Classifies contravariant matrix-valued valuations on polytopes without continuity assumptions.
problem Classifying contravariant matrix-valued valuations on polytopes without continuity assumptions.
method Complete classification of contravariant matrix-valued valuations on polytopes in Rn without continuity assumptions. result The only such valuation is the general Lutwak-Yang-Zhang matrix in dimension n≥4, and a new function in dimension 3. Study G2-metrics from non-integrable special Lagrangian fibrations.
problem Understanding G2-metrics from non-integrable special Lagrangian fibrations. method Decompose SU(3)-structures into solder 1-forms, connection 1-forms, and equivariant matrix-valued functions. result Describe regular parts of G2-manifolds with Lagrangian-type actions. New MVG mechanism improves differential privacy for matrix-valued queries.
problem Lack of optimal methods for matrix-valued queries in differential privacy.
method Proposes MVG mechanism using matrix-variate Gaussian noise.
result Proves MVG mechanism preserves (ε,δ)-differential privacy. 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.
Holomorphic functions from knot complements link to quantum modular forms.
problem Analyzing holomorphic functions from knot complements.
method Matrix-valued holomorphic functions, cocycles, and quantum modularity.
result Identifies a matrix-valued holomorphic quantum modular form.
Study finds weak solutions for complex map flows with optimal lifespan.
problem Existence of weak solutions for two-phase matrix-valued harmonic map flows.
method Modified minimizing movement scheme, discretizing time and interpolating solutions.
result Existence of weak solutions with optimal lifespan for the limiting system.
The paper extends log-Sobolev inequalities to matrix-valued settings using combinatorial methods.
problem Log-Sobolev inequalities for matrix-valued settings.
method Combining noncommutative geometry tools and combinatorial methods.
result Combinatorial methods yield computable lower bounds for matrix-valued log-Sobolev inequalities.
We propose a general matrix-valued multiple kernel learning framework for high-dimensional nonlinear multivariate regression problems. This framework allows a broad class of mixed norm regularizers, including those that induce sparsity, to be imposed on a dictionary of vector-valued Reproducing Kernel Hilbert Spaces. W…
We propose a general matrix-valued multiple kernel learning framework for high-dimensional nonlinear multivariate regression problems. This framework allows a broad class of mixed norm regularizers, including those that induce sparsity, to be imposed on a dictionary of vector-valued Reproducing Kernel Hilbert Spaces. W…
Long term optimal investment problems are studied in a factor model with matrix valued state variables. Explicit parameter restrictions are obtained under which, for an isoelastic investor, the finite horizon value function and optimal strategy converge to their long-run counterparts as the investment horizon approache…
New optimizer designs respect symmetry, improving deep learning models.
problem Optimizers lack respect for symmetry in neural networks.
method Introduce symmetry-compatible principle for optimizer design.
result Symmetry-compatible optimizers improve model performance.
Paper improves matrix-valued data classification using nonparametric LDA.
problem Classification of matrix-valued data in neuroimaging and signal processing.
method Nonparametric LDA based on NPMLE for vectorized and scaled matrices.
result Improves classification performance across various data structures.
Researchers developed a new Riemannian manifold for SPD matrix-valued optimal transport problems.
problem Optimal transport between SPD matrix-valued measures.
method Formulated as a generalized optimal transport problem with block SPD matrices, endowed with a novel Riemannian manifold structure.
result The novel Riemannian manifold allows solving SPD matrix-valued optimal transport problems using Riemannian optimization.
Recovering matrix valued potentials from wave equation data on stationary spacetimes.
problem Recovering a time-dependent matrix valued potential from wave equation data.
method Reduction to non-Abelian light ray transform and study of the transform.
result Sufficient conditions for solving the inverse problem on stationary spacetimes.
Traditional linear methods for forecasting multivariate time series are not able to satisfactorily model the non-linear dependencies that may exist in non-Gaussian series. We build on the theory of learning vector-valued functions in the reproducing kernel Hilbert space and develop a method for learning prediction func…
Enhances SVGD with matrix-valued kernels for faster inference.
problem Efficient approximate inference in complex probability landscapes.
method Integrates geometric information through matrix-valued kernels in SVGD.
result Significant improvement in real-world Bayesian inference tasks.
