790 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Borel versions of the Local Lemma and LOCAL algorithms for graphs of finite asymptotic separation index
Asymptotic separation index is a parameter that measures how easily a Borel
graph can be approximated by its subgraphs with finite components. In contrast
to the more classical notion of hyperfiniteness, asymptotic separation index is
well-suited for combinatorial applications in the Borel setting. The main
result of this paper is a Borel version of the Lov\'asz Local Lemma -- a
powerful general-purpose tool in probabilistic combinatorics -- under a finite
asymptotic separation index assumption. As a consequence, we show that locally
checkable labeling problems that are solvable by efficient randomized
distributed algorithms admit Borel solutions on bounded degree Borel graphs
with finite asymptotic separation index. From this we derive a number of
corollaries, for example a Borel version of Brooks's theorem for graphs with
finite asymptotic separation index
Discrete functional inequalities on lattice graphs
In this thesis, we study problems at the interface of analysis and discrete mathematics. We discuss analogues of well known Hardy-type inequalities and Rearrangement inequalities on the lattice graphs Z^d, with a particular focus on behaviour of sharp constants and optimizers.
In the first half of the thesis, we analyse Hardy inequalities on Z^d, first for d=1 and then for d >= 3. We prove a sharp weighted Hardy inequality on integers with power weights of the form n^\alpha. This is done via two different methods, namely 'super-solution' and 'Fourier method'. We also use Fourier method to prove a weighted Hardy type inequality for higher order operators. After discussing the one dimensional case, we study the Hardy inequality in higher dimensions (d >= 3). In particular, we compute the asymptotic behaviour of the sharp constant in the discrete Hardy inequality, as d \rightarrow \infty. This is done by converting the inequality into a continuous Hardy-type inequality on a torus for functions having zero average. These continuous inequalities are new and interesting in themselves.
In the second half, we focus our attention on analogues of Rearrangement inequalities on lattice graphs. We begin by analysing the situation in dimension one. We define various notions of rearrangements and prove the corresponding Polya-Szego inequality. These inequalities are also applied to prove some weighted Hardy inequalities on integers. Finally, we study Rearrangement inequalities (Polya-Szego) on general graphs, with a particular focus on lattice graphs Z^d, for d >=2. We develop a framework to study these inequalities, using which we derive concrete results in dimension two. In particular, these results develop connections between Polya-Szego inequality and various isoperimetric inequalities on graphs.Open Acces
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Foundations of Node Representation Learning
Low-dimensional node representations, also called node embeddings, are a cornerstone in the modeling and analysis of complex networks. In recent years, advances in deep learning have spurred development of novel neural network-inspired methods for learning node representations which have largely surpassed classical \u27spectral\u27 embeddings in performance. Yet little work asks the central questions of this thesis: Why do these novel deep methods outperform their classical predecessors, and what are their limitations?
We pursue several paths to answering these questions. To further our understanding of deep embedding methods, we explore their relationship with spectral methods, which are better understood, and show that some popular deep methods are equivalent to spectral methods in a certain natural limit. We also introduce the problem of inverting node embeddings in order to probe what information they contain. Further, we propose a simple, non-deep method for node representation learning, and find it to often be competitive with modern deep graph networks in downstream performance.
To better understand the limitations of node embeddings, we prove some upper and lower bounds on their capabilities. Most notably, we prove that node embeddings are capable of exact low-dimensional representation of networks with bounded max degree or arboricity, and we further show that a simple algorithm can find such exact embeddings for real-world networks. By contrast, we also prove inherent bounds on random graph models, including those derived from node embeddings, to capture key structural properties of networks without simply memorizing a given graph
Stability of Homomorphisms, Coverings and Cocycles I: Equivalence
This paper is motivated by recent developments in group stability, high
dimensional expansion, local testability of error correcting codes and
topological property testing. In Part I, we formulate and motivate three
stability problems: 1. Homomorphism stability: Are almost homomorphisms close
to homomorphisms? 2. Covering stability: Are almost coverings of a cell complex
close to genuine coverings of it? 3. Cocycle stability: Are 1-cochains whose
coboundary is small close to 1-cocycles? We then prove that these three
problems are equivalent.Comment: 32 page
Approximate Nearest Neighbor Searching with Non-Euclidean and Weighted Distances
We present a new approach to approximate nearest-neighbor queries in fixed
dimension under a variety of non-Euclidean distances. We are given a set of
points in , an approximation parameter , and
a distance function that satisfies certain smoothness and growth-rate
assumptions. The objective is to preprocess into a data structure so that
for any query point in , it is possible to efficiently report
any point of whose distance from is within a factor of
of the actual closest point.
