The paper explores how the cohomology of certain space arrangements stabilizes as the number of subspaces increases.
problem Stability of cohomology groups of complements of linear subspace arrangements.
method Representation stability in the context of cohomology groups, focusing on arrangements invariant under permutation of coordinates.
result Bounds on stabilization and alternative proof for the stabilization of cohomology groups.
Stability in cohomology of linear subspace arrangements.
problem Stability in cohomology of linear subspace arrangements.
method FI-modules, representation stability, generalized character polynomials.
result Cohomology of linear subspace arrangements satisfies strong form of representation stability.
Quaternionic hyperbolic groups stabilize complex subspaces if trace skew-field is commutative.
problem Characterizing discrete subgroups of quaternionic hyperbolic groups with commutative trace skew-fields.
method Analyzing the trace skew-field of discrete subgroups in Sp(n,1). result Quaternionic hyperbolic groups stabilize complex subspaces if their trace skew-field is commutative.
Stability result for nearly isometric subspaces and Finsler surfaces.
problem Stability of normed spaces and Finsler surfaces under near-isometric conditions.
method Refined topological argument and explicit quantification using Banach-Mazur distance.
result A 2-dimensional surface with near-monochromatic Finsler metric is approximately Riemannian.
We study the stability vis a vis adversarial noise of matrix factorization algorithm for matrix completion. In particular, our results include: (I) we bound the gap between the solution matrix of the factorization method and the ground truth in terms of root mean square error; (II) we treat the matrix factorization as …
New method accelerates neural network training by focusing on flat directions.
problem Improving neural network training speed and stability.
method Bulk-SGD, interpolated gradient methods.
result Updates along the Dominant subspace can accelerate convergence but compromise stability.
New algorithm improves topological stability in non-linear dimensionality reduction.
problem Topological instability in choosing nearest neighbors in Isomap.
method Uses point and its two nearest neighbors to find subspace and orthogonal complement, then adds new points based on distance and angle.
result Improves topological stability and reduces short-circuit errors.
New cyclicity measures defined in weighted Besov spaces, with stability and geometric analysis.
problem Characterizing cyclicity in weighted Besov spaces.
method Defining cyclicity indices based on potential theory and capacity, studying stability under perturbations, and linking zero set structure to cyclicity.
result Novel invariants and conditions for cyclicity in various function spaces.
New method solves saddle-point problems faster than existing methods.
problem Large-scale saddle-point problems in optimization.
method Sequential subspace optimization with proximal regularization.
result Significantly better convergence compared to first-order methods.
New methods for signal reconstruction using guiding sets and frame-less pathways.
problem Signal reconstruction in Hilbert spaces with specified properties.
method Axiomatic approach involving sample consistent and guiding sets, with reconstruction set defined as a shortest pathway.
result Existence and uniqueness of reconstruction set in Hilbert space, with derived stability and error bounds.
We show that the cone-volume measure of a convex body with centroid at the origin satisfies the subspace concentration condition. This implies, among others, a conjectured best possible inequality for the U-functional of a convex body. For both results we provide stronger versions in the sense of stability i…
The paper improves conditions for unique recovery in homomorphic sensing of subspaces.
problem Unique recovery of points in a linear subspace from their images under linear maps.
method Tighter and simpler conditions for unique recovery in single and subspace arrangement cases, extending to noise stability.
result Conditions for unique recovery in homomorphic sensing are improved and unified.
Paper proves linear convergence of SCMS algorithm for directional data.
problem Identifying density ridges in directional data.
method Generalized SCMS algorithm to directional data, derived from SCGA with adaptive step size.
result Linear convergence of the proposed directional SCMS algorithm.
SGD updates align with a low-rank subspace but do not lead to further loss reduction.
problem Understanding the training dynamics of deep neural networks, particularly the role of the dominant subspace.
method Exploring whether neural networks can be trained within the dominant subspace of the loss Hessian.
result SGD updates, when projected onto the dominant subspace, do not decrease the training loss further, suggesting spurious alignment.
In this paper we provide some stability criteria for systems of linear subspaces of V⊗W and for systems of quotient coherent sheaves, using, respectively, the Hilbert-Mumford numerical criterion and moment map. Along the way, we generalize the Gelfand-MacPherson correspondence [11] from point sets to sets of …
Study stabilizes arithmetic statistics of rational maps over finite fields.
problem Stability of arithmetic statistics of rational maps over finite fields.
method Representation stability and arithmetic statistics of spaces of 0-cycles.
result Arithmetic quantities associated to rational maps over finite fields stabilize as degree increases.
