A new method for gradient recovery on manifold data without tangent spaces.
problem Gradient recovery schemes for data on discretized manifolds.
method Parametric Polynomial Preserving Recovery (PPPR) on manifolds.
result Superconvergence and high curvature stability of PPPR.
The paper studies the asymptotic behavior of adversarial training under ℓ ∞ \ell_\infty ℓ ∞ -perturbation.
problem Theoretical guarantees for sparsity-recovery in adversarial training.
method Investigation of the asymptotic distribution of the adversarial training estimator in generalized linear models.
result The asymptotic distribution of the adversarial training estimator under ℓ ∞ \ell_\infty ℓ ∞ -perturbation could have a positive probability mass at 0 when the true parameter is 0. Proof of Gaussian ML estimator consistency in linear auto-regressive models.
problem Consistency of Gaussian maximum likelihood estimator in linear auto-regressive models.
method Information-theoretic proof without stability assumptions.
result Nearly optimal non-asymptotic rates for parameter recovery.
This paper presents the first theoretical results showing that stable identification of overcomplete μ μ μ -coherent dictionaries Φ ∈ R d × K Φ\in \mathbb{R}^{d\times K} Φ ∈ R d × K is locally possible from training signals with sparsity levels S S S up to the order O ( μ − 2 ) O(μ^{-2}) O ( μ − 2 ) and signal to noise ratios up to O ( d ) O(\sqrt{d}) O ( d ) . In particular the di…
Paper explores exact recovery of communities in weighted graphs using Gaussian and exponential distributions.
problem Exact recovery of communities in weighted graphs with Gaussian and exponential distributions.
method Introduces a new semi-metric to describe conditions for exact recovery and analyzes conditions for both complete and incomplete graphs.
result Necessary and sufficient conditions for exact recovery are asymptotically tight and applicable to both complete and incomplete graphs.
Study identifies key differences in convex relaxations for combinatorial penalties.
problem Understanding which structures are preserved by convex relaxations for combinatorial penalties.
method Examined homogeneous and non-homogeneous convex relaxations, introduced lower combinatorial envelope, and proposed adaptive estimator.
result Identified new necessary and sufficient conditions for support recovery in convex monotone regularizers.
Theoretical guarantees for STE, a robust subspace recovery method.
problem Recovering a low-dimensional subspace from corrupted data.
method Subspace-constrained Tyler's estimator (STE) with initialization conditions.
result STE can effectively recover the subspace under certain conditions.
Models predict order book recovery from aggressive trading follows a simple t^1/3 scaling.
problem Understanding order book dynamics in financial markets.
method Developed macroscopic models and solved numerically and asymptotically.
result Order book recovery follows a t^1/3 scaling law.
The paper studies recovering hidden nearest neighbor graphs in large networks.
problem Discovering strong ties in social networks and assembling genome subsequences.
method Maximum likelihood estimator for recovering hidden 2 k 2k 2 k -nearest neighbor graphs. result The maximum likelihood estimator achieves asymptotic recovery guarantees under specific conditions.
Spectral methods improve signal recovery in mixed GLMs with precise asymptotics.
problem Estimating multiple signals from unlabeled observations in mixed GLMs.
method Developed exact asymptotics for spectral methods in a proportional regime.
result Optimized spectral method combined with a linear estimator minimizes estimation error.
Study shows generative priors improve rank-one matrix recovery with optimal sample complexity.
problem Recovering a rank-one signal matrix from noisy data with additional prior information.
method Analysis of a nonlinear least squares objective with a favorable global optimization landscape.
result Established optimal sample complexity for generative priors in rank-one matrix recovery.
Study supports recovery of PDEs from noisy data using a specific regularization method.
problem Support recovery of PDEs from a single noisy trajectory.
method Applying ℓ1-regularized Pseudo-Least Squares model to a given data set.
result Support of ℓ1-c coefficients asymptotically converges to the true signed-support of the PDE.
We study the problem of recovering a hidden community of cardinality K K K from an n × n n \times n n × n symmetric data matrix A A A , where for distinct indices i , j i,j i , j , A i j ∼ P A_{ij} \sim P A ij ∼ P if i , j i, j i , j both belong to the community and A i j ∼ Q A_{ij} \sim Q A ij ∼ Q otherwise, for two known probability distributions P P P and Q Q Q depending on n n n . If $P={\r…
Paper analyzes adaptive Lasso for high-dimensional diffusion processes, improving support recovery and bias.
problem Support recovery for high-dimensional diffusion processes under sparsity constraints.
method Adaptive Lasso estimator for d-dimensional ergodic diffusion process, focusing on linear models.
result Adaptive Lasso achieves support recovery and asymptotic normality for drift parameter under certain conditions.
Unified analysis of reweighted least-squares algorithms for linear models.
problem Recovering unknown signals from linear measurements using reweighted least squares.
method Unified asymptotic analysis of IRLS, lin-RFM, and alternating minimization algorithms.
result The algorithms can achieve favorable performance in a few iterations with appropriate reweighting.
