The traditional Riemann Mapping Theorem can be proved with circle packing techniques. We prove the Combinatorial Riemann Mapping Theorem for tilings of bounded size using circle packings.
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MOCA-HESP optimizes high-dimensional combinatorial and mixed spaces using hyper-ellipsoid partitioning.
problem Challenges in optimizing high-dimensional, combinatorial and mixed spaces.
method MOCA-HESP uses hyper-ellipsoid space partitioning with different categorical encoders and multi-armed bandit for adaptive selection.
result MOCA-HESP outperforms existing methods on various synthetic and real-world benchmarks.
We develop a tighter implementation of basic PL topology, which keeps track of some combinatorial structure beyond PL homeomorphism type. With this technique we clarify some aspects of PL transversality and give combinatorial proofs of a number of known results. New results include a combinatorial characterization of c…
Study infinite combinatorial Ricci flow on spherical surfaces.
problem Investigate infinite combinatorial Ricci flow with spherical background.
method Establish existence and convergence of solution for infinite cellular decompositions.
result Existence and convergence of solution for infinite combinatorial Ricci flow in spherical geometry.
We investigate the combinatorial analogues, in the context of normal surfaces, of taut and transversely measured (codimension 1) foliations of 3-manifolds. We establish that the existence of certain combinatorial structures, a priori weaker than the existence of the corresponding foliation, is sufficient to guarantee t…
The optimization of expensive-to-evaluate black-box functions over combinatorial structures is an ubiquitous task in machine learning, engineering and the natural sciences. The combinatorial explosion of the search space and costly evaluations pose challenges for current techniques in discrete optimization and machine …
New method detects sliceness of knots in a torus.
problem Detecting sliceness of knots in a torus.
method Group-theoretical combinatorial techniques.
result Constructs many sliceness obstructions.
We define combinatorial analogues of stable and unstable minimal surfaces in the setting of weighted pseudomanifolds. We prove that, under mild conditions, such combinatorial minimal surfaces always exist. We use a technique, adapted from work of Johnson and Thompson, called thin position. Thin position is defined usin…
Combining models in appropriate ways to achieve high performance is commonly seen in machine learning fields today. Although a large amount of combinatorial models have been created, little attention is drawn to the commons in different models and their connections. A general modelling technique is thus worth studying …
In the present paper, we show that many combinatorial and topological objects, such as maps, hypermaps, three-dimensional pavings, constellations and branched coverings of the two--sphere admit any given finite automorphism group. This enhances the already known results by Frucht, Cori -- Machì, Širáň -- Škoviera, and …
This paper tackles combinatorial optimization under uncertainty with limited feedback.
problem Tackling combinatorial optimization problems with uncertain or unknown parameters.
method Review of techniques for combinatorial pure exploration with limited bandit feedback.
result Introduction of methods for combinatorial optimization under uncertainty with limited observation.
The paper tackles budget allocation for multiple campaigns using a novel combinatorial bandit approach.
problem Maximizing cumulative returns with limited budgets across various ad lines.
method Formulated as a multi-task combinatorial bandit problem, integrates Bayesian hierarchical models, and uses Thompson sampling.
result Demonstrates robustness and adaptability in maximizing overall cumulative returns.
Algorithm BGLM-OFU minimizes regret in combinatorial causal bandits with binary models.
problem Minimizing expected regret in combinatorial causal bandits with binary generalized linear models.
method BGLM-OFU algorithm based on maximum likelihood estimation for Markovian BGLMs, and causal inference techniques for linear models with hidden variables.
result Achieves regret for binary generalized linear models.
Combinatorial Neural Banditsstat.ML
New algorithms for neural bandits learn from context and arm features.
problem Learning from context and arm features in combinatorial bandit problems.
method Proposed algorithms CN-UCB and CN-TS using deep neural networks.
result Achieved regret guarantees for the first time in combinatorial neural bandits.
New method improves combinatorial optimization by capturing dependencies among solution variables.
problem Performance limitations in solving combinatorial optimization problems using independent solution variables.
method Subgraph tokenization and variational annealing to capture dependencies and improve learning efficiency.
result Empirical evidence shows superior performance of autoregressive methods with tokenization and annealed entropy regularization.
