Convex optimization with expander matrices improves sparse recovery efficiency.
problem Sparse recovery from linear measurements using expander matrices.
method Use of expander matrices for linear sketches in convex optimization to recover block-sparse matrices.
result The recovery error can be expressed in terms of the model-based norm, ensuring the solution is within the model.
We investigate the high-dimensional regression problem using adjacency matrices of unbalanced expander graphs. In this frame, we prove that the ℓ2-prediction error and the ℓ1-risk of the lasso and the Dantzig selector are optimal up to an explicit multiplicative constant. Thus we can estimate a high-dim…
Expands small recommendation datasets to industrial scale.
problem Disconnection between academic and industrial data scales.
method Randomized fractal expansions using Kronecker Graph Theory.
result Generated synthetic data sets with 1.2B ratings, 2.2M users, and 855K items.
TVS-FNNs can approximate any continuous function on expanded input spaces.
problem Processing a broader range of inputs like sequences and matrices.
method Proving a universal approximation theorem for TVS-FNNs.
result TVS-FNNs can approximate any continuous function on expanded input spaces.
The abstract theorem is extended to higher genus surfaces.
problem Generalizing the web trace theorem to higher genus surfaces.
method Geometric derivation and spin geometry of embedded loops.
result Expansion of twisted Kasteleyn matrices for higher genus surfaces.
Generates massive synthetic data sets for recommender systems.
problem Size gap between academic data sets and industrial production systems.
method Expands pre-existing public data sets using Kronecker Graph Theory.
result Preserves higher order statistical properties of user/item interactions.
Deterministic tensor completion using hypergraph expanders with linear sample complexity.
problem Low-rank tensor recovery with minimal samples.
method Minimizing max-quasinorm of tensors using hypergraph expanders.
result Deterministic analysis shows linear sample complexity for tensor recovery.
New derivation shows how a three-factor learning rule is derived from Oja's rule.
problem Deriving a three-factor learning rule from Oja's rule.
method Using frame theory to systematically derive EGHR-PCA from Oja's rule.
result A principled derivation of a biologically plausible learning rule.
We consider high-dimensional quadratic classifiers in non-sparse settings. The target of classification rules is not Bayes error rates in the context. The classifier based on the Mahalanobis distance does not always give a preferable performance even if the populations are normal distributions having known covariance m…
Gradient coding improves distributed learning efficiency.
problem Mitigating straggler issues in distributed learning.
method Gradient coding using cyclic MDS codes and expander graphs.
result Approximate gradient codes enable faster convergence and less computation.
This paper shows universality in spectrum behavior for random inner-product kernel matrices in polynomial regime.
problem Understanding spectrum behavior of random inner-product kernel matrices in polynomial regime.
method Analyzing matrices formed by a nonlinear function applied entrywise to a sample-covariance matrix, considering i.i.d. entries with all finite moments.
result The spectrum of random inner-product kernel matrices is universally described by the free convolution of the semicircular and Marčenko-Pastur distributions, with relative weights given by expanding the nonlinear function in the Hermite basis.
New tail inequalities for sums of random matrices without matrix-dimension terms.
problem Tail behavior of matrix functions in high-dimensional settings.
method Developed new tail inequalities for matrix sums, independent of matrix dimension.
result Tail inequalities for various matrix functions without matrix-dimension terms.
Formula establishes determinant majorization for symmetric matrices.
problem Determining determinant majorization for symmetric matrices.
method Establishes determinant majorization formula for symmetric matrices using invariant Garding-Dirichlet polynomials.
result Formula F(A)N1≥det(A)n1 for symmetric matrices. A method for community detection in multilayer networks using data matrices.
problem Community detection in multilayer networks with various node attributes.
method Data matrix representation and regular decomposition method extension for compression.
result Method identifies community structures well-aligned with real-world network hierarchies.
Classifies cobounded hyperbolic actions of metabelian groups.
problem Classify cobounded hyperbolic actions of metabelian groups.
method Builds connections between hyperbolic geometry and commutative algebra to classify actions.
result Classifies cobounded hyperbolic actions of many abelian-by-cyclic groups.
A Bayesian nonparametric approach for continual learning using neural networks.
problem Catastrophic forgetting in neural networks during sequential task settings.
method Indian Buffet Process (IBP) prior for dynamic model expansion and factorization of weight matrices.
result The method promotes positive knowledge transfer between tasks and allows for dynamic model complexity.
