Paper develops a method for estimating spectral density matrices in high-dimensional time series.
problem Estimating spectral density matrices in high-dimensional time series.
method Thresholded versions of averaged periodograms for regularized estimation.
result Consistent estimation of spectral density matrices possible under high-dimensional regime.
Spectral method detects communities in sparse hypergraphs, achieving detection threshold.
problem Community detection in sparse hypergraphs.
method Non-backtracking operator and spectral approach.
result Spectral method achieves detection threshold for sparse HSBMs.
Solves community detection in sparse hypergraphs above a threshold.
problem Community detection in sparse hypergraphs.
method Generalization of Massoulié's method for sparse random graphs to random hypergraphs.
result Above the threshold, a spectral algorithm constructs a partition correlated with the true partition.
We derive a lower bound to the spectral threshold of the Dirichlet Laplacian in tubular neighbourhoods of constant radius about complete surfaces. This lower bound is given by the lowest eigenvalue of a one-dimensional operator depending on the radius and principal curvatures of the reference surface. Moreover, we show…
The paper finds optimal threshold strategies for insurance companies with a positive terminal value at creeping ruin.
problem Optimizing dividend payments in an insurance company's surplus process with a positive terminal value at creeping ruin.
method Using fluctuation theory, the paper derives explicit formulas for the objective function and shows the optimality of threshold strategies.
result Threshold strategies are optimal for the dividend optimization problem under certain conditions.
New Bethe-Hessian method improves community detection in sparse networks.
problem Detect communities in sparse networks efficiently.
method Spectral clustering using the Bethe-Hessian matrix.
result Bethe-Hessian consistently estimates block number above Kesten-Stigum threshold.
Optimal stopping strategy for a Lévy process near its supremum.
problem Predicting optimal stopping distance for a Lévy process.
method Characterization using scale functions and threshold analysis.
result Non-trivial stopping strategy based on a threshold.
High-dimensional models become unstable when sample size falls below a critical level, leading to a phase transition.
problem Instability in high-dimensional learning models when sample size is insufficient.
method Proved the necessity of a Fisher eigenvalue threshold for stability, introduced Fisher floor for verification.
result A sharp phase transition between reliable concentration and inevitable failure in high-dimensional learning.
Paper studies community detection in censored hypergraphs using information theory.
problem Community detection in censored hypergraphs with missing values.
method Information-theoretic approach, polynomial-time algorithm, spectral algorithm with refinement.
result Derives information-theoretic threshold for exact recovery of community structure.
FSPA bypasses eigenvalue estimation for quantum PCA, achieving optimal complexity and robustness.
problem Quantum PCA eigenvalue estimation is computationally expensive and prone to errors.
method Filtered Spectral Projection Algorithm (FSPA) that projects onto the dominant spectral subspace directly.
result FSPA achieves optimal complexity and robustness, outperforming classical methods.
We study a spectral initialization method that serves a key role in recent work on estimating signals in nonconvex settings. Previous analysis of this method focuses on the phase retrieval problem and provides only performance bounds. In this paper, we consider arbitrary generalized linear sensing models and present a …
Paper improves gas species identification in complex mixtures using neural networks.
problem Identifying gas species in multi-gas mixtures with high accuracy.
method Multi-label neural networks with optimal thresholding for IR spectroscopy.
result Optimal thresholding improves classification performance over conventional methods.
Clustering explores meaningful patterns in the non-labeled data sets. Cluster Ensemble Selection (CES) is a new approach, which can combine individual clustering results for increasing the performance of the final results. Although CES can achieve better final results in comparison with individual clustering algorithms…
Sparse spectral decomposition identifies overlapping communities in networks.
problem Estimating overlapping community memberships in networks where nodes can belong to multiple communities.
method Sparse principal subspace estimation with iterative thresholding.
result The fixed point of the algorithm corresponds to correct node memberships under the stochastic block model.
New method detects communities in complex hypergraphs, matching theoretical limits.
problem Detecting communities in non-uniform hypergraphs with varying hyperedge sizes.
method Developed a spectral theory for weighted non-backtracking operators on non-uniform hypergraphs.
result Achieved the Kesten-Stigum bound for weak recovery in a general class of non-uniform HSBMs.
Optimality of threshold strategies proven for Lévy models with discounting.
problem Proving optimality of threshold strategies in Lévy models with discounting.
method Average problem approach to prove optimality of threshold strategies for Lévy models with continuous additive functional discounting.
result Simpler and neater proofs for qualitative properties of optimal thresholds in recursive optimal stopping problems.
