Study finds saddle connections on random surfaces follow Poisson distribution.
problem Distribution of saddle connections on random translation surfaces.
method Analysis of saddle connections on surfaces of large genus.
result Number of saddle connections in given lengths converges to Poisson distribution.
Randomized neural networks use fixed connections for efficiency.
problem Efficiency in deep learning models.
method Fixed connections in neural networks.
result Deep randomized neural networks achieve state-of-the-art results.
New bounds for convex clustering under graph connectivity.
problem Understanding clustering performance under different graph connectivity structures.
method Random walks and concentration inequalities for random graph models.
result Improved rates of convergence for centroid recovery.
Shared classical randomness improves quantum generative models' output distributions.
problem Improving generative performance of shallow unitary quantum models.
method Introducing stochasticity into unitary quantum models via shared classical randomness.
result Shared classical randomness allows shallow unitary quantum models to represent a strictly larger family of distributions.
We consider the structure learning problem for graphical models that we call loosely connected Markov random fields, in which the number of short paths between any pair of nodes is small, and present a new conditional independence test based algorithm for learning the underlying graph structure. The novel maximization …
Study on connectivity and geometry of random Coxeter groups.
problem Connectivity threshold for square percolation on random graphs.
method Probabilistic combinatorics and techniques from geometric group theory.
result Determines connectivity threshold and cubical coarse median structure for random Coxeter groups.
Estimates intrinsic dimension without distances, outperforming other methods.
problem Estimating intrinsic dimension for efficient data processing.
method Uses binary adjacency matrices based on a random connection model.
result Asymptotic distribution and rate of convergence specified.
New network learns non-parametric invariances from data.
problem Modeling non-parametric invariances in data.
method Introduces PRC-NPTN networks with permanent random connectomes.
result Improves generalization and outperforms existing methods.
New framework models neural systems with random architecture on manifolds.
problem Complex, uncertain systems with non-Gaussian outputs.
method Latent random field on compact manifold generates neural architecture and weights.
result Synthetic neural systems can produce stochastic outputs for deterministic inputs.
A model studies deep neural networks with binary synapses under connection removal.
problem Understanding the mechanism of deep learning from a theoretical perspective.
method Random active path model with diluted binary synapses under removal perturbation.
result A critical value of perturbation separates spin glass and paramagnetic phases, with the latter having poor generalization performance.
The paper studies random Čech complexes on Riemannian manifolds and phase transitions in homological connectivity.
problem Understanding homological connectivity in random Čech complexes on Riemannian manifolds.
method Study of a random Čech complex generated by a homogeneous Poisson process in a compact Riemannian manifold M, focusing on phase transitions for homological connectivity.
result The homology of the complex becomes isomorphic to that of M at a phase transition.
The paper connects bundle curvature to random zero currents.
problem Understanding the relationship between bundle curvature and random zero currents.
method Heat flow on Hermitian line bundles over Riemannian manifolds.
result Random zero currents connect bundle curvature to ground state zero current.
There has been significant interest in the use of fully-connected graphical models and deep-structured graphical models for the purpose of structured inference. However, fully-connected and deep-structured graphical models have been largely explored independently, leaving the unification of these two concepts ripe for …
Randomized graph construction ensures giant component with fewer edges.
problem Efficiently constructing sparse graphs with good connectivity.
method Randomly connecting points to a subset of their nearest neighbors.
result A sparser graph with comparable connectivity properties.
This work employs some techniques in order to filter random noise from the information provided by minimum spanning trees obtained from the correlation matrices of international stock market indices prior to and during times of crisis. The first technique establishes a threshold above which connections are considered a…
Study random walks on sub-Riemannian manifolds using retractions.
problem Modeling random walks on sub-Riemannian manifolds.
method Use retractions to approximate normal geodesics and study convergence to Brownian motion.
result Convergence of geodesic random walks defined with different connections.
Investment diversification affects financial stability, depending on network connectivity.
problem Analyzing stability of financial networks with diversified portfolios.
method Random matrix dynamical model with portfolio rebalancing, considering heterogeneity and diversification effects.
result Stability/instability transition depends on the largest eigenvalue of the random matrix.
