We find an upper bound for geodesic distances associated to monotone Riemannian metrics on positive definite matrices and density matrices.
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48 results for “Density matrices”
Quantum system estimation using trace regression models and low rank density matrices.
problem Estimating unknown density matrices from quantum system measurements.
method Using trace regression models and projections onto the convex set of density matrices, with minimax lower bounds for various distances.
result Minimax lower bounds for low rank density matrices are attained up to logarithmic factors for various distances.
The paper establishes minimax bounds for estimating low-rank quantum density matrices.
problem Estimating low-rank quantum density matrices with optimal error rates.
method Developed minimax lower bounds and upper bounds for least squares estimator with von Neumann entropy penalization.
result Sharp upper and lower bounds for various distances (Kullback-Leibler, Hellinger, Schatten p-norm) are attained.
Kernel density matrices simplify probabilistic deep learning.
problem Representing joint probability distributions of continuous and discrete variables.
method Extending density matrices to a reproducing kernel Hilbert space.
result Versatile representation for marginal and joint probability distributions.
The accurate detection of small deviations in given density matrices is important for quantum information processing. Here we propose a new method based on the concept of data mining. We demonstrate that the proposed method can more accurately detect small erroneous deviations in reconstructed density matrices, which c…
Paper introduces a new anomaly detection framework combining density estimation and deep learning.
problem Detecting anomalies in data with varying dimensions.
method Two versions: shallow approach using adaptive Fourier features and density matrices; deep approach using autoencoder.
result Both methods achieve comparable or superior performance compared to state-of-the-art methods.
The Student ensemble of correlation matrices: eigenvalue spectrum and Kullback-Leibler entropyq-fin.ST
We study a new ensemble of random correlation matrices related to multivariate Student (or more generally elliptic) random variables. We establish the exact density of states of empirical correlation matrices that generalizes the Marcenko-Pastur result. The comparison between the theoretical density of states in the St…
New machine learning method detects quantum separability in large-scale systems.
problem Deciding quantum separability of large-scale bipartite density matrices.
method Frank-Wolfe-based algorithm for finding nearest separable density matrices and classification of density matrices as separable or entangled.
result The method scales up to thousands of density matrices and achieves high quantum entanglement detection accuracy.
Learning HMMs with Nonparametric Emissions via Spectral Decompositions of Continuous Matricesstat.ML
Paper develops a spectral algorithm for nonparametric HMMs with smooth emission densities.
problem Estimating hidden Markov models with nonparametric emission densities.
method Spectral decomposition of continuous matrices for nonparametric density estimation.
result Computational efficiency and competitive performance on synthetic and real problems.
We derive the exact form of the eigenvalue spectra of correlation matrices derived from a set of time-shifted, finite Brownian random walks (time-series). These matrices can be seen as random, real, asymmetric matrices with a special structure superimposed due to the time-shift. We demonstrate that the associated eigen…
Estimates quantum system states using Pauli measurements with improved convergence rates.
problem Estimating low rank density matrices of quantum systems.
method Developed Dantzig estimator for Pauli measurements, proving optimal convergence rates in Schatten norms.
result Improved convergence rates for estimating low rank density matrices, including sharp rates in Kullback-Leibler divergence.
InQMAD detects anomalies in streaming data using quantum measurements and density matrices.
problem Detecting anomalies in streaming data with challenges like conceptual drift and continuous learning.
method Incremental anomaly detection based on random Fourier features and quantum measurements.
result InQMAD outperforms 12 state-of-the-art methods in a systematic evaluation.
Method reduces categorical data to lower dimensions using density matrices.
problem Dimensionality reduction for categorical data.
method Density-matrix construction from class-conditional frequencies; spectral embedding.
result Low-dimensional spectral embeddings with controlled rank.
Estimates spectral density of large implicit matrices efficiently.
problem Estimating eigenvalues of large implicit matrices efficiently.
method Combines randomized estimation techniques to construct unbiased estimators.
result Validated methods on large-scale problems in graph theory and random matrix theory.
Paper develops a method for estimating spectral density matrices in high-dimensional time series.
problem Estimating spectral density matrices in high-dimensional time series.
method Thresholded versions of averaged periodograms for regularized estimation.
result Consistent estimation of spectral density matrices possible under high-dimensional regime.
The paper explores Cholesky decompositions for symmetric matrices and their geometric properties.
problem Understanding the structure and properties of symmetric matrices through Cholesky decompositions.
method Introducing cones of symmetric matrices, proving Cholesky-type factorizations, and showing geometric properties.
result Each symmetric matrix admits an uncountable family of Cholesky-type factorizations, and these cones are isometric Riemannian manifolds.
