Analyzes the structure and rank of neural network Hessians.
problem Understanding redundancy in overparameterized neural networks.
method Theoretical tools to analyze Hessian map range and rank deficiency.
result Exact formulas and tight upper bounds for Hessian rank of deep linear networks.
The study calculates the Smith-Thom deficiency of Hilbert squares and provides conditions for maximality.
problem Calculating the Smith-Thom deficiency of Hilbert squares and conditions for maximality.
method Using Mayer-Vietoris mapping and rank calculations.
result Established necessary and sufficient conditions for maximality of Hilbert squares in projective complete intersections.
A new algorithm solves constrained optimization problems with stochastic gradients.
problem Nonlinear equality constrained optimization with rank-deficient Jacobians.
method Step decomposition strategy combining normal and tangential steps.
result Convergence guarantees in rank-deficient Jacobian cases.
Gradient descent with preconditioning finds global optima in overparameterized nonconvex factorization.
problem Finding global optima in nonconvex Burer-Monteiro factorization.
method Preconditioned gradient descent for overparameterized nonconvex function minimization.
result Gradient descent with preconditioning achieves linear convergence in the overparameterized case.
Regular integer lattices are characterized by k unit vectors that build up their generator matrices. These have rank k for D-lattices, and are rank-deficient for A-lattices, for E_6 and E_7. We count lattice points inside hypercubes centered at the origin for all three types, as if classified by maximum infinity norm i…
New method identifies causal direction with latent confounders.
problem Identifying causal direction in presence of multiple latent variables.
method Use of joint higher-order cumulant matrix properties.
result Causal asymmetry can be seen from rank deficiency properties of cumulant matrices.
New algorithm improves asset ranking for better cross-sectional portfolios.
problem Sub-optimal ranking of assets in cross-sectional systematic strategies.
method Learning-to-rank algorithms to enhance portfolio construction.
result Modern machine learning ranking algorithms boost Sharpe Ratios by approximately threefold.
New algorithms estimate matrix leverage scores using rank revealing and randomization.
problem Estimating leverage scores for matrices of arbitrary rank.
method Combining rank revealing methods with randomized dimensionality reduction.
result Effective estimators for leverage scores, even in rank deficient cases.
Riemannian gradient descent escapes some spurious critical points on low-rank matrix manifold.
problem Spurious critical points on the boundary of low-rank matrix manifold.
method Riemannian gradient descent with dynamical low-rank approximation and rescaled gradient flow.
result Riemannian gradient descent escapes some spurious critical points on the boundary of the manifold.
Bayesian model predicts iron deficiency from multi-source multi-way molecular data.
problem Predicting iron deficiency in rhesus monkeys from multi-source multi-way molecular data.
method Developed a Bayesian approach with a linear model incorporating multi-way dependence and varying signal sizes across sources.
result Model accurately classifies iron deficiency in monkeys and outperforms simpler models.
New method differentiates square-root Kalman filters robustly.
problem Gradient calculation issues in square-root Kalman filters.
method Closed-form chain rule derived from Gramian identity, resolves non-orthogonal and rank-deficient issues.
result Robust automatic differentiation for Kalman filters, resolving numerical stability and gradient issues.
New method reduces summary points for datasets while maintaining quality.
problem Thinning datasets to reduce summary points while maintaining quality.
method Low-rank analysis of sub-Gaussian thinning.
result Guarantees high-quality compression for any distribution and kernel.
Slow feature analysis (SFA) is a method for extracting slowly varying features from a quickly varying multidimensional signal. An open source Matlab-implementation sfa-tk makes SFA easily useable. We show here that under certain circumstances, namely when the covariance matrix of the nonlinearly expanded data does not …
Geometric families of low-rank covariances improve flexibility and tractability in high dimensions.
problem Interpolating and identifying covariance matrices in high dimensions with limited data.
method Differential geometric construction of low-rank covariance families, interpolation on manifolds, and distance minimization for identification.
result Differential geometric covariance families offer significant flexibility and computational tractability.
Generalizing the theorem of Green--Lazarsfeld and Gromov, we classify Kaehler groups of deficiency at least two. As a consequence we see that there are no Kaehler groups of even and strictly positive deficiency. With the same arguments we prove that Kaehler groups that are non-Abelian and are limit groups in the sense …
The paper analyzes a five-factor capital market model and facilitates exact simulation.
problem Analyzing and simulating a five-factor capital market model.
method Using a Vasicek interest rate model, mean-reverting excess return, and realized inflation with expectation, the paper derives the necessary distributional results and describes practical methods to overcome rank deficiency.
result Exact simulation from the model can be achieved by sampling from a seven-dimensional normal distribution.
