New flag area measures constructed via integration over differential forms.
problem Constructing new flag area measures in high-dimensional spaces.
method Using local parallel sets and integration over the normal cycle of differential forms.
result Flag area measures span the space of all smooth SO(n)-covariant flag area measures.
New operators generalize Rankin-Cohen brackets for differential forms.
problem Extending classical operators to differential forms.
method Constructing conformally covariant bi-differential operators for differential forms.
result Generalized operators for differential forms are constructed.
SGDm with fixed step-size diverges under covariate shift, similar to a parametric oscillator.
problem SGDm with fixed step-size diverges under covariate shift.
method Approximated learning system as a time-varying system of ODEs and characterized divergence/convergence modes.
result SGDm with fixed step-size can diverge under covariate shift, similar to resonance in oscillators.
DGCP uses deep neural networks to optimize hyperparameters for Gaussian processes.
problem Optimizing hyperparameters for Gaussian processes in non-uniform input spaces.
method DGCP uses a deep neural network to learn hyperparameters of non-stationary covariance functions.
result DGCP improves Gaussian process predictions in locally adapted or sparse input spaces.
We sketch our recent application of a non-commutative version of the Cartan `moving-frame' formalism to the quantum Euclidean space RqN, the space which is covariant under the action of the quantum group SOq(N). For each of the two covariant differential calculi over RqN based on the R-matrix formalism, we…
Predict covariance from features using convex optimization.
problem Predicting the covariance of a Gaussian vector from another feature vector.
method A generalized linear model with convex optimization for fitting parameters.
result Predicted covariance matrices are symmetric positive definite.
NGD improves multivariate Gaussian inference by optimizing Fisher information.
problem Efficiently optimizing multivariate Gaussian models.
method Natural Gradient Descent applied to multivariate Gaussian parameters.
result NGD updates are more efficient for symmetric covariance matrices.
Reduces high dimensions to improve covariate shift handling.
problem Improving performance of importance weighting methods in high dimensions.
method Optimizes over linear subspaces combining predictive utility and train/test divergence.
result Extreme dimension reduction helps sometimes but not always due to bias.
PACE-GGM uses Gaussian mechanism for private covariance estimation.
problem Private estimation of covariance matrices in high dimensions.
method Data-adaptive selection of entries, Gaussian mechanism, maximum-entropy reconstruction.
result Consistent improvements in estimation error compared to Gaussian mechanism and baselines.
Diagonal transformations preserve independence structures in non-Gaussian distributions.
problem Preserving independence structures in non-Gaussian distributions.
method Diagonal nonlinear transformations of multivariate normal variables.
result Independence structures are preserved in non-Gaussian distributions under diagonal transformations.
We present the FuSSO, a functional analogue to the LASSO, that efficiently finds a sparse set of functional input covariates to regress a real-valued response against. The FuSSO does so in a semi-parametric fashion, making no parametric assumptions about the nature of input functional covariates and assuming a linear f…
The paper shows objective derivatives are covariant derivatives on Riemannian metrics.
problem The definition and interpretation of objective derivatives in continuum mechanics.
method Demonstrates that objective derivatives correspond to covariant derivatives on the manifold of Riemannian metrics.
result Objective derivatives are unified as covariant derivatives on the manifold of Riemannian metrics.
Proves subgaussian distributions are SoS-certifiably subgaussian, enabling efficient algorithms for various statistical tasks.
problem Efficiently learning from subgaussian distributions in high dimensions.
method Universal constant C and polynomial sum of squares (SoS) approach. result Proves subgaussian distributions are SoS-certifiably subgaussian.
The conformally covariant split system generates non-constant mean curvature vacuum initial data.
problem Creating non-constant mean curvature vacuum initial data for the Einstein equations.
method Proved existence of solutions to the conformally covariant split system on compact 3-manifolds using the implicit function theorem.
result The conformally covariant split system provides non-constant mean curvature vacuum initial data for the Einstein equations.
In this work we construct an optimal linear shrinkage estimator for the covariance matrix in high dimensions. The recent results from the random matrix theory allow us to find the asymptotic deterministic equivalents of the optimal shrinkage intensities and estimate them consistently. The developed distribution-free es…
New approach for semi-supervised learning under covariate shifts.
problem Semi-supervised learning under covariate shifts where labeled and unlabeled data distributions differ.
method Information-theoretical approach, addressing covariate shifts.
result Improved performance compared to previous methods.
