Paper proposes neural networks for fundamental matrix estimation without key-point correspondences.
problem Estimating fundamental matrices from noisy and unreliable key-point correspondences.
method End-to-end neural network architectures preserving fundamental matrix properties.
result Neural networks achieve competitive performance on the KITTI dataset without correspondences.
This paper sets fundamental limits for rank-one matrix estimation with varying noise levels.
problem Estimating a rank-one matrix from Gaussian observations with different noise levels across blocks.
method Novel reduction from heterogeneous noise to homogeneous noise, proving asymptotic error bounds.
result Asymptotically exact formulas for minimum mean-squared error in estimating rank-one matrix and factors.
We consider D-branes in string theory and address the issue of how to describe them mathematically as a fundamental object (as opposed to a solitonic object) of string theory in the realm in differential and symplectic geometry. The notion of continuous maps, k-times differentiable maps, and smooth maps from an Azuma…
Optimal rank-adaptive matrix estimation from linear measurements.
problem Estimating high-dimensional matrices from linear measurements with adaptive rank selection.
method Combines Least-Squares estimator with universal singular value thresholding.
result Algorithm performance nearly matches fundamental limits.
A new machine learning model uses matrix exponentials for universal approximation.
problem Developing a robust and efficient machine learning model.
method Introduces a novel architecture using matrix exponentials as the only nonlinearity.
result The model achieves universal approximation properties and outperforms other models on benchmark tasks.
The paper defines and calculates fourth fundamental form and i-th curvatures for hypersurfaces in 4D Euclidean space.
problem Calculating curvatures for hypersurfaces in 4D Euclidean space.
method Defining fourth fundamental form and i-th curvatures for hypersurfaces, calculating them on rotational hypersurface, and studying hypersurfaces satisfying a specific differential equation.
result Fourth fundamental form and i-th curvatures are defined and calculated for hypersurfaces in 4D Euclidean space.
Algorithm finds finite fundamental bikei for virtual knots.
problem Determining which virtual knots have finite fundamental bikei.
method Implemented an algorithm to complete presentation matrices to operation tables.
result Computed fundamental bikei for all prime virtual knots with up to four crossings.
We lay down an elementary yet fundamental lemma concerning a finite algebraicness property of a smooth map from an Azumaya/matrix manifold with a fundamental module to a smooth manifold. This gives us a starting point to build a synthetic (synonymously, C∞-algebraic) symplectic geometry and calibrated geometr…
We want to construct a homological link invariant whose Euler characteristic is MOY polynomial as Khovanov and Rozansky constructed a categorification of HOMFLY polynomial. The present paper gives the first step to construct a categorification of MOY polynomial. For the essential colored planar diagrams with additional…
Paper bounds subspace estimator error from noisy projections.
problem Estimating subspaces from noisy data.
method Derives perturbation bound on optimal subspace estimator.
result Fundamental result with implications in matrix completion and clustering.
Study on limits of detecting a rank-one perturbation in Wigner matrices.
problem Detecting an additive rank-one perturbation in Wigner matrices.
method Gaussian interpolation methods and rigorous incarnation of the cavity method.
result Established the maximal region of contiguity between planted and null models, marking a phase transition for both estimation and detection.
In this Part II of D(11), we introduce new objects: super-Ck-schemes and Azumaya super-Ck-manifolds with a fundamental module (or, synonymously, matrix super-Ck-manifolds with a fundamental module), and extend the study in D(11.1) ([L-Y3], arXiv:1406.0929 [math.DG]) to define the notion of `differentiable maps…
Paper develops new patterns for unique matrix completions.
problem Developing unique completions for non-random matrix patterns.
method Formulated low-rank matrix completion using Plucker coordinates.
result Provides two families of patterns for any rank.
A fast algorithm for generalized matrix regression improves machine learning performance.
problem Efficiently solving generalized matrix regression problems in machine learning.
method Utilizes sketching technique to achieve (1+ε) relative error with sketching sizes of order $\cO(ε^{-1/2})$. result The Fast GMR algorithm achieves better performance in symmetric positive definite matrix approximation and single pass singular value decomposition.
Simplified matrix generator resolves credit migration model calibration issues.
problem Fundamental difficulties in calibrating Markovian credit migration models.
method Simplified matrix generator and elementary ideas from differential geometry.
result Risk-neutral calibration requires volatility information and is unstable.