We prove that the Euler form of a metric connection on real oriented vector bundle E over a compact oriented manifold M can be identified, as a current, with the expectation of the random current defined by the zero-locus of a certain random section of the bundle. We also explain how to reconstruct probabilisticall…
A simplified model shows how wealth distribution can be derived from random exchanges.
problem Understanding wealth inequality and its distribution over time.
method Stylized random exchange model, Markov chain, discrete and continuous stochastic processes, Boltzmann-type kinetic equations.
result Existence of equilibrium distribution in the stylized model.
Sharp threshold for simple connectivity in random 2-complexes.
problem Simple connectivity of random 2-dimensional simplicial complexes.
method Poisson paradigm and Tutte's enumeration of planar triangulations.
result Sharp threshold probability for simple connectivity is p=(γn)−1/2, where γ=44/33. Generative model connects random walk vertices to form networks, tractable for estimation and inference.
problem Modeling network formation with explicit dependence on graph structure.
method Generative model using random walks, maximum likelihood estimation, MCMC for history imputation.
result Model parameters can be recovered from a single graph generated by the model.
Gradient descent finds global minima in deep neural networks with skip-connections.
problem Finding global minima in deep neural networks with skip-connections.
method Analysis of the gradient descent algorithm in over-parametrized deep neural networks with skip-connections.
result Gradient descent can find global minima exponentially fast in the over-parametrized regime.
A new model combines diffusion and random features for better interpretability and comparable performance.
problem Lack of theoretical justification and computational expense in diffusion models, and limited interpretability in random feature models.
method Developed a deep random feature model inspired by diffusion models, derived generalization bounds using score matching.
result The model achieves comparable performance to fully connected neural networks and provides theoretical generalization bounds.
This work estimates edge weights of edge-reinforced random walks using observed data.
problem Statistical estimation of edge weights in edge-reinforced random walks.
method Proposes an estimator based on the generalized method of moments using the magic formula and hyperbolic Gaussian structure.
result Analyzes the sample complexity of the proposed estimator.
Study evaluates neural networks based on random graph structures and finds key performance indicators.
problem Understanding and optimizing neural network architectures using graph theory.
method Evaluation of neural networks with random graph structures, focusing on structural and numerical properties.
result A new numerical graph characteristic selects a set of quasi-1-dimensional graphs that perform well.
New model calculates logarithmic surface diameter.
problem Calculating diameter of random hyperbolic surfaces.
method Exploration process inspired by graph breadth-first search.
result Diameter is logarithmic in surface genus.
Deep GMRFs improve spatial data modeling and prediction.
problem Modeling spatial dependencies in data.
method Established connection between GMRFs and CNNs, allowing for multi-layer architectures.
result Deep GMRFs outperform state-of-the-art models in satellite temperature prediction.
The study connects Kleinian group divergence to random walk recurrence.
problem Understanding the recurrence of random walks on Schreier graphs of Kleinian groups.
method Connecting growth rates of orbits, volume, and Schreier graphs.
result Constructing Kleinian groups of divergence type.
Random forests improve probability estimates through kernel regression.
problem Improving the principled approach to random forest probability estimation.
method Forge a connection between random forests and kernel regression, develop a proximity kernel model.
result Improves statistical footing of random forest probability estimation.
Overview of high-dimensional dynamical systems and their applications to machine learning.
problem Characterizing behavior of high-dimensional dynamical systems driven by random matrices.
method Cavity method arguments, path integrals, dynamical mean field theory (DMFT), and random matrix resolvents.
result Connections between random matrix resolvents and DMFT response, and non-monotonic loss curves in training.
Proposes a link between randomness and compression in deep learning.
problem Improving efficiency in deep learning training.
method Introduces a novel tomographic compression framework called Dual Tomographic Compression (DTC).
result Demonstrates high correlation between learning performance and Gibbs entropy over compression ratios.
We prove that a Gaussian ensemble of smooth random sections of a real vector bundle over compact manifold canonically defines a metric on the bundle together with a connection compatible with it. Additionally, we prove a refined Gauss-Bonnet-Chern theorem stating that if the bundle and the manifold are oriented, then t…
New method improves statistical interpolation for analyzing complex random structures.
problem Analyzing atypical random structures in statistical models.
method Introduces a large deviation upgrade to fully lifted interpolation.
result Allows for easier analysis of atypical random structures.
