Width trees link link invariants and bridge number.
problem Understanding link invariants through geometric structures.
method Associate width trees to links and use their geometric properties to bound link invariants.
result Width trees uniquely realize certain link invariants under specific conditions.
The paper proves link homotopy invariants from tree invariants and explains indeterminacy.
problem Link homotopy invariants from tree invariants.
method Arrow diagrammatic proof and tree stacking operations.
result Indeterminacy of tree invariants is consistent with Milnor's original indeterminacy.
Paper analyzes soft tree ensembles using NTK, finding only leaf count matters.
problem Understanding impact of various tree architectures in ensemble learning.
method Formulated and analyzed Neural Tangent Kernel (NTK) for soft tree ensembles.
result Only the number of leaves at each depth is relevant for tree architecture in ensemble learning.
This paper describes experiments, on two domains, to investigate the effect of averaging over predictions of multiple decision trees, instead of using a single tree. Other authors have pointed out theoretical and commonsense reasons for preferring the multiple tree approach. Ideally, we would like to consider predictio…
TAO outperforms other decision tree algorithms in accuracy.
problem Optimizing decision tree algorithms for better performance.
method Comparison of TAO with established algorithms on various datasets.
result TAO achieves higher accuracy in nearly all datasets.
This paper proves a Faber-Krahn inequality for trees with given matching number.
problem Proving a Faber-Krahn inequality for trees with specific properties.
method Characterization of trees with given matching number.
result The Faber-Krahn inequality holds for trees with given matching number.
Random forest performance varies with number of trees, sometimes peaking before increasing.
problem Optimal number of trees in random forest for best performance.
method Theoretical analysis and application to datasets.
result Classification error rate may have non-monotonic behavior with number of trees.
Subagging improves regression tree performance, especially with many splits.
problem Improving regression tree performance with subsample aggregating.
method Formalized bias and variance dependencies, compared subagging to single trees, and analyzed optimal tree sizes.
result Subagging improves tree performance, especially with many splits.
Study on wind-tree models yields formulas for periodic trajectories.
problem Counting periodic trajectories in wind-tree models.
method Asymptotic formulas and explicit computation of Siegel-Veech constants.
result Asymptotic formulas for closed billiard trajectories in wind-tree models.
Geometrically realized polyhedra from directed trees, including associahedra.
problem Understanding the structure of associative algebras with co-inner products.
method Geometric realization of polyhedra using directed planar trees.
result These polyhedra, including associahedra, are homeomorphic to balls.
New algorithm efficiently learns sparse staged trees.
problem Scalability issue in existing structural learning algorithms for staged trees.
method Searches over a space of models with limited dependencies.
result First scalable structural learning algorithm for staged trees.
This paper provides a relationship between a geometric structure of a suspended tree and the number of link components of the associated link diagram.
Solves weighted bi-colored plane tree enumeration and applies to geometric problems.
problem Weighted bi-colored plane trees with specific vertex counts and edge weights.
method Unified algorithmic counting method.
result Strong Hurwitz number for Riemann spheres with three branched points.
The study proves a conjecture about arborescent links with many twigs.
problem Proving the meridional rank conjecture for arborescent links.
method Using an upper bound on the bridge number in terms of the maximal number of link components of the underlying tree.
result Proves the meridional rank conjecture for arborescent links with specific properties.
Optimal diagram found for complete graphs with linear trees.
problem Finding optimal diagrams for complete graphs.
method Using a linear tree structure to minimize crossing numbers.
result Optimal diagrams without free hamiltonian cycles for odd n≥7. New tree structure for pseudo-Anosovs from interval maps.
problem Understanding pseudo-Anosovs from interval maps.
method Tree structure on pseudo-Anosovs using rational numbers.
result Deepened dictionary between invariants.
The paper improves bounds on skein tree depth and delta-crossing numbers for knots and links.
problem Improving bounds on skein tree depth and delta-crossing numbers for knots and links.
method Theoretical and computational analysis of skein trees and knot invariants.
result New upper and lower bounds on skein tree depth and delta-crossing numbers are derived.