Constructs flow lines connecting unstable to stable self-expanders.
problem Existence of monotone Morse flow lines for expander functionals.
method Constructs a singular Morse flow line connecting unstable to stable self-expanders.
result Constructs a monotone flow line with a small singular set.
Matrix Chernoff bound for Markov chains applied to co-occurrence matrices.
problem Analyzing the behavior of co-occurrence statistics in sequential data.
method Proved a matrix Chernoff-type bound for sums of matrix-valued random variables sampled via a regular Markov chain.
result Achieved exponentially fast convergence rate and sample complexity analysis for co-occurrence matrices.
New expanders found using origami surfaces with spectral gap.
problem Constructing expanders with spectral gap on surfaces of arbitrary genus.
method Affine actions on origami surfaces to achieve spectral gap.
result New expanders distinct from classical ones.
The paper extends rigidity results for λ-self-expanders to hyperplanes, spheres, and cylinders.
problem Characterizing λ-self-expanders as hyperplanes, spheres, and cylinders. method Extending results on self-expanders to λ-self-expanders, proving rigidity results. result Characterizes hyperplanes, spheres, and cylinders as λ-self-expanders. Unified approach to measure expanders using finite graphs.
problem Constructing and understanding measure expanders.
method Defining and analyzing finite graphs approximating actions on measure spaces.
result Graphs form expanders if and only if the action is expanding in measure.
New self-expander found between two given asymptotic ones.
problem Finding new self-expanders between given asymptotic ones.
method Developed a min-max theory for asymptotically conical self-expanders of mean curvature flow.
result Existence of a new asymptotically conical self-expander trapped between two given ones.
This paper describes a fast algorithm for recovering low-rank matrices from their linear measurements contaminated with Poisson noise: the Poisson noise Maximum Likelihood Singular Value thresholding (PMLSV) algorithm. We propose a convex optimization formulation with a cost function consisting of the sum of a likeliho…
Unique expanders found with vanishing entropy.
problem Finding unique expanders with specific entropy properties.
method Adapting White's work and using Bernstein-Wang results.
result Generic uniqueness of expanders with vanishing relative entropy.
Rotational symmetry proven for certain self-expanders.
problem Analyzing self-expanders with decaying principal curvatures.
method Proved a Liouville-type theorem and applied it.
result Rotational symmetry for specific self-expanders.
No expanding breathers found in noncompact Ricci flows with certain curvature conditions.
problem Finding expanding breathers in noncompact Ricci flows.
method Curvature positivity conditions (weak PIC-2 or nonnegative bisectional curvature).
result Complete noncompact expanding breathers are gradient solitons.
Sharp curvature estimates for expanding Ricci solitons in various dimensions.
problem Estimating curvature bounds for expanding Ricci solitons.
method Sharp lower and upper bounds derived for scalar curvature under specific conditions.
result Sharp curvature estimates provided for expanding Ricci solitons in dimensions three and four.
Study shows smooth manifold of self-expanders in mean curvature flow.
problem Understanding the structure of self-expanders in mean curvature flow.
method Analyzes asymptotically conical self-expanders as a smooth Banach manifold.
result Non-degenerate self-expanders are generic.
Study expanding solitons on complex Lie groups with specific algebraic structures.
problem Investigate expanding solitons on complex Lie groups with left-invariant metrics.
method Analyze the algebraic structure of complex Lie groups and their decompositions.
result Show that the Lie algebras of complex Lie groups decompose into semidirect products.
Compactness proven for specific types of self-expanders in mean curvature flow.
problem Proving compactness for asymptotically conical self-expanders of mean curvature flow.
method Analyzing families of self-expanders and showing compactness in locally smooth topology.
result Properness of the projection map for specified classes of self-expanders.
New expanders for mean curvature flow contradict genus-reduction conjecture.
problem Contradicting Ilmanen's genus-reduction conjecture for mean curvature flow.
method Construct new expanders asymptotic to cones arising from shrinkers.
result Existence of expanders of arbitrarily large genus.
New degree theory proves existence of solitons on 4D manifolds.
problem Existence of gradient expanding solitons on 4D manifolds.
method Developed new degree theory for 4D, asymptotically conical gradient expanding solitons.
result Existence of solitons asymptotic to any cone over S^3 with non-negative scalar curvature.
