Analytic proof for minimal rank Sard conjecture.
problem Proving the minimal rank Sard conjecture in the analytic category.
method Using subanalytic abnormal distribution from [4], we establish a proof.
result The set of points accessible through singular horizontal curves of minimal rank has Lebesgue measure zero.
Analytic torsion defined for rank 2 distributions on 5-manifolds.
problem Defining and analyzing analytic torsion for rank 2 distributions.
method Proposed an analytic torsion for Rumin complex associated with rank 2 distributions on 5-manifolds, established anomaly formulas, and showed coincidence with Ray-Singer torsion.
result The proposed torsion coincides with Ray-Singer torsion for certain nilmanifolds.
Analytic torsion matches Ray-Singer for specific nilmanifolds.
problem Matching analytic torsion with Ray-Singer in specific nilmanifolds.
method Examined rank two distributions on 5D nilmanifolds, proving torsion equality.
result Analytic torsion equals Ray-Singer torsion in these specific nilmanifolds.
Convolutional neural networks predict the analytic rank of elliptic curves accurately.
problem Predicting the analytic rank of elliptic curves over Q.
method Applied one-dimensional convolutional neural networks to Frobenius traces.
result High accuracy predictions for analytic rank across various conductors.
Abstract: Proves no non-trivial normal orbits for specific Hamiltonians.
problem Analyzing normal singular geodesics in a conformally generic sub-Riemannian metric.
method Proves the absence of non-trivial normal orbits for specific Hamiltonians.
result No non-trivial normal orbits for the specified Hamiltonians.
A method for ranking items using distance-based learning from positive and unlabeled data.
problem Learning to rank items without an analytic description of what constitutes a good ranking.
method Combining representations using an integer linear program for ranking items based on nominations.
result The method is effective in simulation and real data examples, especially when supervision is light.
A1GM method improves efficiency in reconstructing missing data using KL divergence.
problem Efficiently reconstructing missing data in matrices.
method Fast non-gradient-based rank-1 NMF using KL divergence.
result A1GM outperforms gradient methods in efficiency with competitive reconstruction errors.
Characterizes metabelian distributions and geodesics in sub-Riemannian manifolds.
problem Characterizing metabelian distributions and geodesics in sub-Riemannian manifolds.
method Characterization of metabelian distributions in terms of principal bundle structures. Proof of geodesic properties for rank-2 distributions.
result For rank-2 metabelian distributions, geodesics are of class C1. This paper examines how skip connections prevent rank collapse in sequence models.
problem Rank collapse in sequence models, leading to reduced expressivity and training instabilities.
method Analytical and ablation studies of lambda-skip connections in SSMs.
result A sufficient condition to prevent rank collapse across various architectures.
Study on signal recovery from low-rank matrix with sparse noise.
problem Inference of a rank-one signal in the presence of sparse noise.
method Replica method from statistical physics, recursive distributional equations, population dynamics algorithm.
result Critical signal strength for recovery via top eigenvector identified.
We study subgroups of fundamental groups of real analytic closed 4-manifolds with nonpositive sectional curvature. In particular, we are interested in the following question: if a subgroup of the fundamental group is not virtually free abelian, does it contain a free group of rank two ? The technique involves the theor…
Let (M,g) be a globally symmetric space of noncompact type, of arbitrary rank, and Δ its Laplacian. We prove the existence of a meromorphic continuation of the resolvent $(Δ-\ev)^{-1}$ across the continuous spectrum to a Riemann surface multiply covering the plane. The methods are purely analytic and are adapted fr…
Efficiently reduces rank of non-negative matrices with quadratic time complexity.
problem Efficiently reducing the rank of non-negative matrices.
method Formulated rank reduction as a mean-field approximation using a log-linear model.
result Optimal solution for minimizing KL divergence can be computed in closed form.
We propose the Legendrian web in a contact three manifold as a second order generalization of the planar web. An Abelian relation for a Legendrian web is analogously defined as an additive equation among the first integrals of its foliations. For a class of Legendrian d-webs defined by simple second order ODE's, w…
In this paper, we introduce a new geometric description of the manifolds of matrices of fixed rank. The starting point is a geometric description of the Grassmann manifold Gr(Rk) of linear subspaces of dimension r<k in Rk which avoids the use of equivalence classes. The set $\mathbb{…
In this paper we prove the strong Sard conjecture for sub-Riemannian structures on 3-dimensional analytic manifolds. More precisely, given a totally nonholonomic analytic distribution of rank 2 on a 3-dimensional analytic manifold, we investigate the size of the set of points that can be reached by singular horizontal …
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.
