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.
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.
New bounds on random quadratic forms hold under dependence, useful for adaptive modeling.
problem Need for independence in bounds on random quadratic forms.
method Uniform bounds on random quadratic forms of conditionally independent and sub-Gaussian stochastic processes.
result Bounds hold under general dependencies and sequential design.
MULTIFIT tests independence between two random vectors using multiscale Fisher's test.
problem Detecting local dependence between two random vectors.
method MULTIFIT uses a resampling-free approach to test independence.
result MULTIFIT can easily handle large sample sizes and interpret dependency nature.
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
We propose three measures of mutual dependence between multiple random vectors. All the measures are zero if and only if the random vectors are mutually independent. The first measure generalizes distance covariance from pairwise dependence to mutual dependence, while the other two measures are sums of squared distance…
Gaffke's bound is optimal for a specific parameter ordering in independent random vectors.
problem Finding optimal lower confidence bounds for a scalar parameter in independent random vectors.
method Revisiting classical work on lower confidence bounds, specializing to independent components, and proving optimality with respect to a specific parameter ordering.
result Gaffke's bound is Buehler optimal for the maximum marginal mean parameter.
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…
We show that the error probability of reconstructing kernel matrices from Random Fourier Features for the Gaussian kernel function is at most O(R2/3exp(−D)), where D is the number of random features and R is the diameter of the data domain. We also provide an information-theoretic method-independen…
A concentration graph associated with a random vector is an undirected graph where each vertex corresponds to one random variable in the vector. The absence of an edge between any pair of vertices (or variables) is equivalent to full conditional independence between these two variables given all the other variables. In…
BERET improves binary expansion test for multivariate independence.
problem Testing independence of random vectors in arbitrary dimensions.
method Ensemble approach using sum of squared symmetry statistics and distance correlation.
result Improves power while preserving interpretability.
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
New algorithms learn simple staged trees from data, improving model fit.
problem Complex conditional independences in categorical data vectors.
method Structural learning algorithms for simple staged trees, coalescing the underlying tree.
result Data-learned simple staged trees often outperform Bayesian networks in model fit.
We study the problem of estimating the mean of a random vector X given a sample of N independent, identically distributed points. We introduce a new estimator that achieves a purely sub-Gaussian performance under the only condition that the second moment of X exists. The estimator is based on a novel concept of a…
With increasing concerns about security, the need for highly secure physical biometrics-based authentication systems utilizing \emph{cancelable biometric} technologies is on the rise. Because the problem of cancelable template generation deals with the trade-off between template security and matching performance, many …
In this note, we consider a fixed vector field V on S2 and study the distribution of points which lie on the nodal set (of a random spherical harmonic) where V is also tangent. We show that the expected value of the corresponding counting function is asymptotic to the eigenvalue with a leading coefficient that i…
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.
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.
A method for inferring graph from multivariate time series using ADMM.
problem Inferring conditional independence graph from multivariate Gaussian time series.
method Formulated as multi-attribute graph estimation, used ADMM to minimize penalized negative log-likelihood.
result Proposed method outperforms existing frequency-domain approaches in graph edge detection.
BEGIN network models binary data without parametric assumptions.
problem Conditional independence in non-parametric families of binary data.
method BEGIN network models binary data using sparse linear representations and block factorizations.
result BEGIN network captures conditional independence for arbitrary binary and multinomial variables.
Study predictive performance of linear regression with random functional covariates.
problem Theoretical predictive performance of linear regression with random functional covariates.
method Theoretical analysis of ridge and ridge-less least-squares regression with random functional covariates.
result Probabilistic bounds on predictive excess risk for random functional covariates.
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.
Independent component analysis (ICA) is a method for recovering statistically independent signals from observations of unknown linear combinations of the sources. Some of the most accurate ICA decomposition methods require searching for the inverse transformation which minimizes different approximations of the Mutual I…
Independent component analysis (ICA) is a statistical method for transforming an observable multi-dimensional random vector into components that are as statistically independent as possible from each other. Usually the ICA framework assumes a model according to which the observations are generated (such as a linear tra…
We describe various sets of conditional independence relationships, sufficient for qualitatively comparing non-vanishing squared partial correlations of a Gaussian random vector. These sufficient conditions are satisfied by several graphical Markov models. Rules for comparing degree of association among the vertices of…
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.
A new method for approximating softmax and Gaussian kernels with reduced error.
problem Approximating softmax and Gaussian kernels with low error.
method Simplex Random Features (SimRFs) and SimRFs+.
result SimRFs provide the smallest MSE among weight-independent geometrically-coupled PRF mechanisms.
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 …
We propose a test of independence of two multivariate random vectors, given a sample from the underlying population. Our approach, which we call MINT, is based on the estimation of mutual information, whose decomposition into joint and marginal entropies facilitates the use of recently-developed efficient entropy estim…
The paper studies phase transitions in random matrices and tensor unfolding for detecting signals.
problem Phase transitions in singular values and vectors of large random matrices.
method Analysis of singular values and vectors of long rectangular random matrices, and tensor unfolding algorithm for asymmetric rank-one spiked tensor models.
result An exact threshold for tensor unfolding to detect signals, independent of unfolding procedure.
