Finite resources limit false discovery rate control in structured hypothesis spaces.
problem Controlling false discovery rate in hypothesis testing with finite data and structured hypothesis spaces.
method Framework for exact FDR control and adaptive power maximization.
result Exact FDR control and adaptive power maximization.
New method controls false discoveries in structured hypothesis spaces.
problem Controlling false discoveries in large-scale, interconnected hypothesis spaces.
method Reproducing Kernel Hilbert Space (RKHS) optimization for structured FDR control.
result Unified framework for continuous domains, graphs, and hierarchies.
The paper refutes the manifold hypothesis for image data and proposes the union of manifolds hypothesis.
problem The manifold hypothesis fails to capture the structure of image data.
method Empirical verification of the union of manifolds hypothesis on image datasets.
result Image data lies on a disconnected set with varying intrinsic dimensions.
Paper relaxes assumptions for non-parametric estimation in pairwise learning.
problem Generalization performance of non-parametric estimation for pairwise learning.
method Significantly relaxes restrictive assumptions, constructs structured deep ReLU neural network, and designs targeted hypothesis space.
result Establishes a sharp oracle inequality for empirical minimizer with general hypothesis space for Lipschitz continuous pairwise losses.
Extends linear representation hypothesis to categorical and hierarchical concepts in LLMs.
problem Representing concepts without natural contrasts in large language models.
method Formalizes linear representation hypothesis for categorical and hierarchical concepts, proving relationships between concept hierarchy and representation geometry.
result Validated theoretical results on large language models, estimating representations for 900+ concepts.
Scientific discovery is limited by hypothesis redundancy, and hybrid methods can exploit non-local exploration.
problem Limitation of scientific discovery due to hypothesis redundancy.
method Hybrid discovery systems combining structured local search with LLM-generated non-local proposals.
result Hybrid methods can exploit non-local exploration when three geometric conditions co-occur.
Imputation method respects manifold structure for missing data.
problem Missing data imputation in high-dimensional data.
method Model-based imputation using mixture variational autoencoders and sampling-importance-resampling (SIR).
result Competitive performance and uncertainty quantification in imputations.
Proposes a new model for testing causal structural priors and synthesizing data.
problem Testing and synthesizing causal structural priors using nonparametric knowledge and neural networks.
method Causal Structural Hypothesis Testing (C-SHT) and Causal Structural Variational Hypothesis Testing (C-SVHT) using deep neural networks.
result Demonstrates out-of-distribution generalization error as a proxy for causal structural prior hypothesis testing.
Selecting appropriate regularization coefficients is critical to performance with respect to regularized empirical risk minimization problems. Existing theoretical approaches attempt to determine the coefficients in order for regularized empirical objectives to be upper-bounds of true objectives, uniformly over a hypot…
VAELLS learns latent manifold structure to improve VAE model accuracy.
problem VAEs struggle with mismatched latent structure and global structure.
method Integrates learnable manifold model into latent space of VAE.
result Improves model accuracy by matching prior to data manifold structure.
Given a surface of infinite topological type, there are several Teichmüller spaces associated with it, depending on the basepoint and on the point of view that one uses to compare different complex structures. This paper is about the comparison between the quasiconformal Teichmüller space and the length-spectrum Teichm…
The paper uses RL to verify hypotheses, overcoming existing limitations.
problem Verifying hypotheses using reinforcement learning.
method Formulated hypothesis verification as an RL problem, exploiting hypothesis structure.
result RL agents can successfully verify hypotheses, even those not factorizable.
Simple model explains manifold structure in high-dimensional data.
problem Understanding manifold structure in high-dimensional data.
method Latent Metric Model with latent variables, correlation, and stationarity.
result Establishes statistical explanation for manifold hypothesis.
This paper proves the manifold hypothesis for lower embedding dimensions using osculating hyperspheres.
problem The dataset lies on a low-dimensional submanifold in high-dimensional space.
method Constructing osculating hyperspheres and applying surgery theory to embed the hypersurface.
result The manifold hypothesis holds for embedding dimensionalities up to d−1. Given two étale groupoids $\Cal G$ and $\Cal G'$, we consider the set of pointed morphisms from $\Cal G$ to $\Cal G'$. Under suitable hypothesis we introduce on this set a structure of Banach manifold which can be considered as the space of objects of an étale groupoid whose space of orbits is the space of morphisms fr…
Study risk bounds for distributed ERM with general loss functions and hypothesis spaces.
problem Limited theoretical analysis for distributed ERM with general loss functions and hypothesis spaces.
method Derive tight risk bounds under assumptions on hypothesis space and loss function.
result Developed more general risk bound for distributed ERM without strong convexity restriction.