Randomized HALS for efficient NMF on big data.
problem Challenges in computing nonnegative matrix factorization for big data.
method Randomized hierarchical alternating least squares (HALS) algorithm.
result Efficient nonnegative decomposition for big data applications.
We develop a class of integrals on a manifold M called exponential iterated integrals, an extension of K. T. Chen's iterated integrals. It is shown that the matrix entries of any upper triangular representation of the fundamental group of M can be expressed via these new integrals. The ring of exponential iterated inte…
This paper explores how random sampling and coding can speed up approximate matrix multiplication.
problem Efficiently computing large-scale matrix multiplications in distributed systems.
method Proposes two schemes: coding for recovery and random sampling for approximation.
result Investigates tradeoffs between recovery threshold and approximation error.
New algorithm optimizes matrix reordering for noisy disordered matrices.
problem Optimizing matrix reordering for noisy disordered matrices in single-cell biology and metagenomics.
method Proposed a polynomial-time adaptive sorting algorithm to improve upon spectral seriation.
result Our algorithm achieves superior performance compared to existing methods in real datasets.
New distribution simplifies covariance matrix inference.
problem Efficient inference for covariance matrices in large models.
method Incorporates Inverse G-Wishart distribution for variational message passing.
result Elegant and succinct expression of variational message passing fragments.
A new algorithm speeds up matrix multiplication without actual multiplication.
problem Efficiently multiplying matrices in machine learning.
method Learning-based algorithm that uses hashing, averaging, and byte shuffling.
result Often runs 100x faster than exact matrix products and 10x faster than current approximate methods.
Deep ReLU networks can approximate matrix-vector products with error bounds.
problem Can deep ReLU networks accurately approximate matrix-vector products?
method Derived error bounds in Lebesgue and Sobolev norms for deep ReLU FNNs.
result Developed deep approximation theory with successful applications.
ISEE method efficiently estimates large precision matrices in Gaussian graphical models.
problem Estimating large precision matrices in ultra-large Gaussian graphical models.
method ISEE method combines sparse modeling and large covariance matrix estimation.
result ISEE method can recover graphical structure with significant probability and efficient estimation of link strengths.
Simple matrix formulas for Grassmannian curvatures.
problem Modeling Grassmannian for curvature calculations.
method Symmetric orthogonal matrices and standard matrix operations.
result Explicit, simple formulas for various curvatures.
New CRB derived for curved models using extrinsic geometry.
problem Estimate curved statistical families accurately.
method Vector generalization of CRB with curvature correction using SDP and SOS relaxations.
result Directional curvature correction provides more accurate estimation.
RPCholesky approximates kernel matrices with few evaluations.
problem Approximating kernel matrices efficiently.
method Randomly pivoted partial Cholesky factorization.
result RPCholesky provides nearly optimal low-rank approximations.
Paper finds a lower bound for estimating low-rank matrices in logistic regression.
problem Estimating low-rank coefficient matrices in logistic regression.
method Derives a minimax lower bound on the risk.
result The bound depends on matrix dimensions, rank, and sample size.
The covariance matrix of a p-dimensional random variable is a fundamental quantity in data analysis. Given n i.i.d. observations, it is typically estimated by the sample covariance matrix, at a computational cost of O(np2) operations. When n,p are large, this computation may be prohibitively slow. Moreover, …
New entropy measures reveal information flow in CNNs without approximations.
problem Understanding information flow in convolutional neural networks (CNNs).
method Developed new entropy estimators based on Renyi's α-entropy and applied PID framework.
result Validated fundamental data processing inequalities and revealed properties of CNN training.
Study rigidity of minimal Legendrian submanifolds in spheres via eigenvalues.
problem Rigidity of minimal Legendrian submanifolds in unit Euclidean spheres.
method Using Lu's inequality and eigenvalues of fundamental matrices to establish pinching theorems.
result Optimal pinching theorem and rigidity theorem for submanifolds of all dimensions.
Gradient descent with noise converges to a unique optimum in nonconvex matrix factorization.
problem Gradient descent with noise converges to a unique optimum in nonconvex matrix factorization.
method A perturbed form of gradient descent with arbitrary initialization.
result Gradient descent with noise converges to a unique optimum.
Study reveals limits of PLS in multi-modal learning with correlated signals.
problem Understanding PLS performance in multi-modal learning with correlated signals.
method Random matrix theory analysis of spiked cross-covariance models.
result Identifies SNR and correlation regimes where PLS fails to recover any signal.