We consider actions of non-compact simple Lie groups preserving an analytic rigid geometric structure of algebraic type on a compact manifold. The structure is not assumed to be unimodular, so an invariant measure may not exist. Ergodic stationary measures always exist, and when such a measure has full support, we show…
This paper enhances uplift modeling for multi-treatment marketing campaigns.
problem Optimizing marketing strategies by selecting individuals likely to respond to different treatments.
method Leveraging score ranking and calibration techniques.
result Improves overall performance of marketing campaigns.
Let (M, g) be a simple, real analytic, Riemannian manifold with boundary and of dimension n>=3. In this work, we prove a support theorem for the transverse ray transform of tensor fields of rank 2 defined over such manifolds. More specifically, given a symmetric tensor field f of rank 2, we show that if the transverse …
Matrices of (approximate) low rank are pervasive in data science, appearing in recommender systems, movie preferences, topic models, medical records, and genomics. While there is a vast literature on how to exploit low rank structure in these datasets, there is less attention on explaining why the low rank structure ap…
Analytic torsion and Reidemeister torsion help show exponential growth of torsion in cohomology.
problem Exponential growth of torsion in the cohomology of arithmetic groups.
method Analytic torsion and Reidemeister torsion, applied to fibered cusp ends of manifolds.
result Exponential growth of torsion in the cohomology of arithmetic groups.
Study the distribution for low-rank matrix learning, improving inference methods.
problem Lack of understanding of underlying probability distributions in low-rank matrix learning.
method Analyze the distribution f(X)∝e−λ∥X∥∗, using differential geometry to design an improved MCMC algorithm and learn penalty parameter λ. result Improved MCMC algorithm and penalty parameter learning for low-rank Bayesian inference.
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.
Paper tackles ranking items with a semi-random comparison graph and a monotone adversary.
problem Ranking items based on pairwise comparisons from a semi-random comparison graph with a monotone adversary.
method Developed a weighted maximum likelihood estimator (MLE) and an SDP-based approach to reweight the semi-random graph.
result Achieves near-optimal sample complexity, up to a log^2(n) factor, for identifying the top-K preferred items.
In this paper we continue our program of extending the methods of geometric scattering theory to encompass the analysis of the Laplacian on symmetric spaces of rank greater than one and their geometric perturbations. In our previous work we described the resolvent, and specifically the asymptotic behavior of the Green'…
Unified model combines scores and rankings for grant panel review.
problem Combining scores and rankings for quality assessment in panel review.
method Mallows-Binomial model with tree-search algorithm for exact MLE.
result Model combines scores and rankings to quantify object quality and measure consensus.
An almost complex structure J on a 4-manifold X may be described in terms of a rank 2 vector bundle E. A splitting of J consists of a pair of line bundles spanning E. A hypersurface M in X satisfying a nondegeneracy condition inherits a CR-structure from J and a path geometry from the splitting. Using the Cartan-Kähler…
Low-rank inducing unitarily invariant norms have been introduced to convexify problems with low-rank/sparsity constraint. They are the convex envelope of a unitary invariant norm and the indicator function of an upper bounding rank constraint. The most well-known member of this family is the so-called nuclear norm. To …
We analyze the structure of covariance matrices under graph constraints.
problem Analyzing the structure of covariance matrices under graph constraints.
method We explore the algebraic structure of the solution space of convex optimization problem Constrained Minimum Trace Factor Analysis (CMTFA) under a latent star topology.
result CMTFA can have either a rank 1 or a rank n-1 solution, with conditions for both.
We study the structure of the invariant of K3 surfaces with involution, which we obtained using equivariant analytic torsion. It was known before that the invariant is expressed as the Petersson norm of an automorphic form on the moduli space. When the rank of the invariant sublattice of the K3-lattice with respect to …
The study optimizes supply chain management through a dice-based model to predict cleaner production.
problem Uncertainty in supply chain management and economic predictions.
method A 4-component SC module (environmental, demand, economic, social uncertainties) ranked by weight, using Analytical Hierarchical Process and optimization of a weighted cost function.
result Identifies conditions validating the sustainability of a business venture and optimizes market uncertainty.