Consider the optimal dividend problem for an insurance company whose uncontrolled surplus precess evolves as a spectrally negative Levy process. We assume that dividends are paid to the shareholders according to admissible strategies whose dividend rate is bounded by a constant. The objective is to find a dividend poli…
Gradient descent near stability threshold exhibits sharpness oscillations.
problem Understanding sharpness behavior near stability threshold in non-Euclidean norms.
method Interpreted EoS through Directional Smoothness and generalized sharpness under arbitrary norms.
result Non-Euclidean GD with generalized sharpness shows sharpness oscillations near 2/η. Spectral embedding improves with edge weight transformations.
problem Improving spectral embedding for weighted networks.
method Analyzed edge weight transformations for spectral embedding quality.
result Transformations like tempering or thresholding can significantly enhance spectral embedding.
We consider the problem of clustering a set of high-dimensional data points into sets of low-dimensional linear subspaces. The number of subspaces, their dimensions, and their orientations are unknown. We propose a simple and low-complexity clustering algorithm based on thresholding the correlations between the data po…
Optimal dividend strategy found for risk models with regime switching.
problem Optimal dividend strategy for spectrally negative Markov additive models with regime switching.
method Introduced an auxiliary problem and transformed the original problem into a local optimization problem.
result The refraction-reflection strategy with regime-modulated thresholds is optimal.
Eigenvalues on spheres are compared to the unit round sphere, proving a sharp bound and equality condition.
problem Comparing eigenvalues of spheres under different metrics.
method Analyzing Laplace eigenvalues and using Alexandrov spaces.
result Equality of eigenvalues forces metrics to be isometric to the unit round sphere.
In this paper, we study the sensitivity of the spectral clustering based community detection algorithm subject to a Erdos-Renyi type random noise model. We prove phase transitions in community detectability as a function of the external edge connection probability and the noisy edge presence probability under a general…
Spectrally-truncated KRR outperforms full KRR for large data.
problem Computational intensity of KRR for large datasets.
method Spectrally truncating the kernel matrix to its largest r eigenvalues. result Spectrally-truncated KRR can outperform full KRR for all finite samples above a threshold.
Spectral clustering identifies clusters of multivariate extremes.
problem Analyzing the dependence structure of multivariate extremes.
method Spectral clustering based on a random k-nearest neighbor graph. result Spectral clustering can consistently identify clusters of multivariate extremes under certain conditions.
Optimal spectral method found for inhomogeneous spiked Wigner model.
problem Structured noise in learning scenarios.
method Random matrix theory and spectral analysis.
result Optimal threshold for phase transition in block-structured Wigner model.
Equivalence of norms on manifolds with curvature bounds established.
problem Establishing equivalence of norms on manifolds with bounded sectional curvature.
method Using spectral projector and thickness condition for subsets.
result Constant in equivalence depends only on manifold dimension, curvature bounds, and frequency threshold.
Subspace clustering refers to the problem of clustering high-dimensional data points into a union of low-dimensional linear subspaces, where the number of subspaces, their dimensions and orientations are all unknown. In this paper, we propose a variation of the recently introduced thresholding-based subspace clustering…
Gradient descent near stability threshold shows sharpness oscillations.
problem Understanding sharpness and stability in non-Euclidean norms during gradient descent.
method Interpreted EoS through Directional Smoothness, defined generalized sharpness for arbitrary norms.
result Non-Euclidean GD exhibits sharpness oscillations around the stability threshold.
Study optimal spectral estimator for semi-supervised node classification.
problem Semi-supervised node classification on CSBM with limited labels.
method Spectral estimator inspired by PCA, graph ridge regression, GCN.
result Achieves information-theoretical threshold for exact recovery.
LoRA fine-tuning creates intruder dimensions that can cause forgetting, and a new law predicts when this happens.
problem Predicting when LoRA fine-tuning creates intruder dimensions that can cause catastrophic forgetting.
method Derived a per-layer critical update strength s∗ and an exact secular-equation characterization of the updated spectrum. result The law localizes the empirical threshold within a factor of two on 82% of layers and separates intruder-bearing from intruder-free layers at deployment.
Phase retrieval requires at least d+o(d) measurements to recover signals with high probability.
problem Recovering signals from quadratic measurements with noisy data.
method Used Gaussian sensing vectors and spectral methods to analyze the minimum number of measurements needed.
result A sharp phase transition occurs at n = d+o(d), where a simple spectral estimator achieves positive correlation.
