Note on minimal maps' uniqueness via singular values.
problem Uniqueness of minimal maps into \(\mathbb{R}^n\).
method Using singular values and convexity of area functional, proving local linearity of singular value vectors.
result Improved uniqueness theorem for minimal graphs.
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.
Study inequalities for singular values of rectangular matrices.
problem Inequalities for singular values of rectangular matrices.
method Study convex cones associated to isotropic representations of symmetric spaces.
result Describe inequalities by cohomological conditions.
Study differentiable maps on hypersurface links, finding fold maps with circle singular value sets.
problem Understanding differentiable maps on hypersurface links.
method Restricting holomorphic functions to hypersurface links and analyzing the resulting maps.
result Found fold maps with concentric circle singular value sets.
Study evaluates thresholds for removing noise from DNN weights using random matrix theory.
problem Removing noise from deep neural network weights for better approximation.
method Model weights as signal + noise, use random matrix theory to estimate thresholds, evaluate using cosine similarity.
result Proposed threshold estimation method improves approximation quality.
Study shows XRP price correlates with transaction network metrics.
problem Understanding the relationship between cryptoasset price and network metrics.
method Analysis of correlation tensor spectra, random matrix theory comparison, singular values investigation.
result Distinct correlation between XRP price and singular values during bubble and non-bubble periods.
A low rank matrix X has been contaminated by uniformly distributed noise, missing values, outliers and corrupt entries. Reconstruction of X from the singular values and singular vectors of the contaminated matrix Y is a key problem in machine learning, computer vision and data science. In this paper we show that common…
Deterministic bounds for tensor singular values and vectors, differing from matrix cases.
problem Spectral learning of higher-order orthogonally decomposable tensors.
method Deterministic perturbation bounds for singular values and vectors of orthogonally decomposable tensors.
result Perturbation affects each essential singular value/vector in isolation, independent of multiplicity and distance from other singular values.
We introduce the concept of singular values for the Riemann curvature tensor, a central mathematical tool in Einstein's theory of general relativity. We study the properties related to the singular values, and investigate five typical cases to show its relationship to the Ricci scalar and other invariants.
Study shows singular set of certain graphs has codimension 1.
problem Understanding the singular set of specific graph structures.
method Proved using the area stationarity condition.
result Singular set has codimension 1.
Solves initial value problem for harmonic maps on specific manifolds.
problem Initial value problem for harmonic maps on cohomogeneity one manifolds.
method Setup and solve the initial value problem using equivariant harmonic maps and regular-singular systems.
result Local existence of harmonic maps in a neighborhood of singular orbits.
Improved singular value approximation for convolutional layers.
problem Improving accuracy of singular value approximation for linear convolutional layers.
method Developed a new spectral density matrix method for singular value approximation with improved accuracy and reduced computational complexity.
result Obtained moderate improvement in singular value distribution compared to circular approximation.
The paper solves conditions for extending circle-valued Morse functions.
problem Conditions for extending circle-valued Morse functions on closed orientable surfaces.
method Provided necessary and sufficient conditions for the existence of a non-singular extension.
result Necessary and sufficient conditions for the existence of a non-singular extension of a circle-valued Morse function.
New framework for higher-order singular-value derivatives of rectangular matrices.
problem Challenging to derive higher-order Fréchet derivatives of singular values in real rectangular matrices.
method Using Kato's analytic perturbation theory for self-adjoint operators and embedding rectangular matrices into block self-adjoint operators.
result Closed-form expressions for the n-th order spectral variations of singular values. Study optimal liquidation with multiple regimes using BSDEs with singular terminal values.
problem Optimal liquidation with regime switching in dark pools.
method Introduced a system of BSDEs with jumps and singular terminal values.
result Existence and uniqueness results for the BSDE system are obtained.
