Self-affine arcs without inner weak separation are parabolic segments.
problem Characterizing self-affine Jordan arcs without parabolic segments.
method Analyzing the weak separation property and proving implications for arc types.
result Self-affine Jordan arcs without parabolic segments are attractors of multizippers.
New graph types help identify complex relationships.
problem Understanding complex relationships in data.
method Introducing separable and essentially separable graphs to characterize and identify graphical models.
result Developed algorithms to identify equivalence classes of essentially separable graphs.
Study Poincaré inequality in metric spaces via separating sets.
problem Geometric characterization of Poincaré inequality in metric spaces.
method Properties of separating sets and various notions of energy.
result Equivalence of conditions for 1-Poincaré inequality.
New gaps found in metric curvature.
problem Negative curvature metrics with separated length spectra.
method Topology-based separation of length spectra.
result Exponential gaps in length spectra for negatively curved metrics.
The paper examines subgroup separability for surface and virtual braid groups.
problem Subgroup separability of surface and virtual braid groups.
method Study of subgroup separability (LERF) properties.
result Properties of subgroup separability for surface and virtual braid groups are explored.
New framework for cyclic quantum causal models with graph separation property.
problem Understanding causal relationships in feedback processes and exotic scenarios.
method Introducing a robust probability rule and a novel graph-separation property, p-separation.
result Established graph-separation properties for all consistent cyclic causal models.
Categorical d-separation criterion simplifies probability graph analysis.
problem Detecting causal relationships in probability distributions.
method Introducing categorical definitions for causal models and d-separation.
result Abstract version of d-separation criterion applies to various probability theories.
Develops large-sample theory for non-stationary source separation.
problem Lack of large-sample results for non-stationary source separation methods.
method Large-sample theory for NSS-JD method under specific assumptions.
result Consistency of unmixing estimator and its convergence to Gaussian distribution.
New concept of regular separation for ODEs leads to improved Hardy field results.
problem Understanding solutions of definable ODEs with specific properties.
method Introducing regular separation and proving its implications for ODEs and vector fields.
result The regular separation property leads to improved Hardy field results and non-empty sets of trajectories.
Reservoir computing's success depends on mapping different input time series to separable states.
problem Quantifying the ability of random linear reservoirs to map different input time series.
method Mathematical framework using spectral properties of the connectivity matrix.
result Separation capacity is fully characterized by the spectral properties of the connectivity matrix.
Classifies pro-p PD2 pairs and builds a pro-p curve complex.
problem Classifying pro-p Poincaré duality pairs in dimension two. method Using classification of pro-p PD2 pairs to build a pro-p curve complex. result Established basic properties of the pro-p curve complex. This paper provides a mathematical framework for time-delay reservoir computing.
problem Lack of rigorous mathematical foundations for reservoir computing properties.
method Control-theoretic framework, formal definitions of separation and fading memory, explicit lower bound derivation.
result Established formal definitions and connections to stability notions for time-delay systems.
We prove that the separated curve complex of a closed orientable surface of genus g is (g-3)-connected. We also obtain a connectivity property for a separated curve complex of the open surface that is obtained by removing a finite set from a closed one, but it is then assumed that the removed set is endowed with a part…
Paper develops robust methods for panel data with latent groups, improving inference under group separation violations.
problem Inference in latent group panel models under group separation violations.
method Selective conditional inference approach to derive conditional distribution of coefficients given estimated group structure.
result Valid inference under violations of group separation, superior to traditional asymptotic methods.
Adversarial noises are linearly separable for random neural networks.
problem The challenge of adversarial examples in neural networks.
method Theoretical proof and empirical evidence for two-layer networks with random initialization and neural tangent kernel setup.
result Adversarial noises are linearly separable with corresponding labels.
The complement of a non-separating planar graph contains a K_n minor.
problem Characterizing the structure of complements of planar graphs.
method Analyzing the structure of complements of non-separating planar graphs and using examples to illustrate hypotheses.
result The order 2n-3 is the lowest possible for a non-separating planar graph whose complement contains a K_n minor.
We solve the equivalence problem for the orthogonally separable webs on the three-sphere under the action of the isometry group. This continues a classical project initiated by Olevsky in which he solved the corresponding canonical forms problem. The solution to the equivalence problem together with the results by Olev…
Method separates target signal properties from noisy mixtures.
problem Signal recovery from noisy mixtures with specific statistical properties.
method Statistical component separation method using noise samples and matching statistics.
result Method outperforms standard denoising methods in recovering target signal properties.