Combinatorial linear semi-bandits (CLS) are widely applicable frameworks of sequential decision-making, in which a learner chooses a subset of arms from a given set of arms associated with feature vectors. Existing algorithms work poorly for the clustered case, in which the feature vectors form several large clusters. …
We extend our generic rigidity theory for periodic frameworks in the plane to frameworks with a broader class of crystallographic symmetry. Along the way we introduce a new class of combinatorial matroids and associated linear representation results that may be interesting in their own right. The same techniques immedi…
This paper surveys RL for combinatorial optimization, focusing on TSP.
problem Optimizing solutions for combinatorial optimization problems.
method Reinforcement learning applied to combinatorial optimization problems, specifically the TSP.
result Deep learning mechanisms enhance RL algorithms for near-optimal solutions.
New methods show hyperbolicity of Brunnian links.
problem Determining hyperbolicity of Brunnian links.
method Geometric/combinatorial topology techniques applied to links with at least one unknotted component.
result Determine hyperbolicity for all Brunnian links.
GFlowNets improve combinatorial optimization by efficiently sampling from solution spaces.
problem NP-hard combinatorial optimization problems with structured constraints.
method Design Markov decision processes and train conditional GFlowNets to sample solutions.
result GFlowNet policies find high-quality solutions efficiently on various CO tasks.
We present a novel preconditioning technique for proximal optimization methods that relies on graph algorithms to construct effective preconditioners. Such combinatorial preconditioners arise from partitioning the graph into forests. We prove that certain decompositions lead to a theoretically optimal condition number.…
This work improves privacy in federated combinatorial bandits by balancing regret and privacy.
problem Privacy-preserving learning in competitive online learning settings with quality constraints.
method Proposes P-FCB algorithm for federated combinatorial bandits, balancing regret and privacy.
result Improves regret while maintaining quality constraints and privacy guarantees.
New Ising models improve consensus clustering on specialized hardware.
problem Consensus clustering optimization problems.
method Formulated consensus clustering as Ising models and evaluated on specialized hardware.
result Our Ising models outperform existing techniques on consensus clustering.
A Near-Optimal Change-Detection Based Algorithm for Piecewise-Stationary Combinatorial Semi-Banditscs.LG
We investigate the piecewise-stationary combinatorial semi-bandit problem. Compared to the original combinatorial semi-bandit problem, our setting assumes the reward distributions of base arms may change in a piecewise-stationary manner at unknown time steps. We propose an algorithm, \texttt{GLR-CUCB}, which incorporat…
The paper describes and analyzes a knot concordance invariant ε using grid homology.
problem Understanding and computing the knot concordance invariant ε.
method Combinatorial description and grid homology techniques.
result Computed values and properties of the invariant ε for specific knots.
Hypertoric varieties are hyperkähler analogues of toric varieties, and are constructed as abelian hyperkähler quotients of a quaternionic affine space. Just as symplectic toric orbifolds are determined by labelled polytopes, orbifold hypertoric varieties are intimately related to the combinatorics of hyperplane arrange…
There has been an increased interest in discovering heuristics for combinatorial problems on graphs through machine learning. While existing techniques have primarily focused on obtaining high-quality solutions, scalability to billion-sized graphs has not been adequately addressed. In addition, the impact of budget-con…
Continuous optimization is an important problem in many areas of AI, including vision, robotics, probabilistic inference, and machine learning. Unfortunately, most real-world optimization problems are nonconvex, causing standard convex techniques to find only local optima, even with extensions like random restarts and …
Derives Khovanov homology for 2-strand braids using combinatorial relations.
problem Computational difficulties in Khovanov-Rozansky homology computations.
method Combining state-sum descriptions and combinatorial relations to simplify computations.
result Computation of Khovanov-Rozansky invariant for 2-strand braid links confirmed.
We construct combinatorial volume forms of hyperbolic three manifolds fibering over the circle. These forms define non-trivial classes in bounded cohomology. After introducing a new seminorm on exact bounded cohomology, we use these combinatorial classes to show that, in degree 3, the zero norm subspace of the bounded …
Master-slave architecture tackles combinatorial multi-armed bandits with diversity constraints.
problem Solving top- combinatorial multi-armed bandits with non-linear feedback and diversity constraints.
method Master-slave architecture with six slave models, teacher learning, and policy co-training.
result Significantly outperforms existing algorithms in synthetic and real datasets.
We introduce new combinatorial quantities for concept classes, and prove lower and upper bounds for learning complexity in several models of query learning in terms of various combinatorial quantities. Our approach is flexible and powerful enough to enough to give new and very short proofs of the efficient learnability…
This survey provides an introduction to basic questions and techniques surrounding the topology of the moduli space of stable Higgs bundles on a Riemann surface. Through examples, we demonstrate how the structure of the cohomology ring of the moduli space leads to interesting questions of a combinatorial nature.
