriction on what a proof could be like -- it cannot be too generalizable This paper is a valuable addition to our understanding of a tantalizing question

Anomalous hydrodynamics in a class of scarred frustration-free Hamiltonians

Abstract: Atypical eigenstates in the form of quantum scars and fragmentation of Hilbert space due to conservation laws provide obstructions to thermalization in the absence of disorder. In certain models with dipole and $U(1)$ conservation, the fragmentation results in subdiffusive transport. In this paper we study the interplay between scarring and weak fragmentation giving rise to anomalous hydrodynamics in a class of one-dimensional spin-1 frustration-free projector Hamiltonians, known as deformed Motzkin chain. The ground states and low-lying excitations of these chains exhibit large entanglement and critical slowdown. We show that at high energies the particular form of the projectors causes the emergence of disjoint Krylov subspaces for open boundary conditions, with an exact quantum scar being embedded in each subspace, leading to slow growth of entanglement and localized dynamics for specific out-of-equilibrium initial states. Furthermore, focusing on infinite temperature, we unveil that spin transport is subdiffusive, which we corroborate by simulations of suitable stochastic cellular automaton circuits. Compared to dipole moment conserving systems, the deformed Motzkin chain appears to belong to a different universality class with distinct dynamical transport exponent and only polynomially many Krylov subspaces.

Efficient bi-triangle counting for large bipartite networks

Abstract: A bipartite network is a network with two disjoint vertex sets and its edges only exist between vertices from different sets. It has received much interest since it can be used to model the relatio...

Physical Zero-Knowledge Proof for Numberlink Puzzle and k Vertex-Disjoint Paths Problem

Abstract: Numberlink is a logic puzzle with an objective to connect all pairs of cells with the same number by non-crossing paths in a rectangular grid. In this paper, we propose a physical protocol of zero-knowledge proof for Numberlink using a deck of cards, which allows a prover to convince a verifier that he/she knows a solution without revealing it. In particular, the protocol shows how to physically count the number of elements in a list that are equal to a given secret value without revealing that value, the positions of elements in the list that are equal to it, or the value of any other element in the list. Finally, we show that our protocol can be modified to verify a solution of the well-known k vertex-disjoint paths problem, both the undirected and directed settings.

Cycle Lengths in Expanding Graphs

Abstract: For a positive constant α a graph G on n vertices is called an α-expander if every vertex set U of size at most n/2 has an external neighborhood whose size is at least α|U|. We study cycle lengths in expanding graphs. We first prove that cycle lengths in α-expanders are well distributed. Specifically, we show that for every 0 < α ≤ 1 there exist positive constants n0, C and A = O(1/α) such that for every α-expander G on n≥n0 vertices and every integer $$\ell \in \left[ {C\log n,\tfrac{n}{C}} \right]$$ , G contains a cycle whose length is between $$\ell$$ and $$\ell$$ +A; the order of dependence of the additive error term A on α is optimal. Secondly, we show that every α-expander on n vertices contains $$\Omega \left( {\tfrac{{{\alpha ^3}}}{{\log (1/\alpha )}}} \right)$$ different cycle lengths. Finally, we introduce another expansion-type property, guaranteeing the existence of a linearly long interval in the set of cycle lengths. For β > 0 a graph G on n vertices is called a β-graph if every pair of disjoint sets of size at least βn are connected by an edge. We prove that for every $$\beta < 1/20$$ there exist positive constants $${b_1} = O\left( {\tfrac{1}{{\log (1 - \beta )}}} \right)$$ and b2 = O(β) such that every β-graph G on n vertices contains a cycle of length $$\ell$$ for every integer $$\ell \in \left[ {{b_1}\log n,\left( {1 - {b_2}} \right)n} \right]$$ ; the order of dependence of b1 and b2 on β is optimal.

