Bayesian model compares ML algorithms on various datasets.
problem Comparing multiple machine learning algorithms across multiple datasets.
method Bayesian Bradley-Terry model, defining regions of practical equivalence (ROPE).
result Allows nuanced statements and ROPE definitions for algorithm comparison.
New insights into pseudo-Anosov flows with special periodic orbits.
problem Understanding pseudo-Anosov flows with periodic orbits in 3-manifolds.
method Analyzing the topological features corresponding to trees of scalloped regions and classifying flows with the same free homotopy data.
result Explicit examples of flows with the same free homotopy data but not orbit equivalent.
Two networks are equivalent if they produce the same output for any given input. In this paper, we study the possibility of transforming a deep neural network to another network with a different number of units or layers, which can be either equivalent, a local exact approximation, or a global linear approximation of t…
Convex iso-Delaunay regions found in flat surface strata.
problem Understanding the geometry of flat surfaces.
method Analyzing triangulations and involutions in strata of translation surfaces.
result Convex iso-Delaunay regions in strata of translation surfaces, especially in hyperelliptic components.
Article addresses practical challenges in conformal prediction.
problem Challenges in determining, computing, and controlling conformal prediction regions.
method Proposes a quadratic-polynomial non-conformity measure.
result Allows circumventing three challenges in full conformal prediction framework.
It is well-known that the expressivity of a neural network depends on its architecture, with deeper networks expressing more complex functions. In the case of networks that compute piecewise linear functions, such as those with ReLU activation, the number of distinct linear regions is a natural measure of expressivity.…
A new complexity measure for neural networks improves upon classical methods.
problem Lack of a refined complexity measure for comparing different neural network architectures, especially permutation-invariant ones.
method Introduced an equivalence relation among linear functions and counted them relative to this relation.
result The new complexity measure clearly distinguishes between different models and increases exponentially with depth.
New models reduce regional inequality by adjusting exchange range and asset distribution bias.
problem Reduction of regional inequality in economic systems.
method Proposed new asset exchange models with spatial exchange range and local support bias to adjust asset distribution and circulation rates.
result Achieved asset distribution from over-concentration to exponential and eventually normal, reducing Gini coefficient.
Region-specific linear models are widely used in practical applications because of their non-linear but highly interpretable model representations. One of the key challenges in their use is non-convexity in simultaneous optimization of regions and region-specific models. This paper proposes novel convex region-specific…
Inter-subject registration of cortical areas is necessary in functional imaging (fMRI) studies for making inferences about equivalent brain function across a population. However, many high-level visual brain areas are defined as peaks of functional contrasts whose cortical position is highly variable. As such, most ali…
It has been shown in \cite{DPSU} that, under some additional assumptions, two simple domains with the same scattering data are equivalent. We show that the simplicity of a region can be read from the metric in the boundary and the scattering data. This lets us extend the results in \cite{DPSU} to regions with the same …
Associating distinct groups of objects (clusters) with contiguous regions of high probability density (high-density clusters), is central to many statistical and machine learning approaches to the classification of unlabelled data. We propose a novel hyperplane classifier for clustering and semi-supervised classificati…
GRANITE unifies feature-based explanation methods to reduce disagreement.
problem Disagreement among feature-based explanation methods.
method GRANITE partitions feature space into regions minimizing interaction and distribution influences.
result Unified and consistent feature explanations.
The paper analyzes and improves a deep learning optimization technique using matrix gradient orthogonality.
problem Improving deep learning training through more effective optimization methods.
method Develops a stochastic non-Euclidean trust-region gradient method for deep learning optimization.
result Proves state-of-the-art convergence results for the proposed algorithm in various scenarios.
New method uses conformalization to create classification regions from ambiguous labels.
problem Creating provable guarantees in classification with uncertain labels.
method Conformal methods applied to credal regions for classification problems.
result New method provides smaller and more disentangled prediction sets.
Paper studies the expressivity of Convolutional Neural Networks (CNNs).
problem Understanding the expressivity of CNNs and their superiority in deep learning.
method Mathematical analysis of linear regions in one-layer and multi-layer ReLU CNNs.
result Deeper CNNs and CNNs have more expressivity per parameter than fully-connected NNs.