New expanding Ricci solitons found starting in dimension four.
problem Finding expanding Ricci solitons in specific dimensions.
method Constructing gradient expanding Ricci solitons asymptotic to cones and on trivial vector bundles.
result Continuous families of expanding Ricci solitons on products of Einstein manifolds.
Stable expanding solitons with positive curvature decay proven.
problem Stability of expanding gradient Ricci solitons with positive curvature.
method Proving weak stability with quadratic curvature decay.
result Proven stability of expanding solitons with positive curvature.
Study on the spectrum of drift Laplacian on Ricci expanders.
problem Analyzing the spectrum of the drift Laplacian on Ricci expanders.
method Investigation of discrete spectrum under proper potential function, asymptotic behavior of potential function, and computation of eigenvalues.
result Discrete spectrum of the drift Laplacian on Ricci expanders with bounded Ricci curvature.
Expander graphs have been a focus of attention in computer science in the last four decades. In recent years a high dimensional theory of expanders is emerging. There are several possible generalizations of the theory of expansion to simplicial complexes, among them stand out coboundary expansion and topological expand…
The paper classifies expanding gradient Yamabe solitons based on scalar curvature.
problem Classifying expanding gradient Yamabe solitons based on scalar curvature.
method Rigorous analysis of scalar curvature in both cases: greater than and less than the soliton constant.
result Complete classification of nontrivial complete expanding gradient Yamabe solitons.
Constructs self-expanders of positive genus for cones in R^3.
problem Creating self-expanders of positive genus for cones in R^3.
method Constructs self-expanders asymptotic to cones, uses mean curvature flow.
result Constructs self-expanders with unbounded genus asymptotic to a rotationally symmetric cone.
Study of complete space-like self-expanders in Minkovski space.
problem Characterize complete space-like self-expanders in Minkovski space.
method Use of maximum principle of Omori-Yau type to prove rigidity theorems.
result Classification of 2-dimensional complete space-like self-expanders with constant squared norm of the second fundamental form.
New algorithm learns stable LDSs with lower error and better control performance.
problem Learning stable LDSs from data with minimal reconstruction error and stability constraints.
method Proposes an optimization method using a recent characterization of stable matrices, iteratively improving reconstruction error and ensuring stability.
result Achieves orders-of-magnitude improvement in reconstruction error compared to existing methods.
Study finds unique self-expanders for mean curvature flow.
problem Finding solutions to mean curvature flow.
method Derived equation based on generalized Lawson-Osserman cone and modified equilibria theory.
result Existence and uniqueness of self-expanders proved.
New theory explores high-dimensional expanders.
problem Understanding high-dimensional expanders.
method Exploring new mathematical and computational approaches.
result Developed new methods to study high-dimensional expanders.
Paper develops a decoder for sparse codes without encoder matrix, achieving optimal recovery.
problem Designing a decoder for sparse codes from linear measurements alone.
method Matrix factorization to recover encoder and sparse coding matrices from measurements.
result Decoder-Expander Based Factorisation recovers encoder and sparse coding matrix at optimal measurement rate with high probability.
Study cohomogeneity one expanding Ricci solitons on specific topologies.
problem Characterize and analyze cohomogeneity one expanding Ricci solitons.
method Analyze ODEs, define expander degree, calculate cohomogeneity one expander degree.
result Reconstruct and calculate cohomogeneity one expander degree for specific topologies.
The paper examines properties and rigidity of self-expanders in Euclidean space.
problem Characterizing and estimating properties of self-expanders in Euclidean space.
method Analyzing mean curvature flow, volume growths, and stability of self-expanders.
result Proves the uniqueness of certain self-expanders in 3D space.
Enhances sample diversity in SGMCMC for better uncertainty estimation in BNNs.
problem Limited sample diversity in SGMCMC affects uncertainty estimation and model performance.
method Reparameterizes neural network weights to produce a more diverse set of samples.
result The proposed approach achieves superior performance in image classification tasks, including OOD robustness.
We give a cohomological characterisation of expander graphs, and use it to give a direct proof that expander graphs do not have Yu's property A.
Strict convexity proven for certain self-expanders in high dimensions.
problem Convexity of self-expanders in mean curvature flow.
method Investigation of convexity properties for asymptotically conical self-expanders.
result Strict convexity proven for n-dimensional self-expanders.