Study differential properties of matrix square roots in specific cases.
problem Understanding matrix square roots in semi-simple, symmetric, and orthogonal cases.
method Analysis of differential and metric structures of real square roots of matrices under specific conditions.
result Differential properties of matrix square roots in semi-simple, symmetric, and orthogonal cases.
MuonEq improves training of matrix-valued parameters by rebalancing momentum before orthogonalization.
problem Training matrix-valued parameters with orthogonalized-update optimizers like Muon.
method MuonEq introduces three lightweight pre-orthogonalization equilibration schemes: two-sided row/column normalization (RC), row normalization (R), and column normalization (C).
result Row/column normalization acts as a zeroth-order surrogate for whitening and improves the geometry seen by orthogonalization.
Muon optimizer simplifies matrix optimization with spectral orthogonalization.
problem Matrix optimization challenges, especially with large condition numbers.
method Simplified Muon optimizer using spectral orthogonalization of gradients.
result Simplified Muon converges linearly with independent scalar sequences, outperforming gradient descent and Adam.
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.
Algorithm finds isotropy subgroups of orthogonal similarity on symmetric matrices.
problem Computing isotropy subgroups of orthogonal similarity on symmetric matrices.
method Algorithmic procedure solving a Toeplitz matrix equation.
result Structure of isotropy subgroups described.
Computes isotropy subgroups of orthogonal matrices acting on Hermitian matrices.
problem Computing isotropy subgroups of orthogonal matrices acting on Hermitian matrices.
method Algorithm for solving a matrix equation to compute isotropy subgroups.
result Computed isotropy subgroups of orthogonal matrices acting on Hermitian matrices.
This work proves the asymptotic freeness of layerwise Jacobians in MLPs with Haar orthogonal matrices.
problem Proving the asymptotic freeness of layerwise Jacobians in multilayer perceptrons (MLPs).
method Replacing each layer's parameter matrix with itself multiplied by a Haar orthogonal matrix, and using the invariance of the MLP.
result Proves the asymptotic freeness of layerwise Jacobians in MLPs with Haar orthogonal matrices.
Study optimizes estimation of orthogonal and rotation matrices from noisy data.
problem Estimating orthogonal and rotation matrices from noisy data.
method Iterative polar decomposition algorithm initialized by spectral methods.
result Algorithm achieves optimal error rate of $(1+o(1)) rac{σ^2 d(d-1)}{2np}$.
We develop matrix models for Grassmann, flag, and Stiefel manifolds.
problem Creating efficient models for Grassmann, flag, and Stiefel manifolds.
method Orthogonally-equivariant matrix submanifold models derived for each manifold.
result Exhaustive list of orthogonally-equivariant submanifold models for the lowest dimensions.
We present an intriguing discovery related to Random Fourier Features: in Gaussian kernel approximation, replacing the random Gaussian matrix by a properly scaled random orthogonal matrix significantly decreases kernel approximation error. We call this technique Orthogonal Random Features (ORF), and provide theoretical…
The paper analyzes and improves a deep learning optimization technique using matrix gradient orthogonality.
problem Improving deep learning training through more effective optimization methods.
method Develops a stochastic non-Euclidean trust-region gradient method for deep learning optimization.
result Proves state-of-the-art convergence results for the proposed algorithm in various scenarios.
We orthogonalize the NSS model to condition and diagnose its ill-conditioned parameters.
problem The ill-conditioning of the NSS model's design matrix.
method Exact orthogonal reparametrization via QR decomposition.
result Orthogonalization isolates the conditioning structure and maintains fit uncertainty.
Recurrent neural networks (RNNs) have been successfully used on a wide range of sequential data problems. A well known difficulty in using RNNs is the \textit{vanishing or exploding gradient} problem. Recently, there have been several different RNN architectures that try to mitigate this issue by maintaining an orthogo…
Curious structure of special orthogonal, unitary, and symplectic groups as products of Grassmannians discovered.
problem Understanding the structure of special orthogonal, unitary, and symplectic groups.
method Expressing these groups as products of Grassmannians realized as involution matrices.
result Special orthogonal, special unitary, and symplectic groups can be expressed as products of their corresponding Grassmannians.
A new algorithm POGO optimizes thousands of orthogonal matrices efficiently.
problem Optimizing thousands of orthogonal constraints at scale is computationally expensive.
method Revisits Landing algorithm, uses modern adaptive optimizers, reduces hyperparameters.
result POGO optimizes thousands of orthogonal matrices in minutes, outperforming alternatives.