Shows CM line bundles are ample on K-stable varieties.
problem Ensuring CM line bundles are ample on K-stable varieties.
method Analyzes CM line bundles on K-stable varieties and their families.
result CM line bundles are ample on K-stable varieties with maximal variation.
Study stability of Einstein manifolds with boundary.
problem Stability of Einstein manifolds with geometric boundary conditions.
method Using Ricci flow and calculus of variations, analyze stability with respect to the Einstein-Hilbert action.
result Introduce a new subspace of tensors (TVg tensors) for stability condition due to boundary constraints.
Deep linear networks oscillate beyond the edge of stability in a predictable manner.
problem Understanding oscillations in deep linear networks beyond the edge of stability.
method Theoretical analysis of loss oscillations in deep matrix factorization loss.
result Loss oscillations in deep linear networks follow a period-doubling route to chaos and occur within a small subspace.
Study homological stabilization in Hurwitz spaces using plant complexes.
problem Homological stabilization in Hurwitz spaces with specific properties.
method Introduce plant complexes and generalize Ellenberg-Venkatesh-Westerland result.
result Homological stabilization depends only on zeroth homology groups.
Noise causes learning plateaus in neural networks.
problem Plateau phenomena in online learning due to vanishing gradients.
method Analysis of stochastic gradient descent in multi-layer perceptrons.
result Noise induces synchronisation leading to strong plateaus.
Tyler's M-estimator's phase transition at DS-SNR = 1 is resolved.
problem Robust Subspace Recovery
method Tyler's M-estimator
result TME converges exactly to the true subspace for DS-SNR >= 1 under a new stability condition.
Paper tackles distribution shifts in prediction models with unobserved confounding.
problem Distribution shifts in prediction models with unobserved confounding.
method Linear structural causal model, invariant covariate representations, data-driven representation learning method.
result Optimizes for a lower-dimensional linear subspace and a prediction model confined to that subspace, achieving nearly ideal gap between target and source risk.
We prove that the kernels of the restrictions of symplectic Dirac or symplectic Dirac-Dolbeault operators on natural subspaces of polynomial valued spinor fields are finite dimensional on a compact symplectic manifold. We compute those kernels for the complex projective spaces. We construct injections of subgroups of t…
Local Neural Operators enable efficient system-level analysis of complex PDEs.
problem System-level analysis of large-scale dynamical systems using neural operators.
method Integrating local Neural Operators with Krylov subspace iterative methods for stability and bifurcation analysis.
result Demonstrated effectiveness of local Neural Operators in fixed-point, stability, and bifurcation analysis of nonlinear PDEs.
Classifies stable hypersurfaces in real projective spaces and confirms the isoperimetric conjecture.
problem Volume preserving stability and isoperimetric problem in real projective spaces.
method Classification of stable hypersurfaces and analysis of antipodal invariant hypersurfaces.
result Solutions of the isoperimetric problem are tubular neighborhoods of projective subspaces.
Sparse representations using learned dictionaries are being increasingly used with success in several data processing and machine learning applications. The availability of abundant training data necessitates the development of efficient, robust and provably good dictionary learning algorithms. Algorithmic stability an…
Study investigates non-Kahler Ricci flow singularities converging to Kahler-Ricci solitons.
problem Investigating finite-time Type-I singularities in non-Kahler Ricci flow solutions.
method Examining the convergence of parabolic rescalings at singularities to shrinking Kahler-Ricci solitons.
result Supports the conjecture that the blowdown soliton is stable under Ricci flow and the subspace of Kahler metrics is stable under Ricci flow.
The paper proves conditions for minimal surfaces to be holomorphic and stable.
problem Conditions for stable minimal surfaces to be holomorphic.
method Developed a method of constructing variations to prove the equivalence.
result Holomorphicity and stability conditions for minimal surfaces.
The paper proves a CM line bundle is ample on K-moduli spaces.
problem Proving positivity of the CM line bundle on K-moduli spaces.
method Developed a new invariant for filtrations to test K-stability.
result Proves the CM line bundle is ample on reduced uniformly K-stable Fano varieties.
LASER compresses recursive model activations by exploiting their low-dimensional structure.
problem Understanding and optimizing the geometric structure of recursive reasoning trajectories.
method Dynamic low-rank basis tracking via matrix-free subspace tracking with a fidelity-triggered reset mechanism.
result Recursive activations occupy a linear, low-dimensional subspace that can be compressed efficiently.
PLUMAGE improves large model training efficiency and stability.
problem Accelerator memory and networking constraints during large model training.
method Probabilistic Low rank Unbiased Minimum Variance Gradient Estimator (PLUMAGE) that resolves bias and variance issues.
result PLUMAGE reduces training loss by 28% on average across the GLUE benchmark.