Robust MDPs (RMDPs) can be used to compute policies with provable worst-case guarantees in reinforcement learning. The quality and robustness of an RMDP solution are determined by the ambiguity set---the set of plausible transition probabilities---which is usually constructed as a multi-dimensional confidence region. E…
In quantum geometry, we consider a set of loops, a compact orientable surface and a solid compact spatial region, all inside R×R3≡R4, which forms a triple. We want to define an ambient isotopic equivalence relation on such triples, so that we can obtain equivalence invariant…
Optimizes ellipsoids for uncertainty regions in parameter estimation.
problem Learning minimal volume uncertainty ellipsoids for parameter estimation.
method Differentiable optimization approach using neural networks to approximate optimal ellipsoids.
result Approximately computed ellipsoids are smaller and more accurate than existing methods.
Trust-region methods have yielded state-of-the-art results in policy search. A common approach is to use KL-divergence to bound the region of trust resulting in a natural gradient policy update. We show that the natural gradient and trust region optimization are equivalent if we use the natural parameterization of a st…
MD-split+ creates locally valid prediction regions for complex data.
problem Localized prediction regions for complex data.
method Localized model performance-based partitioning of feature space X.
result MD-split+ creates valid prediction regions that scale to high dimensions.
American options can be equivalent to European options under certain conditions.
problem Determining when American options can be simplified to European options.
method Using methods from Jourdain and Martini, Chrsitensen, and convex duality.
result A first step towards verifying representability of American options.
New bounds for Neyman-Pearson region using f-divergences.
problem Bounding the Neyman-Pearson region for hypothesis testing.
method Establishing novel lower and upper bounds using f-divergences. result Best possible lower bound for the Neyman-Pearson boundary using hockey-stick f-divergences. DL models can outperform regionalized models in hydrology by pooling diverse data.
problem Traditional wisdom in hydrology suggests regionalization improves model performance, but DL models can unify data for better performance.
method Used DL models on pooled data from different regions, showing improved performance compared to regionalized models.
result DL models can improve performance by pooling diverse data, highlighting the 'data synergy' effect.
The neural tangent kernel equivalence theorem fails in practice.
problem Does the neural tangent kernel (NTK) equivalence theorem hold in practical neural network training?
method Rigorously derived NTK and conducted numerical experiments to evaluate the equivalence theorem.
result Adding a layer to a neural network and the corresponding updated NTK do not yield matching changes in predictor error.
Two novel search strategies reduce complexity for target localization with size-dependent noise.
problem Target localization with varying measurement noise based on query region size.
method Proposes dyaPM and hiePM strategies with low complexity and connected query geometry. result Unified analysis shows dyaPM asymptotically optimal in search time, hiePM near-optimal in rate. A new method to measure neural network expressiveness using tighter upper bounds.
problem Measuring the expressiveness of deep neural networks (DNNs).
method Proposes a new tighter upper bound for the number of linear regions in rectifier networks, using matrix computation.
result The proposed upper bound is tighter than existing ones and explains the performance improvements of skip connections and residual structures.
Q-learning adapted to find near-equivalent treatment strategies.
problem Finding optimal treatment sequences in dynamic treatment regimes.
method Introducing a worst-value tolerance criterion to find sets of near-equivalent policies.
result Constructs families of near-equivalent treatment strategies.
Unified perspective on natural gradient methods for GMMs, improving variational inference.
problem Efficiently learning multi-modal approximations of complex distributions.
method Comparison and optimization of VIPS and iBayes-GMM methods for Gaussian mixture models.
result Hybrid approach significantly outperforms both VIPS and iBayes-GMM.
This work generalizes bounds on the number of linear regions in CPWL NNs.
problem Determining the number of linear regions in CPWL neural networks is challenging.
method Generalized bounds on the maximal number of linear regions for arbitrary CPWL activation functions.
result Depth significantly increases the number of linear regions, but not exponentially.
The growing conflicts in and about oil exporting regions and speculations about volatile oil prices during the last decade have renewed the public interest in predictions for the near future oil production and consumption. Unfortunately, studies from only 10 years ago, which tried to forecast the oil production during …
The GI0 distribution is able to characterize different regions in monopolarized SAR imagery. It is indexed by three parameters: the number of looks (which can be estimated in the whole image), a scale parameter and a texture parameter. This paper presents a new proposal for feature extraction and region d…
In this paper we consider the problem of minimizing the relative perimeter under a volume constraint in the interior of a convex body, i.e., a compact convex set in Euclidean space with interior points. We shall not impose any regularity assumption on the boundary of the convex set. Amongst other results, we shall prov…
Paper offers a framework for estimating symmetric properties efficiently.
problem Estimating symmetric properties of distributions from samples.
method General framework using profile maximum likelihood (PML) distribution.
result Optimal sample complexity for many properties, practical algorithms.