A few years ago Kramer and Laubenbacher introduced a discrete notion of homotopy for simplicial complexes. In this paper, we compute the discrete fundamental group of the order complex of the Boolean lattice. As it turns out, it is equivalent to computing the discrete homotopy group of the 1-skeleton of the permutahedr…
In 2003, Ozsváth and Szabó defined the concordance invariant for knots in oriented 3-manifolds as part of the Heegaard Floer homology package. In 2011, Sarkar gave a combinatorial definition of for knots in and a combinatorial proof that gives a lower bound for the slice genus of a knot. Recently, Har…
Algorithm improves movie recommendation efficiency with fairness constraints.
problem Improving movie recommendation efficiency with fairness constraints in combinatorial semi-bandits.
method Adopted Thompson Sampling with beta priors and Bernoulli likelihoods to handle fairness constraints.
result Time-averaged regret upper bounded by $\frac{N}{2η} + O\left(\frac{\sqrt{mNT\ln T}}{T}
ight)$, with fairness constraints satisfied.
Let E be a circle bundle over a Riemann surface that supports a contact structure transverse to the fibers. This paper presents a combinatorial definition of a differential graded algebra (DGA) that is an invariant of Legendrian knots in E. The invariant generalizes Chekanov's combinatorial DGA invariant of Legendrian …
We show that every convex polyhedron admits a simple edge unfolding after an affine transformation. In particular there exists no combinatorial obstruction to a positive resolution of Durer's unfoldability problem, which answers a question of Croft, Falconer, and Guy. Among other techniques, the proof employs a topolog…
New algorithm tackles non-stationary combinatorial semi-bandit problems with optimal regret bounds.
problem Non-stationary combinatorial semi-bandit problems in switching and dynamic environments.
method Developed algorithms for both switching and dynamic cases, achieving nearly optimal regret bounds.
result Achieved nearly optimal regret bounds in both switching and dynamic cases.
We study the problem to extend an immersed circle f in the 2-dimensional sphere to an immersion of the disc. We analyze existence and uniqueness for this problems in terms of the combinatorial structure of a word assigned to f. Our techniques are based on ideas of Blank who studied the extension problem in case of a pl…
A fast ML method solves complex combinatorial auction problems.
problem Solving winner determination in multi-unit combinatorial auctions.
method Graph Neural Network (GNN) with half-convolution operations for bid-item graph modeling.
result Approaches optimal performance with negligible revenue loss and low complexity.
Using techniques from the theories of convex polytopes, lattice paths, and indirect influences on directed manifolds, we construct continuous analogues for the binomial coefficients and the Catalan numbers. Our approach for constructing these analogues can be applied to a wide variety of combinatorial sequences. As an …
Advances combinatorial complexes for better modeling of hierarchical and set-type relations.
problem Lack of effective modeling for complex hierarchical and set-type relations in high-dimensional data.
method Introduces combinatorial complexes as a bridge between cell complexes and hypergraphs, emphasizing their different types of relations.
result Combining set-type and hierarchical relations in a single model can be advantageous in learning tasks.
Study tackles distribution shift in combinatorial settings using matrix completion techniques.
problem Tackling distribution shift in combinatorial settings with rigorous statistical guarantees.
method Develops novel algorithms and theoretical results for extrapolating to test distributions not covered in training.
result Achieves bilinear combinatorial extrapolation under gradual spectral decay in high-dimensional data.
We develop homological techniques for finding explicit combinatorial expressions of finite-type cohomology classes of spaces of knots in generalizing Polyak--Viro formulas for invariants (i.e. 0-dimensional cohomology classes) of knots in . As the first applications we give such formulas for the (r…
The Dehn function of SL(n;Z)math.GR
We prove that when n >= 5, the Dehn function of SL(n;Z) is quadratic. The proof involves decomposing a disc in SL(n;R)/SO(n) into triangles of varying sizes. By mapping these triangles into SL(n;Z) and replacing large elementary matrices by "shortcuts," we obtain words of a particular form, and we use combinatorial tec…
Machine learning reduces combinatorial optimization problem dimensions.
problem Reducing the complexity of large combinatorial optimization problems.
method Generalization of a machine learning model for problem reduction on TSP.
result Machine learning can predict which variables are not part of an optimal solution.
Novel theory combines combinatorial and topological elements.
problem Understanding combinatorial phenomena at the intersection of topology.
method Synthesizes combinatorial and topological approaches with a new framing concept.
result Framed combinatorial spaces exhibit better behavior than classical spaces.