Proposes a new method for kernel density estimation using stagewise minimization and a simple dictionary.
problem Kernel density estimation with data-adaptive weighting parameters and sparse representation.
method Stagewise minimization algorithm based on -divergence and a simple dictionary.
result Develops non-asymptotic error bound for the proposed estimator.
New method reduces density estimation variance for multivariate data.
problem Efficient multivariate density estimation with reduced dimensionality.
method Variance-Reduced Sketching (VRS) framework for multivariate density estimation.
result VRS framework significantly improves density estimation over existing methods.
The Riemannian Bures metric on the space of (normalized) complex positive matrices is used for parameter estimation of mixed quantum states based on repeated measurements just as the Fisher information in classical statistics. It appears also in the concept of purifications of mixed states in quantum physics. Here we d…
Quantum method improves neural density estimation in high dimensions.
problem High-dimensional density estimation with poor performance and high computational complexity.
method Adaptive Fourier features based on quantum density matrices, integrated with neural networks.
result Competitive performance compared to state-of-the-art methods in various datasets.
Bayesian inference on orthogonal matrices using Givens representation.
problem Posterior inference in models with orthogonal matrix parameters.
method Givens representation for posterior inference over the Stiefel manifold.
result Effective methods for transforming densities over the Stiefel manifold into Euclidean space.
Study spectral density of neural networks using resolvent method.
problem Investigate spectral density of neural networks with random feature matrices.
method Use resolvent method from random matrix theory, cumulant expansion.
result Impossible to preserve singular value distribution with additive bias.
Incorporates matrix exponential into generative flows for improved performance.
problem Improving generative flow models for better density estimation.
method Integrates matrix exponential into generative flows, proposing new layers and modifying network architecture.
result The proposed model achieves great performance on density estimation.
We apply random matrix theory to derive spectral density of large sample covariance matrices generated by multivariate VMA(q), VAR(q) and VARMA(q1,q2) processes. In particular, we consider a limit where the number of random variables N and the number of consecutive time measurements T are large but the ratio N/T is fix…
A non-Hermitean extension of paradigmatic Wishart random matrices is introduced to set up a theoretical framework for statistical analysis of (real, complex and real quaternion) stochastic time series representing two "remote" complex systems. The first paper in a series provides a detailed spectral theory of non-Hermi…
We analyze the spectral properties of correlation matrices between distinct statistical systems. Such matrices are intrinsically non symmetric, and lend themselves to extend the spectral analyses usually performed on standard Pearson correlation matrices to the realm of complex eigenvalues. We employ some recent random…
The Sinkhorn-Knopp algorithm converges quickly but the number of iterations is poorly understood.
problem Understanding the number of iterations required for the Sinkhorn-Knopp algorithm to converge.
method Analyzing the Sinkhorn-Knopp algorithm for matrices with a specific density threshold.
result The Sinkhorn-Knopp algorithm requires iterations for matrices with density .
Spectral densities of Wishart-Levy free stable random matrices: Analytical results and Monte Carlo validationcond-mat.stat-mech
Random matrix theory is used to assess the significance of weak correlations and is well established for Gaussian statistics. However, many complex systems, with stock markets as a prominent example, exhibit statistics with power-law tails, that can be modelled with Levy stable distributions. We review comprehensively …
ButterflyFlow uses butterfly matrices for efficient invertible layers in normalizing flows.
problem Building efficient invertible layers for complex probability distributions.
method Proposes butterfly layers for invertible linear layers, leveraging their ability to capture complex structures.
result ButterflyFlow achieves strong density estimation and significantly better log-likelihoods on various datasets.
Algorithm reduces online PCA regret to O(d^(3/2)√T).
problem Online PCA with limited feedback.
method Proposes an algorithm for online PCA with bandit feedback.
result Achieves regret of O(d^(3/2)√T) after T rounds.
Two new methods estimate quantum density matrices using machine learning.
problem Estimating the quantum density matrix for complex systems.
method Quantum Maximum Likelihood and Quantum Variational Inference with quantum flows.
result Improved estimation of quantum density matrices for mixed states.
Learning with symmetric positive definite matrices via generalized Bures-Wasserstein geometrymath.FA
Paper introduces a generalized Bures-Wasserstein geometry for SPD matrices.
problem Understanding the geometry of SPD matrices for machine learning.
method Proposes a generalized Bures-Wasserstein geometry parameterized by a symmetric positive definite matrix.
result The GBW geometry outperforms the BW geometry in machine learning applications.