Paper finds exact recovery threshold in general hypergraph model.
problem Exact recovery of communities in general hypergraph model.
method Developed a two-stage polynomial-time algorithm for exact recovery.
result Sharp threshold for exact recovery in terms of generalized Chernoff-Hellinger divergence.
A fast spectral algorithm detects community structure in evolving graphs.
problem Detecting community structure in time-evolving sparse graphs.
method Extension of the Bethe-Hessian matrix for spectral community detection.
result The algorithm reaches the optimal detectability threshold and outperforms other methods.
Study optimizes inventory restocking for demand processes with exponential replenishment.
problem Optimizing inventory restocking for demand processes with exponential replenishment.
method Developed periodic barrier replenishment policies for spectrally positive Lévy demand processes.
result Optimal policies and value functions are concisely written in terms of scale functions.
We study the spectral gap of the Erdős--Rényi random graph through the connectivity threshold. In particular, we show that for any fixed δ>0 if p≥n(1/2+δ)logn, then the normalized graph Laplacian of an Erdős--Rényi graph has all of its nonzero eigenvalues tightly concentrated around 1. We est…
The interplay between computational efficiency and statistical accuracy in high-dimensional inference has drawn increasing attention in the literature. In this paper, we study computational and statistical boundaries for submatrix localization. Given one observation of (one or multiple non-overlapping) signal submatrix…
Estimates latent inner products from an anisotropic Gaussian graph with improved spectral method.
problem Recovering latent inner products from an anisotropic Gaussian random geometric graph.
method Doubly centered adjacency matrix, rank-d spectral approximation, Hermite expansion, decoupling argument.
result Estimator achieves mean squared error rate matching state of the art for isotropic case and ill-conditioned covariance matrices.
The study reveals the spectral structure of attention layers and its implications for generalization.
problem Understanding the spectral structure and generalization of trained attention layers.
method Empirical risk minimization in a single-head tied-attention layer, using random matrix theory, spin-glass theory, and approximate message passing.
result Exact high-dimensional characterization of training and test error, interpolation and recovery thresholds, and spectrum of the key and query matrices.
New algorithm improves low-rank matrix estimation accuracy.
problem Estimating low-rank matrices with noisy entries.
method Approximate Message Passing (AMP) combined with spectral initialization.
result Achieves Bayes-optimal accuracy above the spectral threshold.
In this paper we study the optimal dividend problem for a company whose surplus process evolves as a spectrally positive Levy process. This model including the dual model of the classical risk model and the dual model with diffusion as special cases. We assume that dividends are paid to the shareholders according to ad…
We study the fundamental limits on learning latent community structure in dynamic networks. Specifically, we study dynamic stochastic block models where nodes change their community membership over time, but where edges are generated independently at each time step. In this setting (which is a special case of several e…
New findings on community recovery in SBM with many communities.
problem Determining community recovery conditions in SBM with more than sqrt(n) communities.
method Constructing motifs and counting them to prove community recovery above the proposed threshold.
result Proving community recovery above the proposed threshold in SBM with K >= sqrt(n) communities.
New findings support a new community recovery threshold for Stochastic Block Model with many communities.
problem Recovering communities in Stochastic Block Model with more than sqrt(n) communities.
method Counting specific motifs to achieve polynomial-time community recovery above a new threshold.
result LDP fails below the new threshold, but polynomial-time recovery is possible above it.
The labeled stochastic block model is a random graph model representing networks with community structure and interactions of multiple types. In its simplest form, it consists of two communities of approximately equal size, and the edges are drawn and labeled at random with probability depending on whether their two en…
Enhanced matrix completion with nonconvex penalties for better predictive performance.
problem Matrix completion from a subset of observed entries with low-rank assumption.
method Proposes nonconvex regularization with spectral thresholding operators and scalable EM-flavored algorithms.
result Nonconvex regularization leads to better predictive performance than nuclear norm methods.
We consider the change-point detection problem of deciding, based on noisy measurements, whether an unknown signal over a given graph is constant or is instead piecewise constant over two connected induced subgraphs of relatively low cut size. We analyze the corresponding generalized likelihood ratio (GLR) statistics a…
This paper considers magnitude, asymptotics and duration of drawdowns for some Lévy processes. First, we revisit some existing results on the magnitude of drawdowns for spectrally negative Lévy processes using an approximation approach. For any spectrally negative Lévy process whose scale functions are well-behaved at …