The paper defines and explores Coxeter type LOTs for ribbon 2-knots.
problem Validity of Whitehead's asphericity conjecture for Ribbon 2-knots.
method Definition and analysis of Coxeter type LOTs and their groups.
result Existence of prime LOTs of Coxeter type with specified rank.
Let L be a lattice in a connected Lie group. We show that besides a few exceptional cases, the deficiency of L is nonpositive.
Factor analysis, a classical multivariate statistical technique is popularly used as a fundamental tool for dimensionality reduction in statistics, econometrics and data science. Estimation is often carried out via the Maximum Likelihood (ML) principle, which seeks to maximize the likelihood under the assumption that t…
In high-dimensional data analysis, regularization methods pursuing sparsity and/or low rank have received a lot of attention recently. To provide a proper amount of shrinkage, it is typical to use a grid search and a model comparison criterion to find the optimal regularization parameters. However, we show that fixing …
We derive an efficient method to perform clustering of nodes in Gaussian graphical models directly from sample data. Nodes are clustered based on the similarity of their network neighborhoods, with edge weights defined by partial correlations. In the limited-data scenario, where the covariance matrix would be rank-defi…
The study shows subgroup separability conditions for specific groups.
problem Conditions for subgroup separability in free-by-cyclic and deficiency 1 groups.
method Analyzes polynomially growing monodromy and asymptotic probability of random groups.
result Random deficiency 1 groups are not subgroup separable with positive probability.
Contextual bandit methods fail with deficient support data.
problem Learning from support-deficient data in contextual bandits.
method Three approaches to IPS-based learning: action space restriction, reward extrapolation, and policy space restriction.
result Systematic analysis and empirical evaluation of approaches to IPS-based learning.
Deep ReLU networks with extra parameters have mostly good loss landscapes.
problem Finding good local minima in the loss landscape of deep neural networks.
method Analyzing shallow and deep ReLU networks with extra parameters on a generic dataset.
result Most activation patterns correspond to regions with no bad local minima.
We consider the teacher-student setting of learning shallow neural networks with quadratic activations and planted weight matrix W∗∈Rm×d, where m is the width of the hidden layer and d≤m is the data dimension. We study the optimization landscape associated with the empirical and the popula…
We study the asymptotic growth of homology groups and the cellular volume of classifying spaces as one passes to normal subgroups Gn<G of increasing finite index in a fixed finitely generated group G, assuming ⋂nGn=1. We focus in particular on finitely presented residually free groups, calculating thei…
MaxVol NMF maximizes the volume of H in NMF for better sparse and interpretable solutions.
problem Finding interpretable and unique NMF solutions.
method Dual approach to MinVol NMF, maximizing the volume of H. result MaxVol NMF solutions correspond to clustering columns in disjoint clusters.
The paper addresses flaws in fixed point assertions for digital images.
problem Deficiencies in previously published works on fixed point assertions for digital images.
method Continues a series of studies to identify and rectify issues in fixed point assertions.
result Identifies and corrects flaws in fixed point assertions for digital images.
Study nondifferentiable metrics in general relativity, resolving causality issues and limits evolution scenarios.
problem Causality issues and evolution scenarios in black hole interiors with closed timelike geodesics.
method Method of equivalence on Courant algebroids to derive new differential invariants.
result Resolved causality issues and limited evolution scenarios for gravitational collapse.
SON-NMF estimates nonnegative rank on-the-fly for NMF.
problem Estimating the nonnegative rank of data in NMF.
method Sum-of-norms (SON) regularization to reduce rank, combined with a first-order BCD algorithm.
result SON-NMF can automatically estimate the rank from data without prior knowledge.
The paper calculates the number of oriented rational links with a given deficiency.
problem Counting oriented rational links with a specific deficiency.
method Derived precise formulas for the number of oriented rational links with crossing number n and deficiency d.
result Precise formulas for the number of oriented rational links with crossing number n and deficiency d.
Soft-Radial Projection solves gradient saturation in constrained deep learning.
problem Gradient saturation in deep learning models when integrating hard constraints.
method Introduces Soft-Radial Projection, a differentiable layer that maps predictions onto constraint boundaries without rank-deficient Jacobians.
result Improves convergence and solution quality over state-of-the-art methods.