New geometric framework for positive semidefinite matrices of fixed rank.
problem Statistical analysis of positive semidefinite matrices of fixed rank.
method Introducing a manifold with Riemannian geometry and Lie group structure.
result Analytical closed forms for geodesics and Fréchet means.
We introduce a new method to handle permutations efficiently using variational inference.
problem Efficient probabilistic reasoning about permutations in high-dimensional spaces.
method We reparameterize the Birkhoff polytope to enable variational inference over permutations.
result Our method enables efficient and accurate Bayesian inference over permutations.
B-NAF is a more compact flow for density estimation and inference.
problem Efficiently modeling complex density functions with fewer parameters.
method Directly models a bijection using a single feed-forward network with block matrices.
result B-NAF uses orders of magnitude fewer parameters while being competitive.
New trust matrix quantifies breakdowns in deep neural networks.
problem Understanding trust breakdowns in deep learning models.
method Introduces trust matrix and conditional trust densities to analyze deep neural networks.
result Trust matrices reveal areas needing improvement for deep neural networks.
New GMM models fit high-dimensional data with fewer parameters.
problem Overparameterization and lack of flexibility in GMMs for high-dimensional data.
method Piecewise-constant covariance eigenvalue profiles, EM and penalized EM algorithms.
result Superior likelihood-parsimony tradeoffs in density fitting, clustering, and denoising.
Study connects spectral clustering to maximum margin and level set estimation.
problem Connecting spectral clustering to maximum margin and level set estimation.
method Obtained bounds on eigenvectors of graph Laplacian matrices in terms of cluster separation and connectivity. Showed sensitivity mitigation by removing outliers and estimating level sets.
result Spectral clustering converges to maximum margin clustering as scaling parameter approaches zero.
In compressed sensing problems, minimization or Basis Pursuit was known to have the best provable phase transition performance of recoverable sparsity among polynomial-time algorithms. It is of great theoretical and practical interest to find alternative polynomial-time algorithms which perform better than $\e…
We develop underdamped diffusion bridges for sampling from unnormalized densities.
problem Sampling from unnormalized densities without direct access to samples.
method Underdamped diffusion bridges with rigorous score matching equivalence.
result State-of-the-art performance in sampling across various problems.
Model credit risk with non-stationary correlations using random matrices.
problem Estimating non-stationary asset correlations for credit risk modeling.
method Random matrix ensemble to model non-stationary correlations, averaging over an ensemble of correlation matrices.
result Explicit results show heavy tails prevail over diversification benefits even with small correlations.
Graph clustering method uses templates to match vertices and outperforms classical methods.
problem Graph clustering with additional structural information.
method Formulates graph clustering as template matching, using orthonormal matrices for embedding.
result Method outperforms classical methods, especially for challenging cases.
New method estimates large matrices' spectra from small sub-matrices.
problem Estimating large matrices' spectra when full matrix-vector products are not available.
method Free decompression based on free probability theory.
result Estimates eigenspectrum of impalpable matrices from small sub-matrices.
We compute, using a formula of Dittmann, the Bures metric tensor (g) for the eight-dimensional convex set of three-level quantum systems, employing a newly-developed Euler angle-based parameterization of the 3 x 3 density matrices. Most of the individual metric elements (g_{ij}) are found to be expressible in relativel…
Deep learning speeds spectral density estimation for large 2D/3D grids.
problem Computational challenges in estimating spectral densities for large grids.
method Deep learning neural network for spectral density estimation.
result Deep learning estimator is a universal approximator and faster than existing methods.
The salient properties of large empirical covariance and correlation matrices are studied for three datasets of size 54, 55 and 330. The covariance is defined as a simple cross product of the returns, with weights that decay logarithmically slowly. The key general properties of the covariance matrices are the following…
The paper connects Riemannian Gaussian distributions to random matrix theory and diffusion kernels.
problem Analyzing Riemannian Gaussian distributions on symmetric spaces.
method Analytical computation of marginals using orthogonal and skew orthogonal polynomials, and diffusion kernels.
result Riemannian Gaussian distributions are random matrix types, and their probability density functions can be computed analytically.
Dynamical Isometry is Achieved in Residual Networks in a Universal Way for any Activation Functionstat.ML
ResNets achieve dynamical isometry regardless of activation function, simplifying learning process.
problem Achieving consistent learning performance in ResNets across different activation functions.
method Deriving a universal formula for spectral density using Free Probability and Random Matrix Theory.
result Dynamical isometry is achieved in ResNets for any activation function, simplifying learning.