JME continually estimates data moments privately and accurately.
problem Private and accurate continual estimation of data moments.
method Uses matrix mechanism and joint sensitivity analysis.
result Improves accuracy in estimating mean and covariance with reduced noise.
Study on Gaussian ensemble of matrix products with mixed moments computed.
problem Understanding the statistical properties of matrix products of Gaussian matrices.
method Analysis of a multi-Wishart ensemble and enumeration of non-crossing pairings.
result Mixed moments of the product matrix are computed and found to be weighted by Fuss-Catalan numbers at large N. Classifies SL(n) covariant matrix-valued valuations on Lp-spaces.
problem Classifying SL(n) covariant matrix-valued valuations on Lp-spaces.
method Established a complete classification for continuous and SL(n) covariant matrix-valued valuations on Lp(Rn,|x|2dx), eliminating matrix symmetry assumption.
result Unique characterization of such valuations by the moment matrix in n>2, rotation matrix in 2D.
Spectral features of the empirical moment matrix constitute a resourceful tool for unveiling properties of a cloud of points, among which, density, support and latent structures. It is already well known that the empirical moment matrix encodes a great deal of subtle attributes of the underlying measure. Starting from …
Classification of SL(n) covariant valuations on Orlicz spaces.
problem Classifying continuous SL(n) covariant valuations on Orlicz spaces.
method Complete classification without symmetric assumptions, focusing on moment matrix and a new functional in dimension two.
result The moment matrix is the only SL(n) covariant valuation for n≥3, and a new functional appears in dimension two.
MOMENT selects and estimates mixed-effects models using moment identities.
problem Selecting and estimating random-effects covariance matrix and fixed-effects coefficients in multiresponse linear mixed-effects models.
method MOMENT is a stage-wise moment-based framework that reduces the random-effects selection problem to a smooth constrained convex optimization problem.
result MOMENT performs competitively and can outperform separate univariate analyses for correlated responses.
Developed moment estimators for affine stochastic volatility models.
problem Estimating parameters of affine stochastic volatility models.
method Introduced recursive equations for moments and proposed moment estimators.
result Established a central limit theorem and derived asymptotic covariance matrix.
This paper deals with the explicit design of strategy formulations to make the best strategic choices from a conventional matrix form of representing strategic choices. The explicit strategy formulation is an analytical model which is targeted to provide a mathematical strategy framework to find the best moment for str…
The paper analyzes stability of random matrix products with Markovian noise.
problem Analyzing stability of random matrix products with Markovian noise.
method Using a super-Lyapunov drift condition and controlled growth of matrix-valued functions, the paper provides an exponential stability result for the p-th moment of random matrix product.
result Finite-time p-th moment bounds for linear stochastic approximation and TD learning algorithms.
Enhances DP linear regression using public data moments.
problem Limited utility of traditional DP methods in linear regression.
method Transform private data using public second-moment matrix for a better OLSE.
result Improved accuracy and robustness of OLSE in DP linear regression.
Study resolvent convergence for random matrices with general covariance profiles.
problem Analyzing resolvent convergence for random matrices with non-identically distributed columns.
method Using moments of quadratic forms and deterministic equivalents, the study provides bounds on the trace of matrix products.
result The trace of matrix products is close to the trace of a deterministic equivalent, controlled by matrix norms.
PMT uses public data moments to make DP feasible for unbounded data.
problem Applying differential privacy to unbounded data distributions.
method Public-moment-guided Truncation (PMT) using second-moments from public data.
result PMT improves the accuracy and stability of DP models.
Independent component analysis (ICA) is the problem of efficiently recovering a matrix A∈Rn×n from i.i.d. observations of X=AS where S∈Rn is a random vector with mutually independent coordinates. This problem has been intensively studied, but all existing efficient algorithms w…
Paper provides unbiased spectral moment estimates from finite data.
problem Challenges in estimating spectral moments from limited data.
method Dynamic programming approach to estimate spectral moments of kernel integral operator.
result Demonstrates consistency with theoretical spectra and practical utility in neural networks.
A new metric assesses latent variable models using data and model moments.
problem Difficulty in assessing the quality of unsupervised learning models.
method A moment-matching metric using matrix norms to compare data and model moments.
result The proposed metric is faster and has less variance than alternative methods.