This work analyzes self-attention matrices using random matrix theory.
problem Understanding the theoretical behavior of self-attention layers in neural networks.
method Asymptotic spectral analysis of the attention matrix, Gaussian equivalence, and linearization.
result The singular value distribution of the attention matrix is asymptotically characterized by a linear model.
New method constructs equivariant neural networks for arbitrary matrix groups.
problem Challenges in constructing equivariant neural networks for complex groups.
method Completely general algorithm for solving equivariant layers of matrix groups.
result Constructs multilayer perceptrons equivariant to multiple groups including O(1,3), O(5), Sp(n), and Rubik's cube group.
SketchyCGM optimizes matrices with optimal storage and low-rank solutions.
problem Optimizing matrices with low-rank solutions efficiently.
method Modifies conditional gradient method to use a small randomized sketch of the matrix variable.
result SketchyCGM converges to a low-rank solution with optimal storage.
We consider computational complexity of problems related to the fundamental group and the first homology group of (embeddable) 2-complexes. We show, as an extension of an earlier work, that computing first homology of 2-complexes is equivalent in computational complexity to matrix diagonalization. That is, the usua…
A new method for SSMF improves upon existing algorithms.
problem Identify identifiable solutions in simplex-structured matrix factorization.
method Dual simplex volume maximization approach.
result The proposed method outperforms state-of-the-art SSMF algorithms.
In this paper, we examine the problem of approximating a general linear dimensionality reduction (LDR) operator, represented as a matrix A∈Rm×n with m<n, by a partial circulant matrix with rows related by circular shifts. Partial circulant matrices admit fast implementations via Fourier tra…
Unified optimization framework for matrix seriation.
problem Discovering latent structure in relational data.
method Mathematical optimization models for seriation.
result Optimization models enhance solution quality and interpretability.
BJMD integrates multi-source data with heterogeneous noise using Bayesian inference.
problem Integrating data from multiple sources with different noise levels.
method BJMD uses a Bayesian framework to model noise heterogeneity and develops scalable algorithms for joint matrix decomposition.
result BJMD outperforms state-of-the-art methods in integrating multi-source data with heterogeneous noise.
Proposes a method to learn a low-rank kernel matrix for graph-based clustering.
problem Challenges in learning an optimal kernel matrix for graph-based clustering.
method Unified framework for graph construction and kernel learning, focusing on a low-rank kernel matrix.
result Efficacy of the proposed method validated through extensive experiments.
New technique stabilizes singular values in concatenated matrices.
problem How singular values of concatenated matrices relate to individual components.
method Developed perturbation technique extending classical results to concatenated matrices.
result Dominant singular values remain stable under small perturbations in submatrices.
The paper uses NMF to detect political communities in Twitter networks.
problem Detecting pure political communities in Twitter networks.
method Developed three NMF frameworks to analyze user connectivity and content.
result User content and endorsement filtered connectivity are complementary.
Study shows generative priors improve rank-one matrix recovery with optimal sample complexity.
problem Recovering a rank-one signal matrix from noisy data with additional prior information.
method Analysis of a nonlinear least squares objective with a favorable global optimization landscape.
result Established optimal sample complexity for generative priors in rank-one matrix recovery.
New method normalizes matrix features for robust low-rank approximation.
problem Robust feature normalization for low-rank matrix approximation.
method Learn quantile normalization operators jointly with matrix factorization.
result Improves quality of low-rank representation of data.
Study optimizes shared singular subspace estimation from noisy matrices.
problem Estimating shared singular subspaces across multiple noisy matrices.
method Low-rank matrix denoising framework with Stack-SVD and novel estimators.
result Stack-SVD achieves minimax rate-optimality for identical shared subspaces, and novel estimators for partial sharing.
New framework explains why nonconvex methods work well in low-rank matrix estimation.
problem Nonconvex low-rank matrix estimation problems in machine learning.
method Developed a theoretical framework revealing a benign regularizer.
result Nonconvex procedures can behave well due to a disguised convexity.
The paper analyzes PLS-SVD in high-dimensional data integration, revealing its strengths and limitations.
problem Understanding the behavior of PLS-SVD in high-dimensional data integration.
method Analysis using random matrix theory and singular value decomposition.
result PLS-SVD exhibits counter-intuitive or limiting behavior in certain regimes and outperforms PCA when detecting common latent subspace.