Independent Component Analysis (ICA) is a statistical tool that decomposes an observed random vector into components that are as statistically independent as possible. ICA over finite fields is a special case of ICA, in which both the observations and the decomposed components take values over a finite alphabet. This p…
Proposes a method to estimate functional graphical models from multivariate random functions.
problem Estimating conditional independence structure of multivariate random functions.
method Neighborhood selection approach combining function-on-function regression and graph recovery.
result Statistical consistency of the method in high-dimensional settings.
A new algorithm FastGM speeds up generating Gumbel-Max variables.
problem Efficiently generating multiple Gumbel-Max variables from high-dimensional vectors.
method FastGM reduces time complexity from O(kn+) to O(klnk+n+) by generating variables in descending order. result Significantly reduces computation time for generating k Gumbel-Max variables. C-MinHash reduces the number of permutations needed for MinHash from thousands to just two.
problem Approximating Jaccard similarity in large binary datasets using many permutations.
method Initial permutation followed by circulant shifting of a second permutation to generate hashes.
result C-MinHash achieves unbiased Jaccard similarity estimation with uniformly smaller variance.
New test for conditional independence using GNNs avoids estimating conditional distributions.
problem Testing conditional independence of X and Y given Z. method Proposes a non-parametric testing procedure using GNNs to sample from marginal conditional distributions.
result Test statistic is doubly robust against GNN approximation errors.
New method detects change-points in population genetics.
problem Identifying homozygosity islands in a population.
method Penalized maximum likelihood approach with dynamic programming and greedy algorithms.
result Consistent detection of homozygosity islands in population genetics.
Suppose that we observe y∈Rn and X∈Rn×m in the following errors-in-variables model: \begin{eqnarray*} y & = & X_0 β^* +ε\\ X & = & X_0 + W, \end{eqnarray*} where X0 is an n×m design matrix with independent subgaussian row vectors, ε∈Rn is a noise vecto…
In Random Forests, proximity distances are a metric representation of data into decision space. By observing how changes in input map to the movement of instances in this space we are able to determine the independent contribution of each feature to the decision-making process. For binary feature vectors, this process …
Estimates mean of random vector with near-optimal error in all directions.
problem Estimating the mean of a random vector with direction-dependent accuracy.
method Proves existence of an estimator with near-optimal error in all directions under certain conditions.
result The estimator satisfies the error bound for all directions, with probability 1-δ.
Paper proposes a differentially private test for joint dependence among random vectors.
problem Detecting joint dependence among sensitive data while maintaining privacy.
method Differentially private permutation methodology for dHSIC test.
result Proposed test attains minimax optimal power across privacy regimes.
Improved guarantees for sparse random embeddings with explicit bounds and empirical superiority.
problem Improving the explicitness and sharpness of guarantees for sparse random embeddings.
method Explicit bounds, tighter estimates for quadratic chaos, extreme properties of sparse linear forms, and improved bounds for sums of independent random variables.
result Significantly outperforms prior works on various real-world datasets.
Study reveals an equivalence principle for the spectrum of random inner-product kernel matrices in polynomial scaling.
problem Understanding the spectrum of random kernel matrices in polynomial scaling regimes.
method Investigates random matrices with nonlinear kernel functions applied to inner products of uniformly distributed vectors.
result The spectrum of the random kernel matrix is asymptotically equivalent to a simpler matrix model through free additive convolution.
A new method tests conditional independence by transforming it into an unconditional problem using transport maps.
problem Testing conditional independence between two random vectors given a third.
method Constructing transport maps to transform conditional independence into unconditional independence, estimating these maps from data using conditional continuous normalizing flow models.
result The proposed method is validated through simulations and real-data analysis, demonstrating practical effectiveness.
Suppose that we observe y∈Rf and X∈Rf×m in the following errors-in-variables model: \begin{eqnarray*} y & = & X_0 β^* + ε\\ X & = & X_0 + W \end{eqnarray*} where X0 is a f×m design matrix with independent subgaussian row vectors, ε∈Rf is a noise vector…
Efficient methods for linear/logistic regression with network-dependent responses.
problem Regression with dependent responses in networked data.
method Projected gradient descent on negative log-likelihood, proving strong convexity and consistency.
result Strong consistency results for vector of coefficients and dependency strength.
Support Vector Data Description (SVDD) is a popular outlier detection technique which constructs a flexible description of the input data. SVDD computation time is high for large training datasets which limits its use in big-data process-monitoring applications. We propose a new iterative sampling-based method for SVDD…
We formulate and analyze a graphical model selection method for inferring the conditional independence graph of a high-dimensional nonstationary Gaussian random process (time series) from a finite-length observation. The observed process samples are assumed uncorrelated over time and having a time-varying marginal dist…
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.