New inequalities for subGaussian vectors, tighter than before.
problem Improving concentration inequalities for subGaussian random vectors.
method Deriving new concentration inequalities for subGaussian norm random vectors.
result Inequalities are tighter up to logarithmic factors.
The paper studies how norms of random vectors are preserved by random projections.
problem Understanding how random matrix affects norms of random vectors.
method Proved the distribution of the norm of random vector is preserved by random projection.
result Random matrix preserves the distribution of the norm of random vectors with i.i.d. entries.
We develop time-uniform confidence spheres for estimating means of random vectors.
problem Sequential mean estimation in high-dimensional spaces.
method Derive time-uniform confidence sphere sequences (CSSs) for various types of random vectors.
result Optimal CSSs for log-concave, sub-Gaussian, and sub-ψ random vectors. Study improves error bounds for sparse regression with heavy-tailed covariates.
problem Estimating sparse coefficients in linear regression with heavy-tailed covariates.
method Employed an ℓ1-penalized Huber regression method. result Error bound identical to Gaussian case for L-subexponential covariates. Paper explores how adding high-dimensional vectors can memorize and solve set membership problems.
problem Set membership problem in high-dimensional vector spaces.
method Utilizes the almost orthogonal property of high-dimensional random vectors to add them efficiently.
result Efficient probabilistic solution to set membership problem.
Random square-tiled surfaces have normal genus distribution and cover all integer vectors.
problem Distribution and properties of random square-tiled surfaces.
method Randomizing model and local central limit theorem for genus.
result The distribution of the genus is asymptotically normal and contains all primitive integer vectors.
The paper introduces a new method for tail bounds of random vectors and matrices.
problem Estimating norms of random vectors and matrices under moment assumptions.
method Variational tail bounds for norms of random vectors and matrices.
result Dimension-free concentration inequalities for various norms of random vectors and matrices.
New estimator for mean of random vector achieves sub-Gaussian performance.
problem Estimating the mean of a random vector with sub-Gaussian performance.
method Introduces a multivariate median-based estimator under the condition of finite second moment.
result Achieves purely sub-Gaussian performance with only second moment condition.
Stochastic trace estimation with tensor train random vectors
problem Stochastic trace estimation for large-scale matrices
method Gaussian random tensor train vectors
result Median-of-means variant achieves dimension-independent guarantees
Study on points where random spherical harmonic nodal set meets tangent vector field.
problem Distribution of points on nodal sets of random spherical harmonics.
method Analysis of expected counting function and eigenvalue asymptotics.
result Asymptotic behavior of counting function is independent of the vector field.
Optimal dictionaries minimize the average squared error in representing random vectors.
problem Finding optimal dictionaries for minimizing ℓ2-norm of coefficients in random vector representations. method Using rank-1 decompositions of symmetric positive semidefinite matrices, explicit descriptions and polynomial-time algorithms for ℓ2-optimal dictionaries are provided. result Explicit descriptions and polynomial-time algorithms for ℓ2-optimal dictionaries are provided. New method recovers sparse vectors from random sinusoidal features.
problem Recovering sparse vectors from random sinusoidal features.
method Proposes a numerically stable algorithm for sparse vector reconstruction.
result Sparse vectors can be reliably recovered from random sinusoidal features.
Generalizes randomized SVD for better matrix approximations using Gaussian vectors.
problem Computing accurate rank-k approximations of matrices with limited data.
method Extends randomized SVD to multivariate Gaussian vectors, incorporating prior knowledge and using Gaussian processes.
result Demonstrates improved accuracy in approximating matrices and Hilbert-Schmidt operators.
New measures quantify mutual dependence between multiple random vectors.
problem Measuring mutual dependence between multiple random vectors.
method Proposes three measures based on generalized distance covariance.
result Empirical and simplified empirical measures effectively test mutual independence.
The paper generalizes product inequalities for random vectors and their applications.
problem Understanding concentration of measure for products of random vectors.
method Develops expressions for the concentration of functionals of random vectors based on product norms.
result Provides generalized Hanson-Wright inequalities and applications to random matrices.
Paper analyzes error bounds for learning with vector-valued RF, improving existing analyses.
problem Learning with vector-valued random features in infinite-dimensional settings.
method Direct analysis of risk functional, avoiding random matrix theory.
result Strong consistency and minimax optimal convergence rates established.