Classifies curvature measures and valuations in Euclidean spaces.
problem Characterizing valuations and curvature measures in Euclidean spaces.
method Classification of curvature measures and valuations using differential forms and representation theory.
result Complete classification of curvature measures and valuations with specific invariance properties.
New criterion improves predictive evaluation in weighted inference scenarios.
problem Improving predictive evaluation in scenarios with different likelihoods for estimation and evaluation.
method Developed the posterior covariance information criterion (PCIC) to handle weighted likelihood inference.
result PCIC is asymptotically unbiased for quasi-Bayesian generalization error in weighted inference.
Anisotropic connections and parallel transport defined in Finsler spacetimes.
problem Defining and characterizing anisotropic connections and parallel transport in Finsler spacetimes.
method Introducing a new covariant derivative and parallel transport, identifying vertically trivial Finsler connections with anisotropic connections, and characterizing the Levi-Civita-Chern anisotropic connection.
result Characterization of the Levi-Civita-Chern anisotropic connection as the one preserving the length of parallely propagated vectors.
Courant algebroid connections help describe string theory equations.
problem Describing equations of motion in string theory.
method Connection on Courant algebroid with curvature tensor.
result Curvature tensor describes string theory equations.
New method prevents posterior collapse in iVAE models.
problem Posterior collapse in iVAE models where observations and ICs are independent given covariates.
method Developed CI-iVAE by considering a mixture of encoder and posterior distributions in the objective function.
result Prevents posterior collapse, resulting in latent representations with more information of the observations.
Bayesian learning from variable-length sequences using Gaussian processes with signature covariances.
problem Learning from sequences of varying lengths and complex sequential structures.
method Gaussian processes with signature kernels, sparse variational approach, combining with LSTM/GRU models.
result Effective learning from sequences of different lengths and complex structures.
The paper calibrates shrinkage covariance estimators for spectral functionals in high dimensions.
problem Calibrating shrinkage covariance estimators for spectral functionals in high dimensions.
method Derives first-order null laws, distribution-free Davis-Kahan bands, and calibrated tests for spectral functionals under shrinkage.
result Calibrated tests and intervals for spectral functionals are provided, addressing the issue of estimation noise and shrinkage bias.
We study the design of portfolios under a minimum risk criterion. The performance of the optimized portfolio relies on the accuracy of the estimated covariance matrix of the portfolio asset returns. For large portfolios, the number of available market returns is often of similar order to the number of assets, so that t…
We study the problem of estimating from data, a sparse approximation to the inverse covariance matrix. Estimating a sparsity constrained inverse covariance matrix is a key component in Gaussian graphical model learning, but one that is numerically very challenging. We address this challenge by developing a new adaptive…
We give a simple explicit algorithm for building multi-factor risk models. It dramatically reduces the number of or altogether eliminates the risk factors for which the factor covariance matrix needs to be computed. This is achieved via a nested "Russian-doll" embedding: the factor covariance matrix itself is modeled v…
Study precise sample covariance error for Gaussian centered data.
problem Precise characterization of sample covariance error for Gaussian data.
method Developed a Random Duality Theory (RDT) framework to determine upper and lower bounds.
result Upper and lower bounds match in large-dimensional contexts, matching the spectral norm's limiting value.
Missing data estimation is an important challenge with high-dimensional data arranged in the form of a matrix. Typically this data matrix is transposable, meaning that either the rows, columns or both can be treated as features. To model transposable data, we present a modification of the matrix-variate normal, the mea…
Paper proposes a new method for sparse covariance Cholesky factor estimation.
problem Estimating sparse covariance matrices for ordered data.
method Matrix loss penalization approach for sparse Cholesky factor estimation.
result The proposed method outperforms existing regression-based approaches in simulations and real data.
The study improves generalization in large-batch training by adding structured covariance noise to gradients.
problem Improving generalization in large-batch training while maintaining optimal convergence.
method Adding covariance noise to the gradients to improve generalization performance.
result The method improves generalization performance without degrading optimization performance and training duration.
In this paper, we investigate community detection in networks in the presence of node covariates. In many instances, covariates and networks individually only give a partial view of the cluster structure. One needs to jointly infer the full cluster structure by considering both. In statistics, an emerging body of work …
Paper proposes a generalized precision matrix for t-Student distributions to improve portfolio optimization.
problem Limitations of inverse covariance matrix in non-Gaussian settings.
method Exploits local dependence function to define generalized precision matrix (GPM) for multivariate t-Student distribution.
result GPM leads to statistically significant lower out-of-sample variances in minimum-variance portfolios.