Independent Component Analysis (ICA) is the problem of learning a square matrix A, given samples of X=AS, where S is a random vector with independent coordinates. Most existing algorithms are provably efficient only when each Si has finite and moderately valued fourth moment. However, there are practical appli…
Study on eigenvalue distribution of correlated time series, showing deformation of Marchenko-Pastur distribution.
problem Eigenvalue distribution of Wishart matrix with temporal correlation.
method Analysis of moments and convergence to deformed Marchenko-Pastur distribution for Gaussian process with temporal correlation.
result Eigenvalue distribution converges to deformed Marchenko-Pastur distribution with longer tail and higher peak.
The paper tackles matrix completion in ultra-sparse sampling, improving imputation accuracy.
problem Matrix completion in ultra-sparse sampling, where each row has only a few entries.
method Estimate row span of matrix or averaged second-moment matrix, normalize and impute missing entries.
result Gradient descent method normalizes and imputes missing entries, achieving low variance and unbiased estimation.
Iteratively reweighted least squares (IRLS) is a widely-used method in machine learning to estimate the parameters in the generalised linear models. In particular, IRLS for L1 minimisation under the linear model provides a closed-form solution in each step, which is a simple multiplication between the inverse of the we…
Spectral learning extends matrix methods to tensors for better latent variable modeling.
problem Limitations of matrix-based spectral methods in capturing non-Gaussian data.
method Extend spectral decomposition to tensor-based methods for higher-order moments.
result Tensor decomposition can identify latent effects missed by matrix methods.
The paper explores tail diversification in financial markets using entropy and mutual information.
problem Tail diversification in financial time series.
method Statistical independence through differential entropy and mutual information, using moments as contrast functions.
result Tail covariance matrix is a key driver of tail diversification.
We prove optimal subspace embedding conjecture up to sub-polylogarithmic factors.
problem Optimal dimension and sparsity of subspace embeddings.
method Iterative decoupling technique to analyze higher-order trace moment bounds.
result Sub-polylogarithmic factors in dimension and sparsity of subspace embeddings.
Proposes a method to identify elements in a skewness matrix for multivariate skew-elliptical distributions.
problem Label switching issue in Bayesian estimation of skewness matrix.
method Imposes a positive lower-triangular constraint and uses Bayesian sparse estimation with horseshoe prior.
result Successfully estimates the true structure of skewness dependency.
In this paper, we study the confounder detection problem in the linear model, where the target variable Y is predicted using its n potential causes Xn=(x1,...,xn)T. Based on an assumption of rotation invariant generating process of the model, recent study shows that the spectral measure induced by the regress…
Corrected whitening restores orthogonality in high-dimensional spherical Gaussian mixtures.
problem In high-dimensional data, standard whitening fails to preserve orthogonality of mixture means.
method Derived exact limits for whitened means dot products using random matrix theory, constructed a corrected whitening matrix.
result Corrected whitening allows for improved estimation of spherical Gaussian mixtures in the large-dimensional regime.
A new portfolio optimization method using the Sherman-Morrison identity.
problem Portfolio optimization with covariance and variance.
method Sherman-Morrison identity applied to replace covariance with second moment matrix.
result Sherman-Morrison-Markowitz portfolio solves standard portfolio optimization problems.
Enhances ROM simulation for multivariate systems with exact Kollo skewness.
problem Modeling multivariate systems with high dimensions and specific higher moments.
method Extends Random Orthogonal Matrix simulation to match target Kollo skewness.
result Established conditions and developed a general approach for constructing admissible values.
Nonnegative matrix factorization (NMF) has been widely used in machine learning and signal processing because of its non-subtractive, part-based property which enhances interpretability. It is often assumed that the latent dimensionality (or the number of components) is given. Despite the large amount of algorithms des…
We study the problem of learning a distribution from samples, when the underlying distribution is a mixture of product distributions over discrete domains. This problem is motivated by several practical applications such as crowd-sourcing, recommendation systems, and learning Boolean functions. The existing solutions e…
This work develops efficient methods for computing moments of Gaussian mixtures.
problem Efficient computation of moments for Gaussian mixtures with large dimensions.
method Theory and numerical methods for implicit computations with moment tensors of Gaussian mixtures.
result Reduced computational and storage costs for moment tensors of Gaussian mixtures.