We introduce a new way to tabulate knots by representing knot diagrams using a pair of planar trees. This pair of trees have their edges labeled by integers, they have no valence 2 vertices, and they have the same number of valence 1 vertices. The number of valence 1 vertices of the trees is called the girth of the kno…
Convex polytope trees expand decision trees with interpretable boundaries.
problem High accuracy often requires many nodes in decision trees, reducing interpretability.
method CPT uses logical disjunction of weighted linear decision-makers, geometrically a convex polytope.
result CPT achieves high accuracy with fewer nodes compared to existing methods.
SNJ recovers latent tree models from similarity matrices.
problem Reconstructing latent tree models from observed data.
method Spectral Neighbor Joining (SNJ) method.
result SNJ is consistent and requires fewer samples for accurate tree recovery.
Multistage Defer Trees improve model accuracy while maintaining interpretability.
problem Balancing model accuracy and interpretability, especially in noisy domains.
method A sequence of sparse decision trees that defer predictions to the next tree or a black box.
result Matches the performance of complex tree-based ensembles while using only one or a few sparse trees.
We introduce community trees to summarize network structures.
problem Stability of community structures in networks.
method Clique percolation method (CPM) and persistent diagrams.
result Total star number (TSN) provides an upper bound on community tree changes.
A new supervised tree-Wasserstein distance improves document classification.
problem Measuring document similarity efficiently and accurately.
method Rewriting Wasserstein distance on tree metric, using contrastive loss for optimization.
result The Supervised Tree-Wasserstein (STW) distance improves document classification accuracy.
Researchers develop methods to identify diffusion sources in tree networks.
problem Identifying the source of a diffusion in regular tree networks.
method Construct confidence sets for the diffusion source with size independent of the number of infected nodes, using probabilistic analysis of Pólya urns.
result It is possible to construct confidence sets for the diffusion source with size independent of the number of infected nodes.
Algorithm removes leaves to find root in uniform trees.
problem Finding the root in large uniform attachment trees.
method Leaf-stripping algorithm recursively removes leaves.
result Set of remaining vertices contains the root with high probability.
In this paper, we investigate adaptive nonlinear regression and introduce tree based piecewise linear regression algorithms that are highly efficient and provide significantly improved performance with guaranteed upper bounds in an individual sequence manner. We use a tree notion in order to partition the space of regr…
The paper uses tensor decompositions to improve neural network models for tree data.
problem Encoding structural knowledge from tree-structured data efficiently.
method Introduces new aggregation functions using Canonical and Tensor-Train decompositions.
result Proposed models outperform traditional methods on tree classification tasks.
The paper tests properties of trees in graphical models using covariance queries.
problem Testing properties of trees in graphical models.
method Covariance queries model, randomized tests for tree properties.
result Efficient testing of global tree properties using sub-quadratic number of queries.
Integrates differentiable decision trees into neural networks for faster training and inference.
problem Combining differentiability and conditional computation in tree ensembles for neural networks.
method Sparse activation function and specialized forward/backward propagation algorithms for efficient training and inference.
result 10x speed-ups and 20x reduction in parameters compared to existing methods, while maintaining performance.
The paper proposes a method to efficiently predict using labeled binary trees and analyzes the number of samples needed.
problem Efficiently predicting using compositional nonparametric models.
method A compositional nonparametric method expressed as a labeled binary tree, with a greedy algorithm for regression validation.
result The sufficient number of samples is O(klog(pq)+log(k!)), and the necessary number of samples is Ω(klog(pq)−log(k!)). Study shows critical exponents for tree-acting groups.
problem Understanding critical exponents of discrete groups on trees.
method Explicit construction of edge-indexed graphs.
result Proven existence of groups with specific critical exponents.
Robustifies tree learning algorithms for corrupted data.
problem Learning latent tree structures with corrupted vector observations.
method Presented robustified algorithms using truncated inner product.
result Optimalities of robust CLRG and NJ verified by sample complexities and impossibility results.
Study on inflection points of plane curve shadows with fixed embedded shapes.
problem Minimum number of inflection points in plane curves with fixed embedded shadows.
method Finite coorientation problem on building polygons, dynamic programming, universal lower bound, tree-necklace shadows.
result Exact formula for minimum number of normalized inflections for tree-like shadows.