Unified derivation of stochastic order conditions for elliptical distributions.
problem Classifying multivariate elliptical distributions based on stochastic orders.
method Established an identity for comparing expectations of functions of elliptical vectors and used it to derive conditions for stochastic orders.
result Unified derivation of conditions for various stochastic orders in multivariate elliptical distributions.
New method estimates matrix trace using machine learning with fewer vectors.
problem Estimating matrix trace when explicit form is not known.
method Uses machine learning to determine a small number of probing vectors for matrix multiplication to a vector.
result Precision of trace estimates with 10 probing vectors is similar to 10000 random vectors.
New method calculates tail probabilities of random vectors under linear transformations.
problem Computing tail probabilities of random vectors under linear transformations.
method Characterization of regular variation on cones in [0,∞)d under random linear transformations. result Allows computation of probabilities of tail events that were previously negligible.
Paper studies tensor models using random matrix theory.
problem Analyzing asymmetric order-d spiked tensor models with Gaussian noise.
method Uses variational definition of singular vectors and values, constructs equivalent spiked symmetric block-wise random matrix from tensor contractions.
result Characterizes asymptotic singular values and alignments of singular vectors with true spike components.
The study examines lower and upper bounds of Wasserstein distances for affine transformations of random vectors.
problem Understanding Wasserstein distances for affine transformations of random vectors.
method Lower and upper bounds for affine transformations of random vectors in Rn are derived using Bures metric and compositions of affine maps. result Concrete lower bounds and upper bounds for affine transformations are derived and applied to various distributions.
Market forecasts converge to true values if some agents are correct.
problem Convergence of market forecasts in dynamic prediction markets.
method Dynamic model of prediction market with agents making forecasts.
result Aggregated market forecasts converge to conditional expectations.
Randomized algorithm solves vector-valued regression problems with low-rank operators.
problem Vector-valued regression problems involving infinite-dimensional spaces.
method Randomized Reduced Rank Regression (R4) using Gaussian sketching for optimization.
result R4 estimators are efficient and accurate, with empirical risk close to optimal.
New proof for convex bounds on random vector sums.
problem Proving convex bounds for random vectors with given marginal distributions.
method Using distortion risk measure and expected utility theories.
result Two results on comonotonic and mutually exclusive random vectors are proven.
Optimal transport is #P-hard when components are independent, even with approximate solutions.
problem Computational complexity of optimal transport with independent marginals.
method Proved #P-hardness and developed a pseudo-polynomial time approximation algorithm.
result Optimal transport is #P-hard even with independent components and approximate solutions.
We introduce a new functional measure of tail dependence for weakly dependent (asymptotically independent) random vectors, termed weak tail dependence function. The new measure is defined at the level of copulas and we compute it for several copula families such as the Gaussian copula, copulas of a class of Gaussian mi…
Paper develops metrics for random dynamical systems using vector-valued RKHSs.
problem Creating metrics for random nonlinear dynamical systems.
method Develops metrics on random dynamical systems using Perron-Frobenius operators in vector-valued reproducing kernel Hilbert spaces (vvRKHSs). Uses operator-valued kernels and time-wise independence criteria.
result Extends existing metrics for deterministic systems and introduces kernel maximal mean discrepancy for random processes.
Deep learning representations of GAN data are like Gaussian mixtures, according to this study.
problem Understanding the statistical nature of deep learning representations of GAN-generated data.
method Using Random Matrix Theory, the study shows that DL representations of GAN data are concentrated random vectors that behave like Gaussian mixtures.
result Deep learning representations of GAN data can be fully described by their first two statistical moments.
Random matrix ensembles yield uniform distributions on manifolds.
problem Understanding distributions of vectors in random matrix ensembles.
method Analyzing eigenvalues, singular values, and Autonne-Takagi vectors of various random matrix ensembles.
result Uniform distributions on specific manifolds for different types of random matrix ensembles.
Active covariance estimation using random sub-sampling of variable subsets.
problem Estimating covariance matrices for partially observed random vectors.
method Unbiased covariance estimator under a model of partially observed variables and active learning framework.
result Derivation of error bounds revealing relations between sub-sampling probabilities and covariance matrix entries.
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.
New methods improve accuracy and scalability for large datasets in multi-class classification.
problem Improving accuracy and scalability for multi-class classification with large datasets.
method Randomized block kernel matrices for approximation of least-squares support vector machines.
result The proposed methods provide good accuracy and reliable scaling for multi-class classification problems with large data sets.