The paper classifies orbits of semisimple elements in real semisimple Lie algebras.
problem Classifying orbits of semisimple elements in real semisimple Lie algebras.
method Case by case analysis of complex numbers and Galois cohomology for real numbers.
result Characterization of orbits with real representatives.
This work ensures stability in POD basis interpolation for pMOR in hyperelasticity.
problem Stability of POD basis interpolation on Grassmann manifolds for pMOR in hyperelasticity.
method Stability conditions derived from Grassmannian Exponential map and principal angles.
result Explicit stability conditions for practical pMOR applications and non-monotonic error behavior.
The study examines stable minimal surfaces in higher dimensions and provides bounds on their properties.
problem Properties of stable minimal surfaces in higher codimension.
method Structural analysis of holomorphic vector bundles and geometric inequalities.
result Explicit bounds on the systole for stable minimal tori and surfaces.
Unified approach to studying hyperbolic groups using stable subspaces and Morse boundaries.
problem Understanding the geometric and algebraic properties of hyperbolic groups.
method Unified approach to viewing geodesic metric spaces as unions of stable subspaces, using quasi-convex subsets and direct limits of Gromov boundaries.
result Unified understanding of stable subgroups and Morse boundaries, leading to new quasi-isometry invariant dimensions.
We provide abelianizations of differentiable actions of finite groups on smooth real manifolds. De Concini-Procesi wonderful models for (local) subspace arrangements and a careful analysis of linear actions on real vector spaces are at the core of our construction. In fact, we show that our abelianizations have stabili…
A new framework selects best outlier detectors locally for improved ensemble performance.
problem Challenges in combining outlier detectors without ground truth.
method Locally Selective Combination in Parallel Outlier Ensembles (LSCP) framework.
result LSCP_AOM variant consistently outperforms other methods on real-world datasets.
Algorithm fine-tunes multi-view weights for streaming data.
problem Streaming data in multi-view learning.
method Fine-tuning combination weights of stable subspaces.
result Algorithm effectively handles streaming views.
The paper shows how to stabilize off-policy reinforcement learning using specific state representations.
problem Stability issues in reinforcement learning with function approximation and off-policy learning.
method Formal analysis of representation learning schemes based on the transition matrix of a policy.
result Schur and orthogonal bases of the Krylov subspace provide stable representations for TD learning.
For a connected abelian Lie group T acting on a Poisson manifold (Y,π) by Poisson isomorphisms, the T-leaves of π in Y are, by definition, the orbits of the symplectic leaves of π under T, and the leaf stabilizer of a T-leaf is the subspace of the Lie algebra of T that is everywhere tangent to all the symplectic leaves…
This paper presents a stochastic behavior analysis of a kernel-based stochastic restricted-gradient descent method. The restricted gradient gives a steepest ascent direction within the so-called dictionary subspace. The analysis provides the transient and steady state performance in the mean squared error criterion. It…
Proposes KMvDA for object recognition from multi-view data.
problem Recognizing objects from different views, even when views are heterogeneous.
method Introduces kernel multi-view discriminant analysis (KMvDA) and uses random Fourier features (RFF) for large-scale learning.
result KMvDA and RFF approximation improve object recognition from multi-view data.
In two previous papers the author developed a second-order price adjustment (tâtonnement) process. This paper extends the approach to include both quantity and price adjustments. We demonstrate three results: a analogue to physical energy, called "activity" arises naturally in the model, and is not conserved in general…
We propose a general framework to study the stability of the subspace spanned by P consecutive eigenvectors of a generic symmetric matrix H0, when a small perturbation is added. This problem is relevant in various contexts, including quantum dissipation (H0 is then the Hamiltonian) and risk control …
New hyperbolic 3-pseudomanifolds with unique properties.
problem Understanding cubulable groups in hyperbolic 3-manifolds.
method Constructing compact hyperbolic 3-manifolds with specific boundary conditions.
result Found groups that are word hyperbolic but not cubulable.
CPCR mitigates bias in PCR for overparameterized models.
problem Bias in Principal Component Regression (PCR) for overparameterized models.
method Calibrated Principal Component Regression (CPCR) learns a low-variance prior in the PC subspace and calibrates the model in the original feature space.
result CPCR outperforms standard PCR in overparameterized settings, improving prediction across multiple problems.
Batching stabilizes risk in high-dimensional linear regression models.
problem Stability and risk behavior in high-dimensional overparameterized linear regression.
method Minimum-norm overparameterized linear regression model with batch-partitioning.
result Optimal batch size is inversely proportional to noise level and overparametrization ratio, leading to stable risk behavior.