Stochastic spectral embedding

Abstract: Constructing approximations that can accurately mimic the behavior of complex models at reduced computational costs is an important aspect of uncertainty quantification. Despite their flexibility and efficiency, classical surrogate models such as Kriging or polynomial chaos expansions tend to struggle with highly non-linear, localized or non-stationary computational models. We hereby propose a novel sequential adaptive surrogate modeling method based on recursively embedding locally spectral expansions. It is achieved by means of disjoint recursive partitioning of the input domain, which consists in sequentially splitting the latter into smaller subdomains, and constructing a simpler local spectral expansions in each, exploiting the trade-off complexity vs. locality. The resulting expansion, which we refer to as "stochastic spectral embedding" (SSE), is a piece-wise continuous approximation of the model response that shows promising approximation capabilities, and good scaling with both the problem dimension and the size of the training set. We finally show how the method compares favorably against state-of-the-art sparse polynomial chaos expansions on a set of models with different complexity and input dimension.

Delocalization Transition for Critical Erdős–Rényi Graphs

Abstract: We analyse the eigenvectors of the adjacency matrix of a critical Erdős–Renyi graph $${\mathbb {G}}(N,d/N)$$ , where d is of order $$\log N$$ . We show that its spectrum splits into two phases: a delocalized phase in the middle of the spectrum, where the eigenvectors are completely delocalized, and a semilocalized phase near the edges of the spectrum, where the eigenvectors are essentially localized on a small number of vertices. In the semilocalized phase the mass of an eigenvector is concentrated in a small number of disjoint balls centred around resonant vertices, in each of which it is a radial exponentially decaying function. The transition between the phases is sharp and is manifested in a discontinuity in the localization exponent $$\gamma (\varvec{\mathrm {w}})$$ of an eigenvector $$\varvec{\mathrm {w}}$$ , defined through $$\Vert \varvec{\mathrm {w}} \Vert _\infty / \Vert \varvec{\mathrm {w}} \Vert _2 = N^{-\gamma (\varvec{\mathrm {w}})}$$ . Our results remain valid throughout the optimal regime $$\sqrt{\log N} \ll d \leqslant O(\log N)$$ .

Ground-Truth Free Multi-Mask Self-Supervised Physics-Guided Deep Learning in Highly Accelerated MRI

Abstract: Deep learning based MRI reconstruction methods typically require databases of fully-sampled data as reference for training. However, fully-sampled acquisitions may be either challenging or impossible in numerous scenarios. Self-supervised learning enables training neural networks for MRI reconstruction without fully-sampled data by splitting available measurements into two disjoint sets. One of them is used in data consistency units in the network, and the other is used to define the loss. However, the performance of self-supervised learning degrades at high acceleration rates due to scarcity of acquired data. We propose a multi-mask self-supervised learning approach, which retrospectively splits available measurements into multiple 2-tuples of disjoint sets. Results on 3D knee and brain MRI shows that the proposed multi-mask self-supervised learning approach significantly improves upon single mask self-supervised learning at high acceleration rates.

A Cheap Feature Selection Approach for the K -Means Algorithm

Abstract: The increase in the number of features that need to be analyzed in a wide variety of areas, such as genome sequencing, computer vision, or sensor networks, represents a challenge for the $K$ -means algorithm. In this regard, different dimensionality reduction approaches for the $K$ -means algorithm have been designed recently, leading to algorithms that have proved to generate competitive clusterings. Unfortunately, most of these techniques tend to have fairly high computational costs and/or might not be easy to parallelize. In this article, we propose a fully parallelizable feature selection technique intended for the $K$ -means algorithm. The proposal is based on a novel feature relevance measure that is closely related to the $K$ -means error of a given clustering. Given a disjoint partition of the features, the technique consists of obtaining a clustering for each subset of features and selecting the $m$ features with the highest relevance measure. The computational cost of this approach is just $\mathcal {O}(m\cdot \max \{n\cdot K,\log m\})$ per subset of features. We additionally provide a theoretical analysis on the quality of the obtained solution via our proposal and empirically analyze its performance with respect to well-known feature selection and feature extraction techniques. Such an analysis shows that our proposal consistently obtains the results with lower $K$ -means error than all the considered feature selection techniques: Laplacian scores, maximum variance, multicluster feature selection, and random selection while also requiring similar or lower computational times than these approaches. Moreover, when compared with feature extraction techniques, such as random projections, the proposed approach also shows a noticeable improvement in both error and computational time.

