New method improves Pham's algorithm for joint diagonalization.
problem Optimizing joint diagonalization of matrices for statistical learning.
method Quasi-Newton method for Pham's diagonalization criterion.
result Proposed method outperforms Pham's algorithm in experiments.
Paper proposes an effective mean-field inference method for NNBMs.
problem Inference in NNBMs is challenging due to their complex structure.
method Uses mean-field method and diagonal consistency method.
result Effective inference method for NNBMs is proposed.
Improved susceptibility propagation for Markov random fields using diagonal matching.
problem Approximate computation of Markov random fields with robustness across network structures.
method Combines belief propagation and linear response method with diagonal matching for inverse Ising problems.
result Proposed method reduces to standard susceptibility propagation and Thouless-Anderson-Palmer equation in specific cases.
This paper optimizes diagonal preconditioning to improve matrix condition numbers.
problem Optimizing diagonal preconditioning to reduce matrix condition numbers.
method Reformulated as a quasi-convex problem, solved with bisection and Newton updates.
result Optimal diagonal preconditioners can significantly improve iterative methods.
Haantjes algebras help in diagonalizing operators on manifolds.
problem Diagonalizing operators on differentiable manifolds.
method Introducing Haantjes algebra, a family of operator fields with vanishing Haantjes torsion and compatibility conditions.
result Simultaneous diagonalization of operators in local coordinates or block-diagonal form in general cases.
Riemannian neural networks outperform standard methods on various datasets.
problem Improving neural network training efficiency and performance.
method Quasi-diagonal Riemannian gradient descent.
result Quasi-diagonal Riemannian algorithms consistently outperform simple stochastic gradient descent.
New theory allows simultaneous block-diagonalization of commuting operator fields.
problem Normal forms of operator fields.
method Generalized Nijenhuis torsions and generalized Haantjes algebra.
result Simultaneous block-diagonalization of commuting operator fields.
Improved sparse Gaussian processes using structured scaling matrices and Power-EP framework.
problem Scaling Gaussian processes for large datasets.
method Structured diagonal scaling matrix and Power-EP framework.
result Structured approximations improve performance without increasing computational cost.
We prove that certain dual pairs of Calabi-Yau manifolds have matching orbifold Euler characteristics.
problem Constructing mirror symmetric Calabi-Yau manifolds with specific symmetry groups.
method Generalized Berglund-Hübsch-Henningson construction to include permutations of variables.
result Reduced orbifold Euler characteristics of dual pairs coincide up to sign under cyclic permutation groups satisfying parity condition.
Researchers found a canonical form for pairs of Hermitian and antilinear operators.
problem Simultaneous normalization of pairs of Hermitian and antilinear operators in differential geometry.
method Finding a canonical form for pairs of Hermitian and antilinear operators.
result Generalized previous results on simultaneous normalization of such pairs.
A new optimization method improves deep learning model training speed.
problem Optimizing large models with natural gradient descent.
method Kronecker-factored eigenbasis for diagonal variance approximation.
result Improves optimization speed for deep network architectures.
This paper solves matrix blind joint block diagonalization with noise.
problem Identifying the diagonalizer and block diagonal structure of matrices under noise.
method Bi-block diagonalization method.
result The method can identify the exact solution under certain conditions.
The paper simplifies the Fisher information matrix for random deep networks, speeding up learning.
problem Learning deep neural networks efficiently with large parameter spaces.
method Statistical neurodynamical method to reveal Fisher information properties, proving unit-wise block diagonal structure and explicit inverse.
result Explicit natural gradient formula without matrix inversion, speeding up learning.
Study on GD and SGD over diagonal networks, focusing on stepsizes and regularisation.
problem Understanding the impact of stochasticity and large stepsizes on gradient descent and SGD solutions.
method Investigation of GD and SGD over diagonal linear networks with macroscopic stepsizes, proving convergence and characterizing solutions.
result Large stepsizes consistently benefit SGD for sparse regression problems, but can hinder GD recovery of sparse solutions, especially in the edge of stability regime.
Develops a novel stochastic algorithm for diagonal estimation of large matrices.
problem Efficient diagonal estimation for large or implicit matrices.
method Adaptive parameter selection in a stochastic algorithm.
result Lower bound on random query vectors needed for estimation.