New approach to Yang-Mills measure on surfaces via Morse theory.
problem Constructing the Yang-Mills measure on compact Riemannian surfaces.
method Morse theoretical approach and resolution of random cohomological equations.
result Definition and computation of the Yang-Mills measure and its partition function.
Topology helps estimate chromatic numbers of random graphs on spheres.
problem Estimating chromatic numbers of random graphs on spheres.
method Topology, specifically connectivity of Lóvasz's neighborhood complex.
result Connectivity bound is useful in dimensions 1 and 2, but generally poor.
HARFE approximates sparse additive functions using random features and ridge regression.
problem Approximating high-dimensional sparse additive functions.
method Hard-ridge random feature expansion with sparse ridge regression and hard-thresholding pursuit.
result HARFE method converges with a given error bound and achieves lower error than other algorithms.
New complexity measures explain overparameterized models' surprising performance.
problem Understanding why overparameterized models generalize well despite fitting training data.
method Reinterpreting classical degrees of freedom in a random-X setting.
result Random-X prediction error better explains generalization in complex models.
Proposes FairRR to improve fairness in machine learning models through randomized response.
problem Achieving group fairness in machine learning models.
method Formulates group fairness as optimizing a design matrix in Randomized Response, proposing FairRR.
result Demonstrates FairRR yields excellent model utility and fairness.
Random projections enhance neural networks by reducing dimensions and speeding up training.
problem Training and expressive power of neural networks with high-dimensional inputs.
method Random projections to embed sparse vectors or low-dimensional manifolds into a smaller space, reducing the number of parameters and speeding up training.
result The number of neurons required for approximating a function depends on sparsity or manifold dimension, not the input vector dimension.
Study on spectral clustering phase transitions in noisy networks.
problem Detecting communities in noisy networks.
method Proved phase transitions using Erdos-Renyi random noise model.
result Critical external edge connection probability for community detectability.
IDPGs extend RDPGs with a Poisson process for random latent positions.
problem Modeling randomness in latent positions for graph structure.
method Introduce IDPGs using Poisson point processes on latent Euclidean space.
result Continuous analogues of adjacency matrices link latent structure to observed graphs.
In this paper we de ne conditional random elds in reproducing kernel Hilbert spaces and show connections to Gaussian Process classi cation. More speci cally, we prove decomposition results for undirected graphical models and we give constructions for kernels. Finally we present e cient means of solving the optimization…
Method quantifies uncertainties in complex MRF models.
problem Uncertainties in MRF predictions due to data, modeling, and approximations.
method Information-based uncertainty quantification using MRF graphical structure.
result Tight bounds on predictions for quantities of interest in MRFs.
Graph connection Laplacian (GCL) is a modern data analysis technique that is starting to be applied for the analysis of high dimensional and massive datasets. Motivated by this technique, we study matrices that are akin to the ones appearing in the null case of GCL, i.e the case where there is no structure in the datas…
The article studies knot distributions in petal diagrams and proves probabilities of specific knot types decay as the number of petals increases.
problem Understanding the probability of specific knot types in petal diagrams as the number of petals grows.
method Established properties of the randomized knot model, proving probabilities decay to zero, improved bounds on crossing number and petal number relationships.
result The n-petal model represents at least exponentially many distinct knots, with probabilities of specific knot types decaying as the number of petals increases.
RF models implicitly regularize kernel methods as feature count increases.
problem Understanding implicit regularization in RF models.
method Random matrix theory applied to Gaussian RF models and KRR.
result The average RF predictor is close to a KRR predictor with an effective ridge.
Random convolutional networks can be fooled with adversarial examples.
problem Existence of adversarial examples for random convolutional networks.
method Utilizing isoperimetric inequalities on the special orthogonal group so(d). result Adversarial examples exist for various random convolutional networks.
Study shows a central limit theorem for random coverings of manifolds with nilpotent groups.
problem Understanding the distribution of connected components in random coverings of manifolds with nilpotent fundamental groups.
method Used sampling homomorphisms from the fundamental group into the symmetric group and subgroup growth zeta functions of nilpotent groups.
result Proved a central limit theorem for the number of connected components of these random coverings.