Collider regression improves predictive performance in regression tasks.
problem Discarding prior causal knowledge in regression tasks.
method Collider regression framework incorporating probabilistic causal knowledge from collider structures.
result Proves positive generalization benefit and provides closed-form estimators.
Paper addresses hypothesis space misspecification in learning from human demonstrations and corrections.
problem Hypothesis space misspecification in learning from human demonstrations and corrections.
method Reason explicitly about how well the robot can explain human inputs given its hypothesis space.
result Demonstrates method on a 7 DOF robot manipulator.
We clarify measurability assumptions in the agnostic PAC learning theorem.
problem Measurability assumptions in the Fundamental Theorem of Statistical Learning.
method Measure-theoretic scrutiny of existing proofs to extract minimal assumptions.
result Sound statement and detailed proof of the Fundamental Theorem in the agnostic setting.
Estimates curvature of network manifolds to understand community structure.
problem Understanding the geometry of network models to infer community structure.
method Develops hypothesis tests to determine manifold type, dimension, and curvature from noisy distance matrices.
result Consistently estimates manifold type, dimension, and curvature from Riemannian manifolds of constant curvature.
The Greenberg-Shalom hypothesis connects subgroup properties to lattice structures in Lie groups.
problem Understanding subgroup properties in Lie groups and their implications.
method Analyzing infinite discrete subgroups of semisimple Lie groups and their commensurators.
result An infinite discrete subgroup of a semisimple Lie group with a dense commensurator is a lattice in a product of some factors.
The paper defines a hypothesis space for deep learning using DNNs.
problem Developing a mathematical framework for deep learning.
method Introducing a Banach space of functions of input variables based on DNNs, proving it's a RKBS, and establishing representer theorems for learning models.
result Solutions to learning problems can be expressed as finite sums of kernel expansions based on training data.
This work constructs a hypothesis test for detecting whether an data-generating function h:Rp→R belongs to a specific reproducing kernel Hilbert space H0 , where the structure of H0 is only partially known. Utilizing the theory of reproducing kernels, we reduce this hypothesis …
In this paper we address the problem of pool based active learning, and provide an algorithm, called UPAL, that works by minimizing the unbiased estimator of the risk of a hypothesis in a given hypothesis space. For the space of linear classifiers and the squared loss we show that UPAL is equivalent to an exponentially…
We consider, in the Euclidean setting, a conformal Yamabe-type equation related to a potential generalization of the classical constant scalar curvature problem and which naturally arises in the study of Ricci solitons structures. We prove existence and nonexistence results, focusing on the radial case, under some gene…
A test for sparsity in Bayesian networks helps choose algorithms.
problem Selecting appropriate structure discovery algorithms for Bayesian networks.
method Developed a hypothesis test using the largest eigenvalue of the normalized inverse covariance matrix.
result The hypothesis test can determine if a BN has max in-degree greater than 1.
We show that the number of unique function mappings in a neural network hypothesis space is inversely proportional to ∏lUl!, where Ul is the number of neurons in the hidden layer l.
This paper introduces localized discrepancy theories for unsupervised domain adaptation.
problem Improving generalization bounds for unsupervised domain adaptation.
method Localized discrepancies defined on the hypothesis space after localization, leading to smaller and asymmetric values.
result Improved generalization bounds and sample complexity reduction.
s-RBFN integrates multiple hypotheses for efficient and diverse prediction.
problem Integrating multiple hypotheses into learning models for regression.
method Structured Radial Basis Function Network (s-RBFN) using Voronoi tessellations and least-squares training.
result s-RBFN achieves superior generalization and efficiency compared to other models.
This paper develops a new theory for ensemble learning beyond variance reduction.
problem Ensemble learning's effectiveness for stable estimators is not fully explained by variance reduction.
method Develops a general weighting theory for ensemble learning, formalizing ensembles as linear operators and introducing geometric and spectral constraints.
result Structured weights can outperform uniform averaging by reshaping approximation geometry and redistributing spectral complexity.
We show that the disagreement coefficient of certain smooth hypothesis classes is O(m), where m is the dimension of the hypothesis space, thereby answering a question posed in \cite{friedman09}.