Essential self-adjointness proved for perturbed quadharmonic operators on Riemannian manifolds.
problem Proving essential self-adjointness for perturbed quadharmonic operators.
method Using bounded geometry assumptions and a non-positive potential function.
result Essential self-adjointness condition established for perturbed quadharmonic operators.
The paper analyzes condition numbers for logistic regression to understand first-order methods' performance.
problem Understanding the performance of first-order methods in logistic regression.
method Introducing condition numbers to measure non-separability and separability of data.
result Condition numbers inform the properties and convergence guarantees of first-order methods.
Gradient descent converges with arbitrary stepsize for separable data under Fenchel-Young losses.
problem Understanding the conditions under which gradient descent converges with arbitrary stepsize.
method Using Fenchel-Young losses and leveraging the classical perceptron argument to derive convergence rates.
result GD converges with arbitrary stepsize for a majority of Fenchel-Young losses, with better rates for specific loss functions.
PeL separates sensory interface optimization from decision learning.
problem Optimizing sensory interfaces without task-specific information.
method Formal separation of perception and decision learning, using metrics for stability, informativeness, and geometry.
result Updates preserving invariants are orthogonal to decision gradients.
Self-shrinkers in Euclidean space intersect if sufficiently separated at infinity.
problem Intersection of self-shrinkers in Euclidean space.
method Localized Reilly formula applied to f-harmonic function.
result Self-shrinkers intersect if sufficiently separated at infinity.
Universal Bayes consistency proved in metric spaces.
problem Proving universal Bayes consistency in metric spaces.
method Extending a multiclass learning algorithm and proving its Bayes-consistency in all metric spaces.
result First learning algorithm universally strongly Bayes-consistent in all metric spaces.
A new measure DCSI quantifies separability for density-based clustering.
problem Quantifying meaningful clusters in data sets.
method Developed a new separability measure DCSI based on separation and connectedness.
result Correctly identifies touching or overlapping classes that do not correspond to meaningful density-based clusters.
New complex connects graph separability to group properties.
problem Understanding separability of graph fundamental groups.
method Introducing separability complex and proving its properties.
result Separability complex has infinite diameter and is nonhyperbolic.
The separability assumption (Donoho & Stodden, 2003; Arora et al., 2012) turns non-negative matrix factorization (NMF) into a tractable problem. Recently, a new class of provably-correct NMF algorithms have emerged under this assumption. In this paper, we reformulate the separable NMF problem as that of finding the ext…
Confocal quadrics lie at the heart of the system of confocal coordinates (also called elliptic coordinates, after Jacobi). We suggest a discretization which respects two crucial properties of confocal coordinates: separability and all two-dimensional coordinate subnets being isothermic surfaces (that is, allowing a con…
We focus on emergence of the power-law cross-correlations from processes with both short and long term memory properties. In the case of correlated error-terms, the power-law decay of the cross-correlation function comes automatically with the characteristics of separate processes. Bivariate Hurst exponent is then equa…
Proves conditions for separating regions in homogeneous spaces without trivial topology.
problem Separating regions in homogeneous, locally compact spaces without trivial topology.
method Analyzes properties of closed subsets and their boundaries in Čech cohomology.
result Conditions for irreducible separation without trivial topology.
We show linear XOR classification is possible and propose equality separation for anomaly detection.
problem Linearly separating XOR data.
method Equality separation, adapting SVM objective for data within/outside margin.
result Equality separation can detect both seen and unseen anomalies.
Let M(Σ,P) be the mapping class group of a punctured oriented surface (Σ,P) (where P may be empty), and let Tp(Σ,P) be the kernel of the action of M(Σ,P) on H1(Σ∖P,Fp). We prove that $\mathcal T_p(Σ, …
Shallow nonlinear networks can separate classes linearly with polynomially scaling width.
problem Understanding the linear separability of deep networks' features.
method Modeling inputs as a union of low-dimensional subspaces and using random weights and quadratic activations.
result Shallow nonlinear networks can achieve linear separation with polynomially scaling width.