This study presents new analytic approximations of the stochastic-alpha-beta-rho (SABR) model. Unlike existing studies that focus on the equivalent Black-Scholes (BS) volatility, we instead derive the equivalent constant-elasticity-of-variance (CEV) volatility. Our approach effectively reduces the approximation error i…
New algorithm quantifies uncertainty in regression models for complex data types.
problem Uncertainty quantification in regression models for complex data types.
method Model-free uncertainty quantification algorithm based on conditional depth measures and kernel mean embeddings.
result Provides faster convergence rates and non-asymptotic guarantees for prediction regions.
Model predicts growth competition on curved surfaces.
problem Growth dynamics of two subsets on Riemannian manifolds.
method Modeling growth rates on spherically symmetric Riemannian manifolds.
result Conditions for bounded or unbounded growth on different manifolds.
This paper presents a new approach for filter design based on stochastic distances and tests between distributions. A window is defined around each pixel, overlapping samples are compared and only those which pass a goodness-of-fit test are used to compute the filtered value. The technique is applied to intensity SAR d…
Algorithm compares Legendrian knots efficiently in some cases.
problem Comparing Legendrian knots efficiently.
method Constructs an algorithm to decide knot equivalence, bypassing time-consuming parts when symmetry is known.
result Efficient comparison possible in many cases with known symmetry.
The goal of this paper is to establish the existence of a foliation of the asymptotic region of an asymptotically flat manifold with nonzero mass by surfaces which are critical points of the Willmore functional subject to an area constraint. Equivalently these surfaces are critical points of the Geroch-Hawking mass. Th…
By using numerical simulation, we confirm that Takayasu--Sato--Takayasu (TST) model which leads Pareto's law satisfies the detailed balance under Gibrat's law. In the simulation, we take an exponential tent-shaped function as the growth rate distribution. We also numerically confirm the reflection law equivalent to the…
BALLET filters a high-confidence region of interest for Bayesian optimization.
problem High-dimensional and non-stationary Bayesian optimization challenges.
method Adaptive level-set estimation using two probabilistic models.
result Ballets can efficiently shrink the search space and exhibit tighter regret bounds.
Study local equivalence of Riemannian submersions using differential invariants.
problem Local equivalence problem for Riemannian submersions under fiber-preserving isometries.
method Analysis of differential invariants for orbit submersions induced by a Killing field.
result Explicit formulas for A and H in terms of base data (gˉ,φ,Ω) and equivalence criterion. This study examines the practical equivalence of Laplace and neural tangent kernels.
problem Understanding the practical equivalence of Laplace and neural tangent kernels.
method The study matches the kernels exactly and by matching posteriors of a Gaussian process. It also analyzes the kernels in R^d and experiments with them in regression tasks.
result The Laplace and neural tangent kernels are practically equivalent.
New forms of multi-marginal POT problem derived for computational efficiency.
problem Optimizing transport between multiple unbalanced measures with limited supports.
method Developed two equivalence forms of the POT problem and an optimization algorithm, ApproxMPOT.
result ApproxMPOT algorithm achieves optimal value with complexity ildeO(m3(n+1)m/ε2). Diffusion models' sampling paths lie in a low-dimensional subspace, resembling boomerangs.
problem Understanding the geometric structure of diffusion-based generative models.
method Characterization of deterministic sampling trajectories using low-dimensional subspace and kernel-estimated data modeling.
result Sampling trajectories in diffusion models are confined to a low-dimensional subspace and exhibit a boomerang shape.
Geodesically equivalent Finsler metrics share invariant volume forms and first integrals.
problem Understanding shared properties of geodesically equivalent Finsler metrics.
method Computing first integrals as coefficients of a characteristic polynomial.
result Geodesically invariant functions are first integrals of geodesically equivalent Finsler metrics.
Paper presents an algorithm to check spacetime equivalence.
problem Determining physical equivalence of spacetimes via coordinate transformations.
method Cartan-Karlhede algorithm for checking spacetime equivalence.
result Illustrates the method through simple examples.