Study recovers community structure from coarse graph measurements.
problem Community recovery from low-resolution graph measurements.
method Formalized coarsening process of graph measurements, developed conditions for perfect recovery.
result Simple and closed-form asymptotic conditions for perfect recovery of coarse graph communities.
Optimal low rank tensor recovery requires a minimum number of entries for accurate reconstruction.
problem Exact recovery of high order tensors of low rank from a subset of their entries.
method Riemannian optimization algorithm with initial value from a spectral method, leveraging tensor restricted isometry property and curvature of the manifold.
result Tensor of size n i m e s n i m e s ⋯ i m e s n n imes n imes \cdots imes n nim es nim es ⋯ im es n of ranks ( r , ⋯ , r ) (r,\cdots,r) ( r , ⋯ , r ) can be reconstructed with high probability from O ( ( r d + d n r ) log ( d ) ) O((r^d+dnr)\log(d)) O (( r d + d n r ) log ( d )) entries. SyncRank recovers global ranking from noisy comparisons with theoretical guarantees.
problem Recovering a global ranking from noisy pairwise comparisons.
method Complex-valued data model and SDP relaxation for exact ranking recovery.
result SyncRank achieves exact ranking recovery with high probability above a critical noise threshold of O(sqrt(n / log n)).
Unfolded ISTA achieves linear convergence for sparse signal recovery.
problem Understanding and improving convergence of iterative algorithms in sparse signal recovery.
method Introducing weight structure and incorporating thresholding in the network.
result Unfolded ISTA can achieve linear convergence, better than sublinear convergence of ISTA/FISTA.
Study improves distributed linear estimation under adversarial conditions.
problem Mean estimation of a random vector with adversarial measurements and asynchrony.
method Two-timescale ℓ1-minimization algorithm with tight convergence rates.
result Unified finite-time characterization of robustness, identifiability, and statistical efficiency.
AMP with Gaussian initialization shows weak-recovery threshold for phase retrieval.
problem Phase retrieval with noiseless data.
method Approximate message passing with random initialization.
result Random initialization attains weak-recovery threshold \( \delta_{ ext{weak}} = 1/2 \).
Enhanced Elastic-Net with box-constraint improves support recovery in noisy measurements.
problem Support recovery of sparse signals from noisy measurements.
method Box-Elastic Net (Box-EN) method with mean squared error and probability of support recovery analysis.
result The Box-Elastic Net outperforms the standard Elastic-Net in support recovery.
Exact recovery method for community detection in Gaussian mixtures with dependent noise.
problem Community detection in Gaussian mixtures with dependent and heterogeneous noise.
method Maximum likelihood estimator (MLE) for constrained quadratic optimization problem, using Σ Σ Σ -whitened separation and local inequalities. result Sharp exact-recovery threshold and no-gap mechanism in the unknown-size setting.
The study examines how side information quality and quantity affect community recovery in graphs.
problem Recovering a hidden community of size K = o ( n ) K=o(n) K = o ( n ) in a graph of size n n n . method Maximum likelihood detection and belief propagation are used to calculate necessary and sufficient conditions for exact and weak recovery. A local voting procedure is also designed and analyzed.
result Tight necessary and sufficient conditions for exact and weak recovery are derived, showing how side information needs to evolve with n n n to improve recovery thresholds. Paper develops a new algorithm for sparse signal recovery.
problem Sparse signal recovery from noisy observations.
method Iterative Stochastic Optimization using Stochastic Mirror Descent.
result Linear convergence during preliminary phase of the routine.
Spectral flow connects manifold geometry to rigidity criteria.
problem Tackling rigidity of simply-connected closed manifolds.
method Spectral deformation flow and invariant-based approach.
result Spherical profile is the unique manifold-compatible asymptotic realization.
Paper analyzes structured matrix recovery using generalized Dantzig selector.
problem Structured matrix recovery for applications like recommender systems and computer vision.
method Non-asymptotic analysis of generalized Dantzig selector for estimation of generally structured matrices.
result Estimation error can be expressed in terms of geometric measures of suitable sets.
Paper uses SDP for community detection with side information.
problem Community detection in graphs with additional non-graph data.
method Formulates SDP relaxation for maximum likelihood node labeling with side information.
result SDP achieves same exact recovery threshold as maximum likelihood with side information.
Paper reconciles minimax rates and optimal recovery rates for noisy observations.
problem Estimating a function from noisy observations.
method Develops NLA minimax rates for Besov classes in L q L_q L q -norms. result NLA minimax rates continuously depend on noise level and match optimal recovery rates as noise decreases.
Efficient algorithm for robust recovery in stochastic block models.
problem Robust recovery in stochastic block models.
method Convex optimization framework, addressing optimization landscape challenges.
result Achieves robust recovery without a price of robustness.
RBM learns in high dimensions via AMP and GD, reaching optimal weak recovery.
problem Learning from high-dimensional data with RBM.
method AMP and GD analysis for RBM training in high dimensions.
result RBM reaches optimal weak recovery threshold in spiked covariance model.
