Proposes PredVAR model for reduced-dimensional dynamics from noisy data.
problem Extracting low-dimensional dynamics from high-dimensional noisy data.
method Probabilistic reduced-dimensional vector autoregressive model with oblique projection.
result Iterative algorithm yields dynamic latent variables with rank-ordered predictability.
Unified theory and debiasing framework for random oblique projections in high dimensions.
problem Systematic statistical bias in random oblique projections induced by sampling.
method Unified non-asymptotic theory and debiasing framework.
result Sharp bias--variance characterizations and improved approximation accuracy.
Develops an oblique projection technique to approximate a foliation for non-normal dynamics.
problem Modeling dynamics far from a primary Spectral Submanifold (SSM) in non-normal systems.
method Oblique projection technique based on experimental data.
result Approximates a stable invariant foliation for non-normal dynamics efficiently.
Decision forests, including Random Forests and Gradient Boosting Trees, have recently demonstrated state-of-the-art performance in a variety of machine learning settings. Decision forests are typically ensembles of axis-aligned decision trees; that is, trees that split only along feature dimensions. In contrast, many r…
Enhanced ODT with Feature Concatenation boosts learning efficiency.
problem Insufficient learning efficiency of ODT due to linear projections not being transmitted to child nodes.
method Feature Concatenation ( exttt{FC-ODT}) to transmit linear projections along decision paths.
result Experiments show exttt{FC-ODT} outperforms state-of-the-art decision trees with a limited tree depth.
New algorithms improve policy evaluation in reinforcement learning.
problem Off-policy stability and on-policy efficiency issues in policy evaluation.
method Introduced novel algorithms using oblique projection method.
result Demonstrated both off-policy stability and on-policy efficiency.
Efficient oblique RSF method improves prediction and interpretability.
problem Limited computational efficiency and difficulty in interpreting oblique RSF ensembles.
method Newton-Raphson scoring for computational efficiency and negation importance for variable importance estimation.
result The method reduces computational overhead by 450 times and improves prediction accuracy.
Oblique BART improves tree-based predictions.
problem Axis-aligned decision rules in BART can be suboptimal.
method Developed an oblique version of BART using data-adaptive hyperplane partitions.
result Oblique BART outperformed axis-aligned BART and other tree methods on benchmarks.
FoLDTree improves oblique decision trees with ULDA, enhancing accuracy and feature selection.
problem Axis-orthogonal splits limit traditional decision trees' performance on oblique decision boundaries.
method Integrates ULDA into decision tree structure for efficient oblique splits, feature selection, and handling missing values.
result FoLDTree outperforms other methods in accuracy and feature selection, comparable to random forest.
This paper optimizes high-dimensional oblique splits for decision trees, enhancing performance and computational efficiency.
problem Enhancing decision tree performance and computational efficiency in high-dimensional data.
method Established Sufficient Impurity Decrease (SID) convergence for s0-sparse oblique splits, proposing progressive trees for iterative refinement. result Demonstrated that SID function class expands with s0-sparsity, enabling capture of complex data-generating processes. The paper simplifies hedging and portfolio allocation in markets without a risk-free asset.
problem Optimal hedging and portfolio allocation in markets without a risk-free asset.
method Establishes equivalence between hedging with and without numeraire change, uses oblique projections.
result Explicit expressions for optimal strategies and efficient frontier computation.
Decision trees are a popular technique in statistical data classification. They recursively partition the feature space into disjoint sub-regions until each sub-region becomes homogeneous with respect to a particular class. The basic Classification and Regression Tree (CART) algorithm partitions the feature space using…
Sparse oblique decision tree improves security rules for renewable power systems.
problem Identifying secure operating conditions in power systems with high renewable energy.
method Sparse weighted oblique decision tree to learn and embed linear security rules.
result The method significantly increases secure states and reduces solution time.
Proposes oblique predictive clustering trees for faster, more efficient learning.
problem Learning time scales poorly with output space dimensionality and cannot exploit data sparsity.
method Design and implement oblique splits using linear combinations of features.
result Achieves performance on-par with state-of-the-art methods and is orders of magnitude faster.
New tree and forest methods use oblique splits for better risk bounds.
problem Improving risk bounds for regression algorithms.
method Randomized decision trees and forests with oblique splits.
result Oblique splits lead to better risk bounds for multi-index models.