Improved neural network inference with eigenvalue correction.
problem Inference of flexible variational posteriors is computationally expensive.
method Eigenvalue correction to matrix-variate Gaussian posterior.
result Empirically, the method outperforms existing algorithms.
Study left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups.
problem Characterize left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups.
method Analyze left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups, classify Lie algebras and groups.
result New results on left-invariant Lorentzian metrics with harmonic curvature and non-parallel Ricci operator.
New adaptive methods improve deep learning performance.
problem Training deep networks efficiently and effectively.
method Block-diagonal matrix adaptation for gradient updates.
result Block-diagonal methods outperform adaptive diagonal methods and vanilla SGD.
Constructs coordinates to diagonalize Toda flow on matrices with simple spectrum.
problem Diagonalizing the Toda flow on matrices with simple spectrum.
method Lie theoretic methods applied to complex semisimple Lie algebras and their real forms.
result Decouples the Toda vector field into simpler components.
Paper proposes ABDR for convex subspace clustering with adaptive block diagonal representation.
problem Subspace clustering with block diagonal structure for noisy data.
method ABDR explicitly pursues block diagonality without sacrificing convexity, using a specially designed convex regularizer.
result Experimental results show ABDR outperforms state-of-the-arts.
Variable screening is a fast dimension reduction technique for assisting high dimensional feature selection. As a preselection method, it selects a moderate size subset of candidate variables for further refining via feature selection to produce the final model. The performance of variable screening depends on both com…
Develops large-sample theory for non-stationary source separation.
problem Lack of large-sample results for non-stationary source separation methods.
method Large-sample theory for NSS-JD method under specific assumptions.
result Consistency of unmixing estimator and its convergence to Gaussian distribution.
New model reduces matrix factorization bias, yielding truly low-rank solutions.
problem Gradient descent's implicit bias in matrix factorization.
method Introducing a new factorization model with constrained factors and diagonal components.
result The new model consistently exhibits a strong implicit bias, yielding truly low-rank solutions.
New methods incorporate alpha signals into portfolio construction, improving performance.
problem Signal-blindness in existing portfolio construction methods.
method Introduces three methods: HRP-μ, HRP-Σμ, and CRISP. result CRISP at intermediate γ consistently outperforms other methods. Modular method simplifies curvature computation in neural nets.
problem Efficient computation of curvature matrices for training neural nets.
method Modular backpropagation for block-diagonal approximations.
result Compact notation and easy integration into machine learning libraries.
T-Rex uses EM to fit robust factor models in noisy data.
problem Robustly fitting factor models in high-dimensional data with heavy tails and outliers.
method Expectation-Maximization (EM) algorithm based on Tyler's M-estimator for elliptical distributions.
result Demonstrates robustness in direction-of-arrival estimation and subspace recovery.
Three methods for tuning HMC diagonal scale matrices compared.
problem Improving Hamiltonian Monte Carlo efficiency with diagonal scale matrices.
method Three approaches: ISG, median crossing frequency, and estimated marginal standard deviations.
result ISG method leads to more efficient sampling in many cases.
In the first quarter of 2006 Chicago Board Options Exchange (CBOE) introduced, as one of the listed products, options on its implied volatility index (VIX). This created the challenge of developing a pricing framework that can simultaneously handle European options, forward-starts, options on the realized variance and …
Randomized block-diagonal preconditioning improves parallel learning convergence.
problem Improving convergence of gradient-based optimization methods in parallel settings.
method Randomization of coordinates during optimization to repartition tasks.
result Randomization significantly improves convergence of block-diagonal preconditioned methods.
Proposes a new metric selection for VM-PG with improved convergence.
problem Improves convergence of VM-PG methods for ill-conditioned problems.
method Diagonal Barzilai-Borwein stepsize for adaptive metric selection.
result Improved convergence results for ill-conditioned problems.
Researchers find non-diagonal Einstein metrics in various signatures.
problem Finding non-diagonal four-dimensional cohomogeneity-one Einstein metrics in different signatures.
method Explicitly seeking and constructing new examples of non-diagonal Einstein metrics, particularly in neutral signature.
result Construct new examples of neutral signature non-diagonal Bianchi type VIII Einstein metrics with self-dual Weyl tensor.
Sparse PCA method for clustering Gaussian mixtures.
problem Clustering Gaussian mixture models.
method Sparse Principal Component Analysis (SPCA) for clustering.
result Comparison with IF-PCA method and discussion of non-diagonal covariance matrices.