Study uncovers complex critical points in tensor decomposition.
problem Nonconvex optimization of symmetric tensor decomposition.
method Utilized symmetry to construct critical points and analyze Hessian.
result Obtained precise analytic estimates on objective function and Hessian.
New method reduces variance in estimating PL model expectations.
problem High variance in Monte Carlo estimates of PL model expectations.
method Combining Gumbel top-k trick with quasi-Monte Carlo sampling.
result More sample-efficient estimators of PL model expectations.
Spirals are not shortest paths in certain sub-Riemannian geometries.
problem Nonminimality of spiral-like curves in sub-Riemannian manifolds.
method Construction of a competing curve to demonstrate non-minimality.
result Spiral-like curves are not length minimizing in sub-Riemannian manifolds.
Using the thermodynamics formalism, we introduce a notion of intersection for projective Anosov representations, show analyticity results for the intersection and the entropy, and rigidity results for the intersection. We use the renormalized intersection to produce a Out(Γ)-invariant Riemannian metric on the smooth …
Gradient descent solves rank-one matrix estimation problem with detailed time evolution analysis.
problem Estimating a rank-one symmetric matrix corrupted by noise.
method Gradient descent on a sphere, using local versions of the semi-circle law.
result Explicit formulas for the time evolution of the estimator and cost function, revealing phase transitions.
Study evaluates sustainability of European banks using a new model.
problem Lack of a framework to evaluate sustainability of banking business models.
method Delphi-Analytic Hierarchy Process method to develop and assess the model.
result Norwegian and German banks have higher sustainability of their business models.
The paper solves an insurance problem using mean-variance and rank-dependent utility theory.
problem Formulating and solving an insurance problem with rank-dependent utility and mean-variance premium principle.
method Formulated as a non-concave maximization problem, then turned into a concave quantile optimization problem, solved using calculus of variations.
result An optimal insurance contract is derived and numerically computed.
The total duration of drawdowns is shown to provide a moment-free, unbiased, efficient and robust estimator of Sharpe ratios both for Gaussian and heavy-tailed price returns. We then use this quantity to infer an analytic expression of the bias of moment-based Sharpe ratio estimators as a function of the return distrib…
Unified approach tackles high-dimensional tensor bandits with convex optimization and weakly decomposable regularizers.
problem Challenges in high-dimensional generalized tensor bandits where existing algorithms fail.
method Proposes a generalized linear tensor bandits algorithm with a unified analytical framework using convex optimization and weakly decomposable regularizers.
result Unified analytical framework provides better bounds and broader applicability compared to existing methods.
Develops complex harmonic maps for Teichmüller theory, proving new theorems.
problem Analyzing complex harmonic maps in Teichmüller theory.
method Complex harmonic maps and Higgs bundles.
result Proves a Bers-type theorem for rank 2 Hitchin components.
The theory of geometric zeta functions for locally symmetric spaces as initialized by Selberg and continued by numerous mathematicians is generalized to the case of higher rank spaces. We show analytic continuation, describe the divisor in terms of tangential cohomology and in terms of group cohomology which generalize…
New method ranks sectors and countries using local and aggregate I-O data.
problem Ranking sectors and countries in global value chains using incomplete I-O tables.
method Rank-1 approximation to I-O tables using local and aggregate information. result Consistently good performance in reconstructing rankings of upstreamness and downstreamness.
For smooth families of projective algebraic curves, we extend the notion of intersection pairing of metrized line bundles to a pairing on line bundles with flat relative connections. In this setting, we prove the existence of a canonical and functorial "intersection" connection on the Deligne pairing. A relationship is…
Researchers compute determinants and torsions of Rumin complex in specific Lie group representations.
problem Computing determinants and torsions of Rumin complex in specific Lie group representations.
method Analyzing Schrodinger and generic representations of the (2,3,5) nilpotent Lie group.
result Computed the spectrum and zeta regularized determinant of Rumin differentials in Schrodinger representations and evaluated their alternating product in generic representations.
The paper decomposes unsupervised learning's generalization error into model, data, and variance components.
problem Understanding the components of unsupervised learning's generalization error.
method Information-geometric decomposition of the Kullback-Leibler generalization error.
result The optimal rank in ε-PCA is the noise floor, balancing model-error gain and data-bias cost. Estimating the dependences between random variables, and ranking them accordingly, is a prevalent problem in machine learning. Pursuing frequentist and information-theoretic approaches, we first show that the p-value and the mutual information can fail even in simplistic situations. We then propose two conditions for r…