Proves continuity and singular set dimension for 2D maps with Q values.
problem Interior regularity of 2D Q-valued maps. method Strong concentration-compactness theorem for equicontinuous maps.
result 2D Q-valued maps are Hölder continuous with singular set dimension ≤1. The complex Lie superalgebras g of type D(2,1;a) - also denoted by osp(4,2;a) - are usually considered for "non-singular" values of the parameter a, for which they are simple. In this paper we introduce five suitable integral forms of g, that are well-defined at singular valu…
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.
Fast and accurate methods for low-rank learning problems.
problem Partial singular value decomposition and numerical rank estimation of huge matrices.
method Krylov subspaces and Ritz vectors for fast and accurate solutions.
result Advantages over traditional methods in accuracy and speed.
In the early 1980's Almgren developed a theory of Dirichlet energy minimizing multi-valued functions, proving that the Hausdorff dimension of the singular set (including branch points) of such a function is at most (n−2), where n is the dimension of its domain. Almgren used this result in an essential way to show t…
The paper studies phase transitions in random matrices and tensor unfolding for detecting signals.
problem Phase transitions in singular values and vectors of large random matrices.
method Analysis of singular values and vectors of long rectangular random matrices, and tensor unfolding algorithm for asymmetric rank-one spiked tensor models.
result An exact threshold for tensor unfolding to detect signals, independent of unfolding procedure.
Proves singular set of certain integral hypercurrents has measure zero.
problem Characterizing singular sets of specific integral hypercurrents.
method Proof based on varifold stationarity.
result Singular set has measure zero.
Paper constructs fold maps with useful singular value sets.
problem Creating fold maps with specific singular value sets.
method Surgery operations to construct fold maps with crossings.
result Fold maps with singular value sets containing crossings.
Despite their prevalence in neural networks we still lack a thorough theoretical characterization of ReLU layers. This paper aims to further our understanding of ReLU layers by studying how the activation function ReLU interacts with the linear component of the layer and what role this interaction plays in the success …
We prove well-posedness and regularity results for elliptic boundary value problems on certain domains with a smooth set of singular points. Our class of domains contains the class of domains with isolated oscillating conical singularities, and hence they generalize the classical results of Kondratiev on domains with c…
Paper proves uniqueness of minimal maps in curved spaces.
problem Proving uniqueness of minimal maps into Cartan-Hadamard manifolds.
method Proof based on convexity of functions in terms of squared singular values.
result Uniqueness theorem for minimal maps into Riemannian manifolds.
Singular Value Decomposition (SVD) constitutes a bridge between the linear algebra concepts and multi-layer neural networks---it is their linear analogy. Besides of this insight, it can be used as a good initial guess for the network parameters, leading to substantially better optimization results.
The paper studies harmonic maps to the circle with complex singular sets.
problem Finding harmonic maps with prescribed singular sets in higher-dimensional spaces.
method Considered variational relaxations of the problem, showing energy convergence to a renormalised volume plus lower-order interaction energy.
result The energy of minimisers converges, after renormalisation, to the volume of the singular set plus a lower-order interaction energy.
Study geometric equations on cohomogeneity one manifolds near singular orbits.
problem Solving geometric equations like Ricci, Einstein, and soliton near singular orbits.
method Special assumption simplifies proof; general case solved in Part II.
result Existence and uniqueness of solutions near singular orbits.
We analyze the local Rademacher complexity of empirical risk minimization (ERM)-based multi-label learning algorithms, and in doing so propose a new algorithm for multi-label learning. Rather than using the trace norm to regularize the multi-label predictor, we instead minimize the tail sum of the singular values of th…
Stochastic gradient descent regularizes least squares problems by smoothing large singular values.
problem Regularization of least squares problems using stochastic gradient descent.
method Analysis of stochastic gradient descent applied to least squares problems, showing a regularization effect.
result Stochastic gradient descent leads to a quick regularization effect, smoothing large singular values.
Quantum SVT reduces credit risk analysis costs.
problem Efficiently estimating credit risk metrics using quantum computing.
method Quantum Singular Value Transformation (QSVT) to reduce state preparation costs.
result Significant reduction in implementation costs for quantum credit risk analysis.