This paper analyzes PCA for noisy, non-uniform data.
problem PCA's performance with heteroscedastic noise.
method Asymptotic analysis of PCA for noisy, non-uniform data.
result Prediction of PCA recovery in noisy, heteroscedastic samples.
Study spectral estimators for multi-index models to recover low-dimensional signal subspaces.
problem Recovering low-dimensional signal subspaces in multi-index models.
method Spectral estimators for multi-index models.
result Precise asymptotic characterization of spectral methods' performance, revealing a phase transition for weak recovery.
New algorithm recovers matrices with unknown correspondences.
problem Recovering matrices from observations with unknown correspondences.
method Solves a nuclear norm minimization problem via proximal gradient with a Max-Oracle.
result Achieves state-of-the-art performance and high accuracy in recovering ground-truth correspondences.
AOLS improves sparse linear regression with lower costs and better recovery.
problem Inferring sparse vectors from random linear combinations.
method Accelerated Orthogonal Least-Squares (AOLS) algorithm.
result AOLS achieves lower sampling complexity and better recovery probability.
This paper sets fundamental limits for rank-one matrix estimation with varying noise levels.
problem Estimating a rank-one matrix from Gaussian observations with different noise levels across blocks.
method Novel reduction from heterogeneous noise to homogeneous noise, proving asymptotic error bounds.
result Asymptotically exact formulas for minimum mean-squared error in estimating rank-one matrix and factors.
Proposes a new method for selecting regularization parameters in sparse precision matrix estimation.
problem Selecting an appropriate regularization parameter for sparse precision matrix estimation.
method Developed a closed-form matrix-valued regularization parameter based on the sampling distribution of optimality conditions.
result The proposed method achieves comparable estimation accuracy and superior support recovery to cross-validation, with significant runtime improvements.
New method selects variables in groups with few nonzeros, improving support recovery.
problem Structured variable selection with sparse patterns across groups.
method Composite norm and proximal algorithm for exclusive group sparsity.
result Asymptotic consistency in signed support recovery under conventional assumptions.
The binary symmetric stochastic block model deals with a random graph of n n n vertices partitioned into two equal-sized clusters, such that each pair of vertices is connected independently with probability p p p within clusters and q q q across clusters. In the asymptotic regime of p = a log n / n p=a \log n/n p = a log n / n and q = b log n / n q=b \log n/n q = b log n / n for fixe…
The paper explores how language models can provide reliable state measurements without being interpreted as beliefs.
problem How to use language models to reliably infer states without misinterpreting them as beliefs.
method Developed a semantic map and semiparametric inverse to link language probabilities to state probabilities, avoiding hidden models.
result Conditions for existence, identification, stable recovery, and uniform stability of posterior states from observable language probabilities.
For the problems of low-rank matrix completion, the efficiency of the widely-used nuclear norm technique may be challenged under many circumstances, especially when certain basis coefficients are fixed, for example, the low-rank correlation matrix completion in various fields such as the financial market and the low-ra…
Geometric framework links clustering accuracy to structural recovery.
problem Understanding the trade-off between robustness and sensitivity in clustering.
method Develops a clustering condition number to compare within-cluster scale to the minimum loss increase required to move a point across a cluster boundary.
result Sharp phase transitions for exact recovery under different objectives, providing geometric principle for interpreting low objective values.
New method avoids spurious critical points for low-rank matrix recovery.
problem Low-rank matrix recovery problems on Riemannian manifold.
method Riemannian gradient descent with random initialization.
result Riemannian gradient descent avoids spurious critical points and converges nearly linearly.
Recovering hidden influence networks from cascade data using Jacobian-based machine learning.
problem Recovering influence networks behind dynamic cascades.
method CascadeNet, a Jacobian-based machine learning framework.
result CascadeNet achieves high accuracy in network recovery.
Study sparse function recovery from indirect noisy observations using ℓ 1 \ell^1 ℓ 1 -regularization.
problem Recovering sparse functions from indirect, noisy observations.
method Proposes an ℓ 1 \ell^1 ℓ 1 -regularized empirical risk minimizer and analyzes its statistical properties. result Established almost-sure consistency and derived high-probability convergence rates in prediction and ℓ 1 \ell^1 ℓ 1 norms. AMP algorithm for matrix tensor product model provides recovery conditions.
problem Generalization of standard spiked matrix models with multiple pairwise observations.
method Approximate message passing with optimal weighing and combining of estimates.
result Asymptotically exact performance description and necessary/sufficient recovery conditions.
This paper addresses the problem of inferring sparse causal networks modeled by multivariate auto-regressive (MAR) processes. Conditions are derived under which the Group Lasso (gLasso) procedure consistently estimates sparse network structure. The key condition involves a "false connection score." In particular, we sh…
Paper uses Stochastic Mirror Descent for large-scale sparse recovery problems.
problem Statistical estimation of high-dimensional sparse parameters.
method Non-Euclidean Composite Stochastic Mirror Descent (CSMD) algorithm for solving penalized stochastic optimization problems.
result The proposed algorithm achieves optimal convergence in sparse Generalized Linear Regression problems.