We study relations between the Alexander-Conway polynomial ∇L and Milnor higher linking numbers of links from the point of view of finite-type (Vassiliev) invariants. We give a formula for the first non-vanishing coefficient of ∇L of an m-component link L all of whose Milnor numbers μi1...ip van…
A new method for automatic gradient tree boosting using information theory.
problem Automatic selection of tree complexity and number in gradient boosting.
method Optimism of greedy leaf splitting procedure modeled as a Cox-Ingersoll-Ross process, leading to an information criterion for model selection.
result The method achieves significant speedups (10-1400) compared to xgboost without sacrificing predictive power.
New findings on diffusion rates in wind-tree model with rational parameters.
problem Understanding diffusion rates in the wind-tree model with rational parameters.
method Analyzing real numbers in [0,1) as diffusion rates and providing a criterion for Lyapunov spectrum.
result Exhibit an infinite family of wind-tree billiards with the interior of the Lyapunov spectrum being the full square (0,1)^2.
Alexander polynomial equals spanning tree count at t=1.
problem Alexander polynomial for spatial graphs.
method Combinatorial constructions generalized to weighted graphs.
result Value of Alexander polynomial at t=1 equals weighted spanning tree count.
The notion of friendliness between trees first appeared in solution of Lando's problem on intersection of polyhedra in 3-space. A tree is friendly to a path graph if edges of the tree can be numbered so that for each k,s the path between the edges k and k+1 contains either both or none of the edges k+2s,k+2s+1. Theorem…
The number of BMW groups on tree products is bounded.
problem Counting and understanding BMW groups on tree products.
method Introduced a random model and used group theory.
result Bounds on the number of commensurability classes of BMW groups.
For a fully irreducible automorphism φof the free group F_k we compute the asymptotics of the intersection number n \mapsto i(T,T'φ^n) for trees T,T' in Outer space. We also obtain qualitative information about the geometry of the Guirardel core for the trees T and T'φ^n for n large.
ForestPrune optimizes tree ensemble pruning for compactness and speed.
problem Large tree ensembles in predictive models consume excessive memory and reduce interpretability.
method Developed a specialized optimization algorithm to efficiently prune tree ensembles by depth layers.
result ForestPrune produces compact, high-performing models that outperform existing post-processing methods.
SMAC method optimizes tree-boosting hyperparameters best.
problem Optimizing hyperparameters for tree-boosting to improve model accuracy.
method Compared and evaluated various hyperparameter optimization methods.
result SMAC method outperforms other methods for hyperparameter tuning.
dtControl uses decision trees to represent controllers efficiently and explainably.
problem Representing controllers concisely and explainably.
method dtControl uses decision tree learning algorithms to represent controllers. Novel techniques for determinizing controllers are introduced.
result Novel techniques for determinizing controllers during decision tree construction are extremely efficient, yielding small decision trees.
Tree prototypes improve tree ensemble interpretability.
problem Making tree ensembles interpretable.
method Introducing prototypes, a new distance for GBTs, and adaptive selection methods.
result Prototypes can perform as well as or better than original tree ensembles when used as nearest-prototype classifiers.
New method improves statistical performance on large output classification problems.
problem Multiclass, multilabel classification with large number of outputs.
method Hierarchical spectral method using tree decomposition and eigenvalue technique.
result Computational efficiency with good statistical performance.
Evolutionary algorithms improve decision tree ensembles.
problem Improving predictive performance of decision trees.
method Real-valued vector representation of decision trees, evolutionary algorithms (Differential evolution, Evolution strategies).
result Proposed methods outperform classical decision tree induction algorithms.
BART's performance improves with more trees, converging to a Gaussian process.
problem Understanding and explaining BART's superior performance in prediction and causal inference.
method Analyzing BART as the number of trees grows towards infinity, showing convergence to a Gaussian process.
result BART converges to a Gaussian process with favorable inferential properties, explaining its excellent performance.
Flexible tree ensemble learning framework supports arbitrary loss functions and multi-task learning.
problem Limited modeling capabilities of existing tree ensemble learning toolkits.
method Differentiable tree ensembles with tensor-based formulation for efficient training.
result Our framework leads to 100x more compact and 23% more expressive tree ensembles.