Both neural networks and decision trees are popular machine learning methods and are widely used to solve problems from diverse domains. These two classifiers are commonly used base classifiers in an ensemble framework. In this paper, we first present a new variant of oblique decision tree based on a linear classifier,…
NR retraction approximates geodesics on submanifolds efficiently.
problem Efficiently approximating geodesics on submanifolds for practical algorithms.
method Introducing Newton retraction (NR) as a class of retractions on submanifolds induced by a foliation of the ambient manifold.
result NR is more stable and computationally cheaper than oblique projection, with superlinear convergence regions.
Resource-efficient oblique trees reduce neural signal classification costs.
problem Implementing efficient neural signal classifiers on resource-constrained devices.
method Integrating model compression, probabilistic routing, and cost-aware learning.
result Significant reduction in model size and feature extraction cost compared to state-of-the-art models.
We show how neural models can be used to realize piece-wise constant functions such as decision trees. The proposed architecture, which we call locally constant networks, builds on ReLU networks that are piece-wise linear and hence their associated gradients with respect to the inputs are locally constant. We formally …
Surface area and mean width of a cylinder (the convex hull of two parallel disks) in R^3 are computed. It is more difficult to obtain analogous results for a cone (the convex hull of a disk D and a point p). Oblique formulas for mean width, as well as those for mean curvature, are new. Let L denote the unique diameter …
Chimera state refers to coexistence of coherent and non-coherent phases in identically coupled dynamical units found in various complex dynamical systems. Identification of Chimera, on one hand is essential due to its applicability in various areas including neuroscience, and on other hand is challenging due to its wid…
Paper proposes a novel SVM method for creating survival trees.
problem Creating non-linear survival trees for right-censored data.
method L2-regularized dipole splitting criteria with kernel methods.
result Non-linear splits using polynomial and Gaussian kernels show similar predictive power but often smaller tree sizes.
This paper presents four different ways of looking at the well-known Least Squares Temporal Differences (LSTD) algorithm for computing the value function of a Markov Reward Process, each of them leading to different insights: the operator-theory approach via the Galerkin method, the statistical approach via instrumenta…
New random forest variants achieve optimal performance in high dimensions.
problem Handling dependencies between features in high-dimensional data.
method Using oblique splits in random forests with general split directions.
result Achieved minimax optimal convergence rates in arbitrary dimension.
New method learns nonlinear projections for reduced-order modeling of complex dynamical systems.
problem Modeling transient dynamics near a manifold in nonlinear systems.
method Constrained autoencoder neural networks with invertible activation functions and biorthogonal weight matrices.
result Demonstrated effectiveness on a vortex shedding model, learning oblique fibers for fast dynamics.
Study integrates reliability constraints into generation planning models.
problem Challenges in integrating reliability constraints with generation planning models.
method Leverages a weighted oblique decision tree (WODT) technique to embed reliability verification constraints.
result Demonstrates effectiveness in achieving reliable and optimal planning solutions.
The study calculates the average genus of 2-bridge knots based on their crossing numbers.
problem Determining the average genus of 2-bridge knots with a given crossing number.
method Analytical approach focusing on the properties of 2-bridge knots.
result Obtained the oblique asymptote of the average genus as crossing numbers increase.
Conventional decision trees have a number of favorable properties, including interpretability, a small computational footprint and the ability to learn from little training data. However, they lack a key quality that has helped fuel the deep learning revolution: that of being end-to-end trainable, and to learn from scr…
We introduce canonical correlation forests (CCFs), a new decision tree ensemble method for classification and regression. Individual canonical correlation trees are binary decision trees with hyperplane splits based on local canonical correlation coefficients calculated during training. Unlike axis-aligned alternatives…
A new topology design improves zero-shot classification performance in contrastive learning.
problem Improving zero-shot classification performance in contrastive visual-textual alignment.
method Proposed an alternative topology design using multiple class tokens and an oblique manifold with negative inner product.
result Improves zero-shot classification performance by an average of 6.1%.
NSOTree combines neural networks and trees for better survival analysis interpretability.
problem Survival analysis event prediction with interpretability.
method Proposes NSOTree integrating neural networks and trees for better interpretability.
result NSOTree improves both performance and interpretability in survival analysis.