The paper connects decision tree interpretability and robustness through separation.
problem Empirical observation of a connection between robustness and interpretability in decision trees.
method Investigation of the connection through decision trees and l∞-perturbation robustness, proving bounds on tree size. result First algorithm with guarantees on robustness, interpretability, and accuracy for decision trees.
Generalizes underlap coefficient for multivariate group separation.
problem Quantifying distributional separation across groups in statistical learning.
method Generalizes underlap coefficient (UNL) to multivariate variables, establishes key properties, interprets as dependence measure, proposes efficient estimator.
result Highlights the UNL's utility in clustering for evaluating group structure dependence on covariates.
The paper defines Fenchel conjugate and biconjugate on Hadamard manifolds.
problem Defining Fenchel conjugate and biconjugate on curved spaces.
method Introduced a new definition of Fenchel conjugate and biconjugate on Hadamard manifolds based on the tangent bundle.
result Developed a Fenchel-Moreau Theorem for geodesically convex functions on Hadamard manifolds.
Estimates intrinsic dimensionality of biological datasets using Fisher separability.
problem High-dimensional biological datasets with complex structures.
method Fisher separability analysis to estimate intrinsic dimensionality.
result The method performs competitively with state-of-the-art measures and is robust to noise.
New NMF method tackles nonnegative data with separability relaxed.
problem Nonnegative matrix factorization for nonnegative data.
method Generalized separability assumption, convex optimization model, gradient method, heuristic algorithm.
result Effective in synthetic, document, and image data sets.
New techniques reveal subgroup properties in Coxeter groups.
problem Characterizing and understanding subgroups of right-angled Coxeter groups.
method Using cube complexes and Stallings-like techniques to study subgroups.
result Reflection and one-ended subgroups are quasiconvex.
Hierarchical clustering is a popular method for analyzing data which associates a tree to a dataset. Hartigan consistency has been used extensively as a framework to analyze such clustering algorithms from a statistical point of view. Still, as we show in the paper, a tree which is Hartigan consistent with a given dens…
Non-negative matrix factorization (NMF) is a natural model of admixture and is widely used in science and engineering. A plethora of algorithms have been developed to tackle NMF, but due to the non-convex nature of the problem, there is little guarantee on how well these methods work. Recently a surge of research have …
This work presents a novel approach to train invertible linear layers by adding rank-one perturbations.
problem Training invertible linear layers during optimization with gradient-based methods is challenging.
method Train rank-one perturbations and add them to weight matrices infrequently, keeping track of inverses and determinants.
result Invertible linear layers improve mixing and mode separation in normalizing flows.
The paper studies conditions for exact posterior modeling in Bayesian networks.
problem Exact modeling of posterior distributions in Bayesian networks.
method Derives conditions for a recognition network to model the true posterior distribution exactly.
result Perfectness of the recognition network is crucial for local conditions to hold.
We answer a question of Aschenbrenner and Friedl regarding virtual p-efficiency for 3-manifold groups. We then study conjugacy p-separability and prove results for Fuchsian groups, Seifert fibre spaces and graph manifolds.
Develops a new framework for causal models on cyclic graphs, solving unique solvability issues.
problem Challenges in specifying unique probability distributions for cyclic functional causal models.
method Introduces a new probability rule and graph-separation property (p-separation) for cyclic fCMs.
result Proves p-separation is sound and complete for all consistent cyclic fCMs, recovering d-separation for DAGs.
Gradient descent-based adversarial training converges to robust classifiers on linearly separable data.
problem Understanding the inductive bias of adversarial training for robustness.
method Gradient descent on binary classification tasks with linearly separable data, focusing on inductive bias and convergence rates.
result Gradient descent-based adversarial training converges to the maximum margin classifier at a faster rate than clean data training.
Thirty years after the birth of foliations in the 1950's, André Haefliger has introduced a special property satisfied by holonomy pseudogroups of foliations on compact manifolds, called compact generation. Up to now, this is the only general property known about holonomy on compact manifolds. In this article, we give a…
New non-separable covariance kernels for spatiotemporal data derived from harmonic oscillator physics.
problem Capturing complex spatiotemporal dependencies in Gaussian processes.
method Hybrid spectral method based on the harmonic oscillator, deriving explicit covariance kernels.
result Explicit non-separable covariance kernels with space-time interactions.