FVNNs use graph convolutions on fair covariance estimates to improve fairness in machine learning.
problem Data-driven methods can encode biases in sample covariance matrices, leading to unfair treatment of different subpopulations.
method FVNNs perform graph convolutions on fair covariance estimates and use a fairness regularizer in the loss function.
result FVNNs provide a flexible model that is intrinsically fairer than PCA approaches and can handle low sample regimes.
A new method uses nearest neighbors for importance weighting.
problem Data covariate shift problems in machine learning.
method Nearest neighbor classification scheme for determining importance weights.
result Demonstrated effectiveness through comparative experiments on various classification tasks.
The paper proposes a new method for covariate balancing using IPM to improve causal inference.
problem Covariate imbalance in causal inference weighting methods, especially when models are not correctly specified.
method The integral probability metric (IPM) is used to determine optimal weights for treated and control groups.
result The proposed method can be consistent without specifying either the propensity score or outcome regression model.
In the high-dimensional regression model a response variable is linearly related to p covariates, but the sample size n is smaller than p. We assume that only a small subset of covariates is `active' (i.e., the corresponding coefficients are non-zero), and consider the model-selection problem of identifying the a…
Optimizes sample reweighting to match laws under covariate shift using Wasserstein distance.
problem Matching laws of samples with different distributions under covariate shift.
method Minimizes Wasserstein distance between empirical measures of samples using Nearest Neighbors weights.
result Consistent reweighting leads to asymptotic convergence of empirical measures.
The paper addresses statistical issues in high-dimensional financial data.
problem Standard statistical techniques fail in high-dimensional data.
method Exploring modifications and new statistical methods for high-dimensional covariance matrix, regression, PCA, multiple testing, and classification.
result Development of fast algorithms for practical application.
New spectral mixture kernels improve MOGP cross-covariance interpretation.
problem Limited parametric interpretation of cross-covariances in MOGPs.
method Complex-valued cross-spectral densities, Cramér's Theorem, phase shifts, delays.
result Improved expressive and interpretable multivariate covariance functions.
This paper compares HMC and RNN expressivity using SRT.
problem Comparing expressivity of HMC and RNN models.
method Embed HMC and RNN in a GUM, use SRT to compare structured covariance series.
result Conditions for realizing covariance series by GUM, HMC, or RNN.
Classical mean-variance portfolio theory tells us how to construct a portfolio of assets which has the greatest expected return for a given level of return volatility. Utility theory then allows an investor to choose the point along this efficient frontier which optimally balances her desire for excess expected return …
Clustering stocks reduces estimation error in global minimum variance portfolio.
problem High estimation error in covariance matrix estimation.
method Bounded clustering to limit maximum cluster size.
result Reduction in out-of-sample volatility and gap between in-sample and out-of-sample volatility.
We construct a covariant functor from the topological torus bundles to the so-called Cuntz-Krieger algebras; the functor maps homeomorphic bundles into the stably isomorphic Cuntz-Krieger algebras. It is shown, that the K-theory of the Cuntz-Krieger algebra encodes torsion of the first homology group of the bundle. We …
Improved variational inference by enforcing consistency with linear response.
problem Inconsistent variational approximations in inference methods.
method Introducing constraints on covariance to ensure consistency with linear response.
result Improvement in marginal probability distribution inference.
We consider the estimation of large covariance and precision matrices from high-dimensional sub-Gaussian or heavier-tailed observations with slowly decaying temporal dependence. The temporal dependence is allowed to be long-range so with longer memory than those considered in the current literature. We show that severa…
Extends FJS analysis to general label spaces, including classification and regression.
problem Distribution shift in general label spaces, including covariate and label shifts.
method Proposes a framework for analyzing FJS in general label spaces and generalizes existing results.
result Generalizes FJS analysis to general label spaces, including classification and regression.
We provide a method to prepare covariance matrices for quantum datasets.
problem No concrete protocol for preparing covariance matrices for quantum datasets.
method Amplitude encoding of data, exploiting global phase symmetry to center the dataset.
result Covariance matrix can be prepared for arbitrary quantum datasets or centered classical datasets.
Study Gaussian processes on permutations for ranking problems.
problem Learning real functions on permutation groups.
method Develop harmonic analysis for covariance operators.
result Propose and study Gaussian processes models on permutations.