We extend Kyle's model to include stochastic liquidity and multiple assets.
problem Modeling informed trading with stochastic liquidity and multiple assets.
method Developed a variational formulation and derived a matrix-valued martingale depth process.
result A linear-Gaussian equilibrium with stochastic matrix-valued price impact.
Improved hypothesis testing and change-point detection using diffusion-based methods.
problem Limited power of score-based hypothesis tests and change-point detection.
method Extending score-based Fisher divergence to diffusion-divergence by multiplying score functions with a matrix-valued function or weight matrix.
result Theoretical quantification and demonstration of optimal performance of diffusion-based algorithms.
The paper estimates matrix-valued functions with low rank using penalized estimators.
problem Estimating matrix-valued functions with low rank from incomplete data.
method Innovative nuclear norm penalized local polynomial estimator and bias-reducing kernels.
result Optimal rates of convergence for various matrix norms.
Proposes a novel linear discriminant analysis for matrix-valued data.
problem Classification of high-dimensional matrix-valued data from imaging studies.
method Efficient nuclear norm penalized regression with low-rank structure.
result Superior performance compared to existing methods in simulations and EEG data.
Paper develops a new test for high-dimensional matrix-valued data.
problem Hypothesis testing for mean of matrix-valued data in high-dimensional settings.
method Proposes a new test statistic for high-dimensional matrix rank testing.
result Develops a novel approach for sparse singular value decomposition (SVD) estimation.
We give a complete classification of conformally covariant differential operators between the spaces of i-forms on the sphere Sn and j-forms on the totally geodesic hypersphere Sn−1. Moreover, we find explicit formulæ for these new matrix-valued operators in the flat coordinates in terms of basic operators …
A new method estimates SDEs using occupation kernels.
problem Learning multivariate stochastic differential equations (SDEs).
method Two-step procedure: estimate drift, then diffusion. Occupation kernels used in RKHS.
result Validated on simulated and real-world data.
Defines an equivariant Ruelle dynamical zeta function for flows on manifolds.
problem Defining a zeta function for equivariant flows on manifolds.
method Equivariant generalization of Guillemin's trace formula.
result Computes the equivariant Ruelle zeta function in various examples.
Algorithm predicts performance of learning in multi-layer networks with matrix-valued hidden variables.
problem Signal recovery and learning in multi-layer neural networks with matrix-valued hidden variables.
method Unified approximation algorithm for MAP and MMSE inference, extending ML-VAMP to handle matrix-valued unknowns.
result Performance of ML-Mat-VAMP algorithm can be predicted in a random large-system limit.
The paper develops methods for calculating equivariant homology from Morse functions.
problem Calculating equivariant homology from equivariant Morse functions.
method Alter equivariant Morse functions to stable ones, use generic equivariant metrics, and analyze the Morse spectral sequence.
result Equivariant Morse functions induce a filtration that computes equivariant homology.
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.
Study proposes efficient estimators for matrix-valued linear regression under sparsity assumptions.
problem Estimation of parameters in matrix-valued linear regression models.
method Explicit optimization-free estimators for matrix-valued linear regression models with sparsity assumptions.
result Established non-asymptotic convergence rates for the proposed estimators.
We consider a model for linear transient price impact for multiple assets that takes cross-asset impact into account. Our main goal is to single out properties that need to be imposed on the decay kernel so that the model admits well-behaved optimal trade execution strategies. We first show that the existence of such s…
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.
The paper proves a formula for complex Monge-Ampère equations on manifolds.
problem Proving a mean value formula for complex Monge-Ampère equations.
method Using subharmonic Hermitian matrix valued functions and Liouville type theorems.
result Obtained a Liouville theorem for complex Monge-Ampère equations.
Extends DCP framework to Hadamard manifolds for geodesically convex functions.
problem Verifying convexity in nonlinear programs on Hadamard manifolds.
method Introduces Disciplined Geodesically Convex Programming (DGCP) framework, defining compositions and transformations for geodesically convex functions.
result Allows verification of geodesic convexity for a broader range of functions, including statistical estimators and matrix-valued optimization.