We present a new way of constructing an ensemble classifier, named the Guided Random Forest (GRAF) in the sequel. GRAF extends the idea of building oblique decision trees with localized partitioning to obtain a global partitioning. We show that global partitioning bridges the gap between decision trees and boosting alg…
Generalizes Hoeffding's decomposition for dependent inputs under mild conditions.
problem Performing global sensitivity analysis on black-box models with dependent inputs.
method Proposes a novel framework based on probability theory, functional analysis, and combinatorics to handle dependencies.
result Any square-integrable, real-valued function of random elements with mild dependence assumptions can be uniquely additively decomposed.
Decision forests (Forests), in particular random forests and gradient boosting trees, have demonstrated state-of-the-art accuracy compared to other methods in many supervised learning scenarios. In particular, Forests dominate other methods in tabular data, that is, when the feature space is unstructured, so that the s…
Adapts Stein's method for geometric inequalities, addressing boundary terms.
problem Geometric inequalities and their stability under constraints.
method Uses elliptic PDE with oblique boundary condition to handle boundary terms.
result Stability results for various geometric inequalities with respect to a new distance.
We study an agent-based stock market model with heterogeneous agents and friction. Our model is based on that of Foellmer-Schweizer(1993): The process of a stock price in a discrete-time framework is determined by temporary equilibria via agents' excess demand functions, and the diffusion approximation approach is appl…
The Mondrian process represents an elegant and powerful approach for space partition modelling. However, as it restricts the partitions to be axis-aligned, its modelling flexibility is limited. In this work, we propose a self-consistent Binary Space Partitioning (BSP)-Tree process to generalize the Mondrian process. Th…
Single tree outperforms random forest in testing accuracy.
problem The challenge of improving single decision tree performance.
method Gradient-based entire tree optimization framework, scaled sigmoid approximation, numerical stability algorithm, subtree polish strategy.
result Optimized single tree outperforms classic random forest by 2.03% on average.
New spectral estimates for minimal surfaces with boundary conditions.
problem Quantifying the Morse index of free boundary minimal surfaces.
method Adapted Montiel-Ros partitioning methods to compact manifolds with boundary, accounting for mixed and group actions.
result Explicit two-sided linear bounds on the Morse index for minimal surfaces.
Study geodesics on neck-degenerate manifolds, focusing and winding behavior observed.
problem Geodesics behavior on neck-degenerate manifolds with cuspidal singularities.
method Detailed multiscale analysis, blow-up techniques.
result Geodesics exhibit focussing and winding behavior as the neck degenerates.
Dimensionality reduction techniques play an essential role in data analytics, signal processing and machine learning. Dimensionality reduction is usually performed in a preprocessing stage that is separate from subsequent data analysis, such as clustering or classification. Finding reduced-dimension representations tha…
Unified theory solves strain compatibility and elasticity of origami metamaterials.
problem Understanding and controlling the morphing paths of origami metamaterials.
method Unified theory for a wide array of origami tessellations, solving strain compatibility and elasticity.
result Origami metamaterials exhibit equal but opposite in-plane and out-of-plane Poisson's ratios and bending energy depends on strain gradient.
In this paper we propose a synergistic melting of neural networks and decision trees (DT) we call neural decision trees (NDT). NDT is an architecture a la decision tree where each splitting node is an independent multilayer perceptron allowing oblique decision functions or arbritrary nonlinear decision function if more…
Model for cross-border markets with limited transmission capacities.
problem Limited transmission capacities between two countries' markets.
method Developed a regime-switching process model with high-frequency approximation.
result Analytic tractability allows computation of key market quantities.
The paper defines projective structures for Lie bialgebras and Poisson-Lie groups.
problem Defining projective analogues of Lie bialgebras and Poisson-Lie groups.
method Introducing projective tensor products and adapting classical notions to these structures.
result Every quasi-triangular projective r-matrix gives rise to a projective Banach Lie bialgebra.
Characterizes minor-minimal separating projective planar graphs and their generalizations.
problem Understanding projective planar graphs and their properties.
method Analyzing minors, embeddings, and specific link types.
result Partial characterization of minor-minimal separating projective planar graphs and their generalizations.
Legendrian knots can be represented by projections with multi-crossings.
problem Representing Legendrian knots with multi-crossings.
method Investigating übercrossing and petal projections in front and Lagrangian projections.
result Legendrian knots with übercrossing projections in front are smoothly isotopic to the unknot.
The study classifies holomorphic projective connections on complex threefolds.
problem Characterizing holomorphic projective connections on complex threefolds.
method Analyzing properties of holomorphic projective connections on complex projective threefolds.
result Holomorphic projective connections on complex threefolds are either flat or translation invariant on abelian threefolds.