A story of diameter, radius and (almost) Helly property

Abstract: We present new algorithmic results for the class of Helly graphs, i.e., for the discrete analogues of hyperconvex metric spaces. Specifically, an undirected unweighted graph is Helly if every family of pairwise intersecting balls has a nonempty common intersection. It is known that every graph isometrically embeds into a Helly graph, that makes of the latter an important class of graphs in Metric Graph Theory. We study diameter and radius computations within the Helly graphs, and related graph classes. This is in part motivated by a conjecture on the fine-grained complexity of these two distance problems within the graph classes of bounded fractional Helly number-that contain as particular cases the proper minor-closed graph classes and the bounded clique-width graphs. Note that under plausible complexity assumptions, neither the diameter nor the radius can be computed in truly subquadratic time on general graphs. In contrast to these negative results, we first present algorithms which given an n-vertex m-edge Helly graph G as input, compute with high probability (w.h.p.) its radius and its diameter in O(m √ n) time (i.e., subquadratic in n + m). Our algorithms are based on the Helly property and on the unimodality of the eccentricity function in Helly graphs: every vertex of locally minimum eccentricity is a central vertex. Then, we improve our results for the C 4-free Helly graphs, that are exactly the Helly graphs whose balls are convex. For this subclass, we present linear-time algorithms for computing the eccentricity of all vertices. Doing so, we generalize previous results on strongly chordal graphs to a much larger subclass, that includes, among others, all the bridged Helly graphs and the hereditary Helly graphs. Lastly, we derive approximate versions of our results for the class of chordal graphs: with the latter satisfying an almost-Helly-type property, and a stronger (induced-path) convex-ity property than the C 4-free Helly graphs. For the chordal graphs, we can compute in quasi linear time the eccentricity of all vertices with an additive one-sided error of at most one, which is best possible under the Strong Exponential-Time Hypothesis (SETH). This answers an open question of [Dragan, IPL 2019]. In fact, we obtain this last result as a byproduct from a more general reduction: from diameter computation on chordal graphs to the Disjoint Sets problem. Roughly, it implies that the split graphs are the only hard instances for diameter computation on chordal graphs. We also get from our reduction that on any subclass of chordal graphs with constant VC-dimension (and so, for undirected path graphs), the diameter can be computed in truly subquadratic time.

Quantum Online Streaming Algorithms with Logarithmic Memory

Abstract: We consider quantum and classical (deterministic or randomize) streaming online algorithms with respect to competitive ratio. We show that there is a problem that can be solved by a quantum online streaming algorithm better than by classical ones in the case of logarithmic memory. The problem is an online version of the Disjointness problem (Checking weather two sets are disjoint or not).

Turán numbers for disjoint paths

Abstract: The Tur\\'{a}n number of a graph $H$, $ex(n,H)$, is the maximum number of edges in any graph of order $n$ which does not contain $H$ as a subgraph. Lidick\\'{y}, Liu and Palmer determined $ex(n, F_m)$ for $n$ sufficiently large and proved that the extremal graph is unique, where $F_m$ is disjoint paths of $P_{k_1}, \\ldots, P_{k_m}$ [Lidick\\'{y},B., Liu,H. and Palmer,C. (2013). On the Tur\\'{a}n number of forests. Electron. J. Combin. 20(2) Paper 62, 13 pp]. In this paper, by mean of a different approach, we determine $ex(n, F_m)$ for all integers $n$ with minor conditions, which extends their partial results. Furthermore, we partly confirm the conjecture proposed by Bushaw and Kettle for $ex(n, k\\cdot P_l)$ [Bushaw,N. and Kttle,N. (2011) Tur\\'{a}n numbers of multiple paths and equibipartite forests. Combin. Probab. Comput. 20 837-853]. Moreover, we show that there exist two family graphs $F_m$ and $F_m^{\\prime}$ such that $ex(n, F_m)=ex(n, F_m^{\\prime})$ for all integers $n$, which is related to an old problem of Erd\\H{o}s and Simonovits.