New algorithm speeds up group equivariant neural networks computations.
problem Challenging computations in group equivariant neural networks.
method Diagrammatic framework based on category theory for matrix multiplication.
result Exponential improvement in time complexity for matrix multiplication.
Pion optimizes LLMs by preserving weight matrix singular values.
problem Training large language models (LLMs) with standard optimizers leads to unstable weight matrices.
method Pion uses orthogonal transformations to update weight matrices, preserving their singular values.
result Pion offers a stable alternative to standard optimizers for LLM pretraining and finetuning.
New method relaxes PCA orthogonality constraints using explained variance of correlated components.
problem Difficulty in using PCA for sparse design due to orthogonality constraints and non-differentiable penalty.
method Introduce expvar(Y) to measure variance explained by correlated components, relax orthogonality constraints.
result Two expvar(Y) definitions suitable for block PCA formulations without orthogonality constraints.
OSA overcomes instability in skipless Transformers.
problem Instability in skipless Transformers using Softmax Self-Attention.
method OSA parametrizes attention matrix to be orthogonal via skew-symmetric matrix exponential.
result OSA allows for training non-causal Transformers without skip connections and normalisation layers.
A recent theoretical analysis shows the equivalence between non-negative matrix factorization (NMF) and spectral clustering based approach to subspace clustering. As NMF and many of its variants are essentially linear, we introduce a nonlinear NMF with explicit orthogonality and derive general kernel-based orthogonal m…
Recurrent Neural Networks (RNNs) are designed to handle sequential data but suffer from vanishing or exploding gradients. Recent work on Unitary Recurrent Neural Networks (uRNNs) have been used to address this issue and in some cases, exceed the capabilities of Long Short-Term Memory networks (LSTMs). We propose a simp…
New ONMF model minimizes KL divergence for better sparse data modeling.
problem Clustering and data modeling with sparse vectors.
method Developed KL-ONMF algorithm based on alternating optimization.
result KL-ONMF outperforms Frobenius-norm ONMF for document classification and hyperspectral image unmixing.
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.
A new method for efficiently computing derivatives of skew-symmetric matrix exponentials.
problem Efficient computation of derivatives for skew-symmetric matrices.
method Characterization of invertibility, construction of nearby logarithm, and efficient implementation.
result Explicit formulae for differentiation and its inverse of skew-symmetric matrix exponentials.
OMIC improves matrix completion with orthonormal side information and nuclear-norm regularization.
problem Matrix completion with improved interpretability and adaptability.
method OMIC combines orthonormal side information and nuclear-norm regularization, optimized by a converging algorithm.
result OMIC outperforms state-of-the-art methods in synthetic and real-world datasets.
Efficiently optimizes orthogonal and Stiefel matrices on parallel units.
problem Optimization over orthogonal groups on parallel units.
method CWY and T-CWY transforms for parametrization and optimization.
result CWY and T-CWY methods lead to convergence on parallel units.
HD algorithm simulates dynamics on random matrix ensembles without generating full matrices.
problem Simulating dynamics on dense random matrix ensembles with high space and time complexity.
method Householder reflectors for adaptive and recursive construction, deferring decisions.
result Significant reductions in runtime and memory footprint for practical T≪n. This handbook simplifies Grassmann manifold geometry for matrix-based algorithms.
problem Modeling linear subspaces in various applications.
method Expository work on Grassmann manifold geometry, including new algorithms and formulas.
result Improved understanding and computational tools for the Grassmann manifold.
Orthogonal random features approximate a Bessel kernel, offering sharper bounds than random Fourier features.
problem Approximating Gaussian kernel efficiently for large datasets.
method Use of Haar orthogonal matrices to construct orthogonal random features and analyze their bias and variance.
result Orthogonal random features approximate a Bessel kernel, not the Gaussian kernel, with sharper bounds.
Study real logarithms of semi-simple matrices, focusing on differential structure.
problem Understanding the differential structure of real logarithms of semi-simple matrices.
method Examines the differential structure of real logarithms of semi-simple matrices under specific matrix types.
result Characterizes the differential structure of real logarithms of semi-simple matrices.