Study solutions of elliptic sinh-Gordon equation, focusing on spectral genus two.
problem Investigate solutions of the elliptic sinh-Gordon equation of spectral genus g<3.
method Parametrize solutions by complex matrix-valued polynomials, analyze flows and isospectral sets.
result Construct three conformal maps to R^4 on every elliptic curve, constrained Willmore.
Online graph learning from matrix-valued time series data.
problem Identifying dependency structure among sensors in a network.
method Extends VAR models to matrix-variate models, proposes online procedures for graph learning, and introduces Lasso-type approaches.
result Demonstrates effectiveness of online graph learning methods in both synthetic and real data.
Deep neural networks can approximate invariant/equivariant functions with fewer parameters.
problem Approximating functions that respect group symmetries with neural networks.
method Constructing deep neural networks with G-actions and G-equivariant/invariant affine transformations. result Deep neural networks can approximate G-invariant/equivariant functions with exponentially fewer parameters. Optimal portfolio choice with cross-impact propagators, solving complex equations.
problem Maximizing revenue-risk in a continuous-time portfolio choice problem with cross-impact.
method Formulated as a maximization problem, solved explicitly using operator resolvents and stochastic Fredholm equations.
result Sufficient conditions for the absence of price manipulation, providing financial insights.
Study stabilizes Morse-Bott cohomology for equivariant manifolds.
problem Equivariant cohomology of manifolds with group actions.
method Stabilization technique to construct Morse-Bott functions.
result Realization of equivariant transversality and orientability.
New model closes gap in understanding equivariant set functions.
problem Understanding universality of equivariant set functions.
method Proves PointNet not equivariant universal and introduces PointNetST.
result PointNetST is the simplest permutation equivariant universal model.
An impossibility result shows limitations in learning symmetries and equivariant functions.
problem Learning symmetries and equivariant functions simultaneously is impossible under certain conditions.
method Careful study of approximation for groups and semigroups, analysis of neural networks.
result Linearly equivariant networks can be used to learn equivariant functions, but group-convolutional networks have limitations.
New method clusters matrix-valued data by latent variables.
problem Clustering matrix-valued data with hidden structure.
method Latent variable model with hierarchical clustering.
result Algorithm attains clustering consistency in high dimensions.
Guillemin trace formula adapted for group actions.
problem Distributional trace for proper, cocompact group actions.
method Developing an equivariant version of the distributional trace.
result Equivariant Guillemin trace formula for group actions.
Study non-commutative function algebras using contact geometry.
problem Quantize non-commutative function algebras in several variables.
method Contact geometry and rational differential operators.
result Generalizes known constructions of classical equivariants.
Paper develops a classification method using matrix-variate t-distributions.
problem Classifying matrix-valued observations with dependence structure.
method Develops an Expectation-Maximization algorithm for discriminant analysis.
result Method shows promise on various datasets.
The paper computes metrics and Einstein tensors on even-dimensional manifolds.
problem Computing metrics and Einstein tensors on even-dimensional Riemannian manifolds.
method Development of spectral Einstein functionals and equivariant Bismut Laplacian.
result Explicit computation of the equivariant noncommutative residue density.
Equivariant CNNs improve RL performance in symmetric environments.
problem Learning equivariant representations for RL in symmetric environments.
method Proposed and studied equivariant CNNs for RL.
result Equivariant CNNs enhance RL performance and sample efficiency.
Universal MLPs with a single hidden layer can learn any function.
problem Learning on various data structures like sequences, images, sets, and graphs.
method Using group theory, the paper proves the universality of a broad class of equivariant MLPs with a single hidden layer.
result Having a hidden layer on which the group acts regularly is sufficient for universal equivariance (invariance).
Formula proves symmetry breaking operators for differential forms.
problem Symmetry breaking operators between differential forms on spheres and their hyperplanes.
method Explicit residue formula for meromorphic continuation of operators.
result Simple construction of symmetry breaking operators and determination of zeros.
LoRA and privacy: Random projections help but not always.
problem Ensuring differential privacy in LoRA fine-tuning.
method Wishart projection mechanism and noisy variants.
result LoRA is not inherently private, but low-rank fine-tuning can be more private.