Maximum $ H $-index of bipartite network with some given parameters

Abstract: A network is an abstract structure that consists of nodes that are connected by links. A bipartite network is a type of networks where the set of nodes can be divided into two disjoint sets in a way that each link connects a node from one partition with a node from the other partition. In this paper, we first determine the maximum $ H $-index of networks in the class of all $ n $-node connected bipartite network with matching number $ t $. We obtain that the maximum $ H $-index of a bipartite network with a given matching number is $ K_{t, n-t} $. Secondly, we characterize the network with the maximum $ H $-index in the class of all the $ n $-vertex connected bipartite network of given diameter. Based on our obtain results, we establish the unique bipartite network with maximum $ H $-index among bipartite networks with a given independence number and cover of a network.

(Theta, triangle)‐free and (even hole, K4)‐free graphs—Part 1: Layered wheels

Abstract: We present a construction called layered wheel. Layered wheels are graphs of arbitrarily large treewidth and girth. They might be an outcome for a possible theorem characterizing graphs with large treewidth in term of their induced subgraphs (while such a characterization is well understood in term of minors). They also provide examples of graphs of large treewidth and large rankwidth in well studied classes, such as (theta, triangle)-free graphs and even-hole-free graphs with no $K_4$ (where a hole is a chordless cycle of length at least~4, a theta is a graph made of three internally vertex disjoint paths of length at least~2 linking two vertices, and $K_4$ is the complete graph on~4 vertices).

Three Types of Two-Disjoint-Cycle-Cover Pancyclicity and Their Applications to Cycle Embedding in Locally Twisted Cubes

Abstract: A graph $G=(V,E)$ is two-disjoint-cycle-cover $[r_1,r_2]$-pancyclic if for any integer $l$ satisfying $r_1 \\leq l \\leq r_2$, there exist two vertex-disjoint cycles $C_1$ and $C_2$ in $G$ such that the lengths of $C_1$ and $C_2$ are $l$ and $|V(G)| - l$, respectively, where $|V(G)|$ denotes the total number of vertices in $G$. On the basis of this definition, we further propose Ore-type conditions for graphs to be two-disjoint-cycle-cover vertex/edge $[r_1,r_2]$-pancyclic. In addition, we study cycle embedding in the $n$-dimensional locally twisted cube $LTQ_n$ under the consideration of two-disjoint-cycle-cover vertex/edge pancyclicity.

SU(3)-skein algebras and webs on surfaces

Abstract: The $$SU_3$$ -skein algebra of a surface F is spanned by isotopy classes of certain framed graphs in $$F\times I$$ called 3-webs subject to the skein relations encapsulating relations between $$U_q(sl(3))$$ -representations. It is expected that their theory parallels that of the Kauffman bracket skein algebras. We make the first step towards developing that theory by proving that the reduced $$SU_3$$ -skein algebra of any surface of finite type is finitely generated. We achieve that result by developing a theory of canonical forms of webs in surfaces. Specifically, we show that for any ideal triangulation of F every reduced 3-web can be uniquely decomposed into unions of pyramid formations of hexagons and disjoint arcs in the faces of the triangulation with possible additional “crossbars” connecting their edges along the ideal triangulation. We show that such canonical position is unique up to “crossbar moves”. That leads us to an associated system of coordinates for webs in triangulated surfaces (counting intersections of the web with the edges of the triangulation and their rotation numbers inside of the faces of the triangulation) which determine a reduced web uniquely. Finally, we relate our skein algebras to $${\mathcal {A}}$$ -varieties of Fock–Goncharov and to $$\text {Loc}_{SL(3)}$$ -varieties of Goncharov–Shen. We believe that our coordinate system for webs is a manifestation of a (quantum) mirror symmetry conjectured by Goncharov–Shen.