Study tackles ranking fraud in online platforms by learning robust rankings.
problem Fraudulent fake users manipulate product rankings.
method Developed algorithms for robust ranking in two informational environments.
result Our algorithms converge to optimal rankings, robust to fake users.
Motivated by problems in topology, we explore the complexity of balanced group presentations. We obtain large lower bounds on the complexity of Andrews-Curtis trivialisations, beginning in rank 4. Our results are based on a new understanding of how Dehn functions of groups behave under certain kinds of push-outs. We co…
DBKs enable scalable GPs with tractable inference for large datasets.
problem Scaling Gaussian processes to large and complex datasets while maintaining tractable inference.
method DBKs constructed from neural-network-parameterized basis functions with explicit low-rank structure, enabling linear-complexity inference.
result DBKs provide a unified perspective and improve predictive accuracy, uncertainty quantification, and computational efficiency.
Kernel approximation using randomized feature maps has recently gained a lot of interest. In this work, we identify that previous approaches for polynomial kernel approximation create maps that are rank deficient, and therefore do not utilize the capacity of the projected feature space effectively. To address this chal…
New method corrects Laplace/BIC errors in singular models, revealing effective dimension.
problem Laplace/BIC errors in singular models due to incorrect effective dimension assumption.
method RLCT (real log canonical threshold) to correct effective dimension in linear models.
result Correct evidence slope and effective dimension estimation in linear settings.
We examine certain symmetries in the deficiencies of a rational surgery on a knot in S3 by comparing the Spinc-structures on the rational surgery with those on a related integral surgery. We then provide an application of these symmetries in the form of a theorem that obstructs Dehn surgeries in S3. Thi…
Specialists tolerate defects to gain flexibility, which can be removed when needed.
problem The economic benefits and limitations of deliberately tolerating defects in decision-making.
method Analyzes the conditions under which defects can be kept and removed, using economic models and structural analysis.
result A defect is profitably removable if certain conditions are met, and the premium is the support function of the class's ROC set.
Specialists tolerate defects to gain flexibility, which can be removed when needed.
problem The economic benefits and limits of deliberately tolerating defects in decision-making.
method Analyzes the economic position of keeping and removing defects, using a coupling lemma and structural economic models.
result A defect is profitably removable if the detector-relevant distinction survives a restriction and the advantage condition holds.
Study uses DHS to classify anemia types using CBC indices.
problem Anemia classification for medical purposes.
method Dynamic Harmony Search (DHS) applied to CBC indices.
result DHS outperforms other models in anemia classification.
Proves one-relator groups are fundamental groups of Sasakian manifolds.
problem Characterizing groups that can be fundamental groups of Sasakian manifolds.
method Analyzes one-relator groups and classifies groups of deficiency at least two.
result Proves sufficient and necessary conditions for one-relator groups to be fundamental groups of Sasakian manifolds.
"Deep Learning" methods attempt to learn generic features in an unsupervised fashion from a large unlabelled data set. These generic features should perform as well as the best hand crafted features for any learning problem that makes use of this data. We provide a definition of generic features, characterize when it i…
Proposes a tensor Laplacian-based method for better subspace clustering of non-uniformly distributed data.
problem LRR's inability to handle non-uniform data distribution and local information loss.
method Tensor Laplacian Regularized Low-Rank Representation (TLRR) using hypergraph model and tensor Laplacian algorithm.
result Higher accuracy and precision in subspace clustering compared to state-of-the-art methods.
For every N > 0 there exists a group of deficiency less than -N that arises as the fundamental group of a smooth homology 4-sphere and also as the fundamental group of the complement of a compact contractible submanifold of the 4-sphere. A group is the fundamental group of the complement of a contractible submanifold o…
We address two fundamental and well-known problems of Gromov and Lyndon: \demo{Problem A} (Gromov, see [5]). Consider a category Mn of closed manifolds of dimension n with nonzero-degree ways as morphisms. Study a partial order M≥N⇔Mor(M,N)=φ. For which N the degrees of maps $f: M \t…
A new method combines online and offline learning to tackle contextual bandits with missing action support.
problem Learning optimal policies with logged data when the logging policy has deficient support.
method Hybrid approach using online exploration to exploit supported actions and offline learning to avoid unnecessary explorations.
result Determines an optimal policy with theoretical guarantees using minimal online explorations.
Magnetoencephalography and electroencephalography (M/EEG) can reveal neuronal dynamics non-invasively in real-time and are therefore appreciated methods in medicine and neuroscience. Recent advances in modeling brain-behavior relationships have highlighted the effectiveness of Riemannian geometry for summarizing the sp…