New method cleans cross-covariance matrices for better financial forecasting.
problem Asymptotically optimal cross-covariance cleaners fail in real-world, time-varying markets.
method Physics-informed neural network that learns from empirical singular values.
result Trained model outperforms analytical cleaners in out-of-sample cross-covariance prediction.
We regard pre-trained residual networks (ResNets) as nonlinear systems and use linearization, a common method used in the qualitative analysis of nonlinear systems, to understand the behavior of the networks under small perturbations of the input images. We work with ResNet-56 and ResNet-110 trained on the CIFAR-10 dat…
The paper updates SVD of evolving matrices using projection techniques.
problem Updating the rank-k truncated SVD of evolving matrices.
method Projection viewpoint, building subspaces to approximate singular vectors.
result The proposed algorithm leads to higher accuracy, especially for large singular values.
Randomized SVD shows phase transitions in noisy data.
problem Noise sensitivity of randomized SVD in large rank matrices.
method Analyzed R-SVD under low-rank signal plus noise model.
result R-SVD exhibits BBP-like phase transition with outliers above detectability threshold.
In this paper we study the singular set of Dirichlet-minimizing Q-valued maps from Rm into a smooth compact manifold N without boundary. Similarly to what happens in the case of single valued minimizing harmonic maps, we show that this set is always (m−3)-rectifiable with uniform Minkowski b…
New nonconvex regularizer speeds up low-rank matrix completion.
problem Low-rank matrix completion with good theoretical and empirical performance.
method Proposes a new nonconvex regularizer with adaptive shrinkage, scalable, and fast optimization.
result Proposed method achieves state-of-the-art recovery performance and is the fastest.
Study examines how crypto arbitrage affects XRP price and network correlation.
problem Impact of crypto arbitrage on XRP price and network correlation.
method Examined XRP price fluctuations and correlation tensor spectra of transaction networks across crypto exchanges.
result Arbitrage opportunities across crypto exchanges anti-correlate with XRP price during bubble periods.
We describe how to compute topological objects associated to a polynomial map of several complex variables with isolated singularities. These objects are: the affine critical values, the affine Milnor numbers for all irregular fibers, the critical values at infinity, and the Milnor numbers at infinity for all irregular…
Let X be the moduli space of SL(n,C), SU(n), GL(n,C), or U(n)-valued representations of a rank r free group. We classify the algebraic singular stratification of X. This comes down to showing that the singular locus corresponds exactly to reducible representations if there exist singularities at all. Then by relating a…
The truncated singular value decomposition (SVD) of the measurement matrix is the optimal solution to the_representation_ problem of how to best approximate a noisy measurement matrix using a low-rank matrix. Here, we consider the (unobservable)_denoising_ problem of how to best approximate a low-rank signal matrix bur…
Singularities of even smooth functions are studied. A classification of singular points which appear in typical parametric families of even functions with at most five parameters is given. Bifurcations of singular points near a caustic value of the parameter are also studied. A determinant for singularity types and con…
Defines tensor eigenvalues and singular values without basis, simplifying analysis.
problem Defines tensor eigenvalues and singular values without basis.
method Intrinsic definition of tensor eigenvalues and singular values using concepts from pure mathematics.
result Shows the relationship between tensor analysis and pure mathematics.
Study estimates gaps in semigroup products, proving embedding properties.
problem Estimating singular value gaps in semigroup products.
method Lower estimates for singular value gaps of free products of semigroups in ping-pong position.
result Groups generated by semigroups in ping-pong position are quasi-isometrically embedded.
Ranky solves SVD for large sparse matrices in distributed systems.
problem Rank problem in large sparse matrices for SVD.
method Distributed approach to solve rank problem.
result Recovers SVD with negligible error for large sparse matrices.
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.
It is well known that the initialization of weights in deep neural networks can have a dramatic impact on learning speed. For example, ensuring the mean squared singular value of a network's input-output Jacobian is O(1) is essential for avoiding the exponential vanishing or explosion of gradients. The stronger condi…