Time homogeneous polynomial processes are Markov processes whose moments can be calculated easily through matrix exponentials. In this work, we develop a notion of time inhomogeneous polynomial processes where the coeffiecients of the process may depend on time. A full characterization of this model class is given by m…
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.
DGMM improves Gaussian mixture modeling efficiency and stability.
problem Efficiently estimating Gaussian mixtures in high dimensions.
method Diagonally-weighted generalized method of moments (DGMM).
result DGMM achieves smaller estimation errors with shorter runtime.
Estimates rank-one spikes from heavy-tailed noise using self-avoiding walks.
problem Estimating rank-one spikes from heavy-tailed noise.
method Self-avoiding walks to count and estimate the spikes.
result Optimal estimation up to the BBP threshold for heavy-tailed noise.
Efficiently estimates mean of symmetric distributions without moments.
problem Estimating mean of symmetric distributions without moment assumptions.
method Generalization of filtering technique, Huber-loss-based techniques, SoS proofs.
result Achieves optimal error bounds for various symmetric distributions.
Consider a random vector with finite second moments. If its precision matrix is an M-matrix, then all partial correlations are non-negative. If that random vector is additionally Gaussian, the corresponding Markov random field (GMRF) is called attractive. We study estimation of M-matrices taking the role of inverse sec…
New method estimates log-determinant using trace powers, avoiding classical limitations.
problem Estimating log-determinant of large matrices efficiently and accurately.
method Interpolating moment-generating function and its derivative at zero using trace powers.
result No continuous estimator using finite moments can be uniformly accurate over unbounded conditioning.
In the setting of polynomial jump-diffusion dynamics, we provide an explicit formula for computing correlators, namely, cross-moments of the process at different time points along its path. The formula appears as a linear combination of exponentials of the generator matrix, extending the well-known moment formula for p…
We study the column subset selection problem with respect to the entrywise ℓ1-norm loss. It is known that in the worst case, to obtain a good rank-k approximation to a matrix, one needs an arbitrarily large nΩ(1) number of columns to obtain a (1+ε)-approximation to the best entrywise ℓ1-norm low ra…
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.
The asymptotic distribution of the Markowitz portfolio is derived, for the general case (assuming fourth moments of returns exist), and for the case of multivariate normal returns. The derivation allows for inference which is robust to heteroskedasticity and autocorrelation of moments up to order four. As a side effect…
Robustly estimates linear regression coefficients with adversarial and noisy data.
problem Estimating robust linear regression coefficients with adversarial and noisy data.
method Adversarial robust weighted Huber regression with polynomial computational complexity.
result Derives an estimation error bound that depends on the stable rank and condition number of the covariance matrix.
Corrected moment-based methods improve inference in topic model regression.
problem Inferential difficulties in topic model plug-in workflow for regression.
method Corrected spectral moment methods for LDA, response-weighted word moments.
result Direct identification of regression coefficients without estimating topic shares.
The learning of domain-invariant representations in the context of domain adaptation with neural networks is considered. We propose a new regularization method that minimizes the discrepancy between domain-specific latent feature representations directly in the hidden activation space. Although some standard distributi…
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.
Second-order optimization speeds up deep hedging for complex options.
problem Hedging exotic options with market frictions in realistic markets.
method Second-order optimization scheme leveraging pathwise differentiability and Kronecker-factoring.
result Our method optimizes the policy in 1/4 the steps of standard optimization.
A new neural network initialization method is proposed for faster and more accurate training.
problem Efficient initialization for training multi-layer feedforward neural networks.
method Initialization based on Stein's identity, using eigenvectors of cross-moment matrix.
result The SteinGLM method is faster and more accurate than other initialization methods.
Gradient descent on Hadamard manifolds converges to boundary points, solving optimization problems.
problem Optimization on Hadamard manifolds with unbounded convex functions.
method Gradient descent, duality theorem, moment-weight inequality.
result Gradient descent converges to boundary points, solving optimization problems.