Prior to this work, the most efficient data structures for approximate
nearest-neighbor searching in spaces of constant dimensionality applied only to
the Euclidean metric. This paper overcomes this limitation through a method
called convexification. For admissible distance functions, the proposed data
structures answer queries in logarithmic time using space, nearly matching the best known bounds for the
Euclidean metric. These results apply to both convex scaling distance functions
(including the Mahalanobis distance and weighted Minkowski metrics) and Bregman
divergences (including the Kullback-Leibler divergence and the Itakura-Saito
distance)
Gradual Domain Adaptation: Theory and Algorithms
Unsupervised domain adaptation (UDA) adapts a model from a labeled source
domain to an unlabeled target domain in a one-off way. Though widely applied,
UDA faces a great challenge whenever the distribution shift between the source
and the target is large. Gradual domain adaptation (GDA) mitigates this
limitation by using intermediate domains to gradually adapt from the source to
the target domain. In this work, we first theoretically analyze gradual
self-training, a popular GDA algorithm, and provide a significantly improved
generalization bound compared with Kumar et al. (2020). Our theoretical
analysis leads to an interesting insight: to minimize the generalization error
on the target domain, the sequence of intermediate domains should be placed
uniformly along the Wasserstein geodesic between the source and target domains.
The insight is particularly useful under the situation where intermediate
domains are missing or scarce, which is often the case in real-world
applications. Based on the insight, we propose enerative Gradual
Dmain daptation with Optimal ransport
(GOAT), an algorithmic framework that can generate intermediate domains in a
data-dependent way. More concretely, we first generate intermediate domains
along the Wasserstein geodesic between two given consecutive domains in a
feature space, then apply gradual self-training to adapt the source-trained
classifier to the target along the sequence of intermediate domains.
Empirically, we demonstrate that our GOAT framework can improve the performance
of standard GDA when the given intermediate domains are scarce, significantly
broadening the real-world application scenarios of GDA. Our code is available
at https://github.com/yifei-he/GOAT.Comment: arXiv admin note: substantial text overlap with arXiv:2204.0820
Open Problems in (Hyper)Graph Decomposition
Large networks are useful in a wide range of applications. Sometimes problem
instances are composed of billions of entities. Decomposing and analyzing these
structures helps us gain new insights about our surroundings. Even if the final
application concerns a different problem (such as traversal, finding paths,
trees, and flows), decomposing large graphs is often an important subproblem
for complexity reduction or parallelization. This report is a summary of
discussions that happened at Dagstuhl seminar 23331 on "Recent Trends in Graph
Decomposition" and presents currently open problems and future directions in
the area of (hyper)graph decomposition
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
On Pairwise Graph Connectivity
A graph on at least k+1 vertices is said to have global connectivity k if any two of its vertices are connected by k independent paths. The local connectivity of two vertices is the number of independent paths between those specific vertices. This dissertation is concerned with pairwise connectivity notions, meaning that the focus is on local connectivity relations that are required for a number of or all pairs of vertices. We give a detailed overview about how uniformly k-connected and uniformly k-edge-connected graphs are related and provide a complete constructive characterization of uniformly 3-connected graphs, complementing classical characterizations by Tutte. Besides a tight bound on the number of vertices of degree three in uniformly 3-connected graphs, we give results on how the crossing number and treewidth behaves under the constructions at hand. The second central concern is to introduce and study cut sequences of graphs. Such a sequence is the multiset of edge weights of a corresponding Gomory-Hu tree. The main result in that context is a constructive scheme that allows to generate graphs with prescribed cut sequence if that sequence satisfies a shifted variant of the classical Erdős-Gallai inequalities. A complete characterization of realizable cut sequences remains open. The third central goal is to investigate the spectral properties of matrices whose entries represent a graph's local connectivities. We explore how the spectral parameters of these matrices are related to the structure of the corresponding graphs, prove bounds on eigenvalues and related energies, which are sums of absolute values of all eigenvalues, and determine the attaining graphs. Furthermore, we show how these results translate to ultrametric distance matrices and touch on a Laplace analogue for connectivity matrices and a related isoperimetric inequality
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