We present extremal constructions connected with the property of simplicial collapsibility. (1) For each d≥2, there are collapsible (and shellable) simplicial d-complexes with only one free face. Also, there are non-evasive d-complexes with only two free faces. (Both results are optimal in all dimensions.) (2…
The paper shows vector-valued risk measures ignore dependence structures.
problem Defining capital allocation rules for random vectors with dependence.
method Defined vector-valued risk measures by axioms and showed their properties.
result Vector-valued risk measures ignore dependence structures, unlike set-valued measures.
New tree-structured Markov fields with Poisson marginals for counting variables.
problem Counting variables with complex dependencies.
method Tree-structured Markov random fields with Poisson marginals.
result Straightforward sampling and joint probability calculations.
Reconstructing signature features from randomized vector fields in differential equations.
problem Reconstructing signature features from controlled differential equations with random vector fields.
method Using controlled ordinary differential equations driven by continuous bounded variation curves, the study explores the extent to which signature features can be reconstructed from the non-linear flow of these equations.
result The number of signature features that can be reconstructed from the non-linear flow of controlled ordinary differential equations with random vector fields is exponential in the hidden dimension, under certain conditions.
Study on nodal components of random band-limited functions on surfaces, finding a universal law.
problem Distribution of tangencies of nodal components to a vector field on surfaces.
method Analysis of random band-limited functions on smooth compact Riemannian surfaces with vector fields.
result The distribution of tangencies to a vector field on nodal components of random band-limited functions on surfaces follows a universal deterministic law.
Bayesian approach approximates probability functions of Gaussian mixtures.
problem Approximating probability functions of non-spherical Gaussian mixtures.
method Bayesian decomposition, spherical radial decomposition, random sampling.
result Established differentiability and integral representation of gradient for probability functions.
Study analyzes perturbations in singular subspaces under random noise.
problem Understanding singular vector and subspace changes in signal-plus-noise models.
method Generalized Davis-Kahan-Wedin theorem for any unitarily invariant norm, considering ℓ∞ and ℓ2,∞ bounds. result Fine-grained insights into singular vector and subspace perturbations, including ℓ∞ and ℓ2,∞ bounds. New method reduces deep learning training costs by approximating vector-jacobian products.
problem Efficiently training deep neural networks with reduced computational and memory costs.
method Randomized, unbiased approximations of vector-jacobian products during backpropagation.
result Validated potential for reducing deep learning training costs through unbiased estimates.
Monotone aggregation of dependent random vectors has an absolutely continuous distribution under certain conditions.
problem Monotone aggregation of dependent random vectors
method Coordinatewise monotonicity and uniform lower-increment conditions
result One-dimensional push-forwards of dependent random vectors have an absolutely continuous distribution
A new type of random forest improves robustness against noisy data.
problem Noise in test samples damages random forest performance.
method Introduces denoising autoencoders into random forests to identify and correct incorrect decisions.
result Improves estimation accuracy by considering multiple traversal paths for incorrect nodes.
The Kaczmarz algorithm is popular for iteratively solving an overdetermined system of linear equations. The traditional Kaczmarz algorithm can approximate the solution in few sweeps through the equations but a randomized version of the Kaczmarz algorithm was shown to converge exponentially and independent of number of …
New method estimates Gaussian vector functions more efficiently.
problem Estimating functions of Gaussian vectors with high dimensions.
method Combines randomized dimension reduction and PCA.
result Algorithm outperforms Monte Carlo method by a factor of d.
The paper shows context vectors are half the dimensions of word vectors.
problem Understanding the relationship between word and context embeddings.
method Starting from probabilistic assumptions, the paper shows context vectors are reflections of word vectors in half the dimensions.
result Context vectors are reflections of word vectors in approximately half the dimensions.
New ensemble SVM model reduces prediction error without choosing best kernel.
problem Reducing prediction error in regression problems.
method Bagged-weighted support vector regression model with random machines.
result Regression Random Machines achieve lower generalization error.
This paper analyzes the variability of Concept Activation Vectors (CAVs).
problem The variability of CAVs in explaining AI models.
method Theoretical analysis and experiments on real-life datasets to quantify CAVs variability.
result The variance of CAVs decreases as 1/N, where N is the number of random examples.
New insights into tail behavior of heavy-tailed random vectors and processes.
problem Understanding tail behavior of aggregates of heavy-tailed random vectors.
method Analyzing multivariate regularly varying random vectors and Lévy processes.
result More than one large jump can determine tail behavior of aggregates.