While statistical learning methods have proved powerful tools for predictive modeling, the black-box nature of the models they produce can severely limit their interpretability and the ability to conduct formal inference. However, the natural structure of ensemble learners like bagged trees and random forests has been …
Cellular Electron CryoTomography (CECT) is a 3D imaging technique that captures information about the structure and spatial organization of macromolecular complexes within single cells, in near-native state and at sub-molecular resolution. Although template matching is often used to locate macromolecules in a CECT imag…
This letter presents a novel Block Bayesian Hypothesis Testing Algorithm (Block-BHTA) for reconstructing block sparse signals with unknown block structures. The Block-BHTA comprises the detection and recovery of the supports, and the estimation of the amplitudes of the block sparse signal. The support detection and rec…
Generative models can approximate high-dimensional data from lower dimensions without needing a latent dimension equal to or greater than the data's intrinsic dimension.
problem Theoretical limitations on the latent dimension required for generative models to approximate high-dimensional data distributions.
method Inspired by space-filling curves, the work demonstrates that generative networks can approximate distributions on d-dimensional manifolds from inputs of any arbitrary dimension, even lower than d. result Generative models can approximate high-dimensional data distributions from lower-dimensional inputs without needing a latent dimension equal to or greater than the data's intrinsic dimension.
Improved algorithm for selecting a hypothesis locally privately with fewer queries.
problem Locally private hypothesis selection with minimal privacy-preserving queries.
method Introduces a Scheffé graph to reduce query complexity for hypothesis selection.
result Algorithm performs O~(k3/2) queries, improving on previous methods. New method estimates causal effects in complex spaces using topological structures.
problem Challenges in estimating causal effects in non-Euclidean spaces.
method Developed a topological causal inference framework using power-weighted silhouette functions of persistence diagrams.
result Successfully quantifies topological treatment effects across various complex outcomes.
Study on metrics with singularities on spheres, showing moduli space structure.
problem Constant Q-curvature metrics on spheres with singular points.
method Analysis of moduli space, Gromov-Hausdorff topology, symplectic structure construction.
result Moduli space structure is a real analytic variety with formal dimension equal to the number of punctures.
New algorithms for multitask learning with long-term memory.
problem Learning from tasks partitioned into unknown segments with associated hypotheses.
method Online multitask learning algorithms exploiting segmentation and hypothesis association.
result Regret bounds and efficient algorithms for various hypothesis classes.
We develop a novel computationally efficient and general framework for robust hypothesis testing. The new framework features a new way to construct uncertainty sets under the null and the alternative distributions, which are sets centered around the empirical distribution defined via Wasserstein metric, thus our approa…
In this paper, we consider a supervised learning setting where side knowledge is provided about the labels of unlabeled examples. The side knowledge has the effect of reducing the hypothesis space, leading to tighter generalization bounds, and thus possibly better generalization. We consider several types of side knowl…
Learning robot objective functions from human input has become increasingly important, but state-of-the-art techniques assume that the human's desired objective lies within the robot's hypothesis space. When this is not true, even methods that keep track of uncertainty over the objective fail because they reason about …
Paper hypothesizes MLP layers in LLMs can be approximated by sparse Mixture of Experts.
problem Understanding dense MLP layers in LLMs.
method Theoretical connection between MoE models and SAE structure in activation space.
result MLP layers in LLMs can be well approximated by sparse Mixture of Experts.
Unified framework for large-scale hypothesis testing with confounders.
problem Bias in large-scale hypothesis testing due to unmeasured confounders.
method Unified statistical estimation and inference framework that disentangles confounding effects and jointly estimates latent and primary effects.
result Effective Type-I error control and power in hypothesis testing.
We propose a methodology for testing linear hypothesis in high-dimensional linear models. The proposed test does not impose any restriction on the size of the model, i.e. model sparsity or the loading vector representing the hypothesis. Providing asymptotically valid methods for testing general linear functions of the …
To reduce the label complexity in Agnostic Active Learning (A^2 algorithm), volume-splitting splits the hypothesis edges to reduce the Vapnik-Chervonenkis (VC) dimension in version space. However, the effectiveness of volume-splitting critically depends on the initial hypothesis and this problem is also known as target…
This article develops a framework for testing general hypothesis in high-dimensional models where the number of variables may far exceed the number of observations. Existing literature has considered less than a handful of hypotheses, such as testing individual coordinates of the model parameter. However, the problem o…
The paper analyzes the generalization of deep neural networks for metric and similarity learning.
problem Lack of rigorous understanding of generalization performance in metric and similarity learning.
method Derive explicit form of true metric, construct structured deep ReLU neural network, establish excess risk bounds.
result Explicit excess risk bounds for metric and similarity learning are derived.