Emergency Department (ED) crowding is a worldwide issue that affects the efficiency of hospital management and the quality of patient care. This occurs when the request for an admit ward-bed to receive a patient is delayed until an admission decision is made by a doctor. To reduce the overcrowding and waiting time of E…
In this paper, we propose a scalable algorithm for spectral embedding. The latter is a standard tool for graph clustering. However, its computational bottleneck is the eigendecomposition of the graph Laplacian matrix, which prevents its application to large-scale graphs. Our contribution consists of reformulating spect…
Study finds Calabi-Yau models' operator spectra match random matrix theory.
problem Understanding spectra of Calabi-Yau sigma models.
method Numerical methods for Ricci-flat metrics, averaging over complex structure moduli space.
result Spectrum matches Gaussian orthogonal ensemble of random matrix theory.
Optimizing over the set of orthogonal matrices is a central component in problems like sparse-PCA or tensor decomposition. Unfortunately, such optimization is hard since simple operations on orthogonal matrices easily break orthogonality, and correcting orthogonality usually costs a large amount of computation. Here we…
Study connects curvature to graph theory and reveals differences.
problem Exploring differences between Quadratic Orthogonal Bisectional Curvature and Real Bisectional Curvature.
method Real (1,1)--forms and Weitzenböck curvature operator used to represent graph Dirichlet energy.
result Curvature differences illuminated between Quadratic Orthogonal Bisectional Curvature and Real Bisectional Curvature.
A new algorithm avoids retractions to optimize orthogonal matrices efficiently.
problem Optimizing functions over the manifold of orthogonal matrices efficiently.
method Landing algorithm that avoids retractions using potential energy.
result The landing algorithm is faster and less prone to numerical errors than retraction-based methods.
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.
New algorithm for signal estimation in noisy matrix models.
problem Signal estimation in rectangular spiked matrix models with rotationally invariant noise.
method Orthogonal Approximate Message Passing (OAMP) algorithm for signal estimation.
result Optimal OAMP algorithm minimizes mean-squared error and achieves Bayes-optimal performance.
Nonnegative matrix factorization (NMF) is a popular method for audio spectral unmixing. While NMF is traditionally applied to off-the-shelf time-frequency representations based on the short-time Fourier or Cosine transforms, the ability to learn transforms from raw data attracts increasing attention. However, this adds…
Paper derives local Plücker formulas for special orthogonal groups.
problem Deriving Plücker formulas for special orthogonal groups.
method Reduction to classical A_n case.
result Local Plücker formulas for special orthogonal groups derived.
In this paper, we propose an efficient and scalable low rank matrix completion algorithm. The key idea is to extend orthogonal matching pursuit method from the vector case to the matrix case. We further propose an economic version of our algorithm by introducing a novel weight updating rule to reduce the time and stora…
Several variants of recurrent neural networks (RNNs) with orthogonal or unitary recurrent matrices have recently been developed to mitigate the vanishing/exploding gradient problem and to model long-term dependencies of sequences. However, with the eigenvalues of the recurrent matrix on the unit circle, the recurrent s…
Random matrix ensembles yield uniform distributions on manifolds.
problem Understanding distributions of vectors in random matrix ensembles.
method Analyzing eigenvalues, singular values, and Autonne-Takagi vectors of various random matrix ensembles.
result Uniform distributions on specific manifolds for different types of random matrix ensembles.
Paper proposes a method to recover point configurations from noisy distance data.
problem Recovering point configurations from noisy distance data.
method Robust Euclidean Distance Geometry via Dual Basis (RoDEoDB) algorithm.
result Exact recovery guarantees for point configuration and Gram matrix under mild conditions.
New differential geometry perspective on orthogonal RNNs.
problem Mitigating exploding and vanishing gradients in RNNs.
method Using tools from differential geometry, parameterizing vector fields via directional derivatives of scalar functions.
result Our approach achieves comparable or better results on benchmark tasks.
A new method for sparse PCA using orthogonal rotations and soft-thresholding.
problem Sparse PCA with a new basis using orthogonal rotations.
method Initialize with leading principal components, apply kimesk orthogonal rotation, and soft-threshold the rotated components. result The proposed method is more stable and explains more variance compared to alternatives.
OLinear forecasts time series more efficiently by transforming data orthogonally.
problem Efficiently forecasting time series with entangled dependencies.
method OLinear uses OrthoTrans to transform data orthogonally, then applies NormLin for linear layer.
result OLinear achieves state-of-the-art performance with high efficiency.
Characterizes the local diffeomorphism structure of the exponential in the set of skew-symmetric matrices.
problem Characterizing the local diffeomorphism structure of the exponential in the set of skew-symmetric matrices.
method Introduce the diffeomorphic logarithm of special orthogonal matrices and an efficient algorithm.
result The region containing the principal logarithm has a special multiplicity structure.