Diagonalizes metrics of 3D Lorentzian manifolds.
problem Diagonalizing metrics of 3D Lorentzian manifolds.
method Applying the technique of moving frames.
result Every smooth Lorentzian 3-manifold admits an atlas with a diagonal metric.
adaQN improves training RNNs with low cost and good performance.
problem Training RNNs is computationally difficult due to vanishing/exploding gradient issues.
method Stochastic quasi-Newton algorithm with L-BFGS updating, low per-iteration cost.
result adaQN is competitive with popular RNN training algorithms on language modeling tasks.
New method improves Latent Dirichlet Allocation using ICA techniques.
problem Improving Latent Dirichlet Allocation (LDA) estimation.
method Moment matching techniques linking LDA to discrete ICA, using joint diagonalization of tensors.
result New combination of tensors and orthogonal joint diagonalization outperforms existing methods.
Gradient methods work well on overparameterized diagonal linear networks.
problem Understanding why gradient-based methods work well in overparameterized models.
method Study of Deep Diagonal Linear Networks with gradient flow analysis.
result Gradient flow on layer parameters induces a mirror-flow dynamic in the effective parameter space, leading to explicit convergence guarantees.
An irreducible representation of the free group on two generators X,Y into SL(2,C) is determined up to conjugation by the traces of X,Y and XY. We study the diagonal slice of representations for which X,Y and XY have equal trace. Using the three-fold symmetry and Keen-Series pleating rays we locate those groups which a…
Study Ricci vector fields on 2D space with diagonal metrics.
problem Understanding Ricci vector fields on 2D space with specific metrics.
method Examined Ricci vector fields on R2 with a diagonal metric. result Characterized Ricci vector fields on R2 with a diagonal metric. New MCMC method learns sparse preconditioner for high-dimensional problems.
problem High-dimensional sampling with complex correlation structures.
method Adaptive MCMC with sparse preconditioner using online PCA.
result Significant reduction in computational complexity and improved performance.
Octagon map accelerates diagonal changes algorithm.
problem Improving the efficiency of diagonal changes algorithm.
method Octagon Farey map as an acceleration.
result Octagon map accelerates diagonal changes algorithm.
New diagonal knots found with non-torus structure.
problem Identifying knots with diagonal grid diagrams.
method Analysis of knots represented by diagonal grid diagrams.
result All diagonal knots are positive, and a new non-torus example is found.
Study finds symmetries in a special 3D space with a diagonal metric.
problem Identifying symmetries in a specific 3D space.
method Determining Killing vector fields on a diagonal metric in R3. result Killing vector fields on the space R3 with a diagonal metric have been identified. A new method solves diagonally constrained SDPs quickly and accurately.
problem Solving large-scale diagonally constrained SDPs efficiently.
method Combines momentum from convex optimization with coordinate descent and matrix factorization.
result Local linear convergence and first-order critical point convergence proved.
We use mathematical induction to prove that the horizontal composition in the class of coherently diagonal complexes is indeed a binary operation. That is to say, the embedding of two coherently diagonal complexes in an alternating planar diagram produces a coherently diagonal complex.
Extends method for solving certain hydrodynamic systems.
problem Solving non-diagonalisable integrable systems of hydrodynamic type.
method Generalised hodograph method applied to F-manifolds with compatible connections.
result Provides general solution under certain assumptions.
The paper studies Bergman kernels for analytic Kähler potentials on compact Kähler manifolds.
problem Analyzing the asymptotic behavior of Bergman kernels for analytic Kähler potentials.
method Using the method of Berman-Berndtsson-Sjöstrand to find upper bounds of Bergman coefficients.
result Improved asymptotic expansion of Bergman kernels for analytic Kähler potentials, with a shrinking neighborhood size of $k^{- rac14}$.
New unitary RNN architecture using complex Cayley transform outperforms existing methods.
problem Vanishing or exploding gradient problem in RNNs.
method Developed a unitary RNN architecture based on a complex scaled Cayley transform.
result scuRNN achieves comparable or better results than existing unitary RNNs.
New method improves deep learning model robustness and accuracy for long sequences.
problem Challenges in learning long-range sequence tasks using state-space models.
method Proposes a perturb-then-diagonalize (PTD) methodology to address ill-posed diagonalization problems in SSMs.
result Demonstrates improved robustness and accuracy of S5-PTD model on Long-Range Arena benchmark.