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Found 16 papers, 4 papers with code
Statistically Optimal K-means Clustering via Nonnegative Low-rank Semidefinite Programming
In contrast, nonnegative matrix factorization (NMF) is a simple clustering algorithm widely used by machine learning practitioners, but it lacks a solid statistical underpinning and theoretical guarantees.
Fast and Accurate Estimation of Low-Rank Matrices from Noisy Measurements via Preconditioned Non-Convex Gradient Descent
Recently, the technique of preconditioning was shown to be highly effective at accelerating the local convergence of non-convex gradient descent when the measurements are noiseless.
Tight Certification of Adversarially Trained Neural Networks via Nonconvex Low-Rank Semidefinite Relaxations
Nevertheless, once a model has been adversarially trained, one often desires a certification that the model is truly robust against all future attacks.
Simple Alternating Minimization Provably Solves Complete Dictionary Learning
This paper focuses on complete dictionary learning problem, where the goal is to reparametrize a set of given signals as linear combinations of atoms from a learned dictionary.
Accelerating SGD for Highly Ill-Conditioned Huge-Scale Online Matrix Completion
The matrix completion problem seeks to recover a $d\times d$ ground truth matrix of low rank $r\ll d$ from observations of its individual elements.
Improved Global Guarantees for the Nonconvex Burer--Monteiro Factorization via Rank Overparameterization
We consider minimizing a twice-differentiable, $L$-smooth, and $\mu$-strongly convex objective $\phi$ over an $n\times n$ positive semidefinite matrix $M\succeq0$, under the assumption that the minimizer $M^{\star}$ has low rank $r^{\star}\ll n$.
Preconditioned Gradient Descent for Overparameterized Nonconvex Burer--Monteiro Factorization with Global Optimality Certification
We consider using gradient descent to minimize the nonconvex function $f(X)=\phi(XX^{T})$ over an $n\times r$ factor matrix $X$, in which $\phi$ is an underlying smooth convex cost function defined over $n\times n$ matrices.
Sharp Global Guarantees for Nonconvex Low-Rank Matrix Recovery in the Overparameterized Regime
Under the restricted isometry property (RIP), we prove, for the general overparameterized regime with $r^{\star}\le r$, that an RIP constant of $\delta<1/(1+\sqrt{r^{\star}/r})$ is sufficient for the inexistence of spurious local minima, and that $\delta<1/(1+1/\sqrt{r-r^{\star}+1})$ is necessary due to existence of counterexamples.
How Many Samples is a Good Initial Point Worth in Low-rank Matrix Recovery?
Optimizing the threshold over regions of the landscape, we see that for initial points around the ground truth, a linear improvement in the quality of the initial guess amounts to a constant factor improvement in the sample complexity.
On the Tightness of Semidefinite Relaxations for Certifying Robustness to Adversarial Examples
If the relaxation is loose, however, then the resulting certificate can be too conservative to be practically useful.
Large-Scale Traffic Signal Offset Optimization
The offset optimization problem seeks to coordinate and synchronize the timing of traffic signals throughout a network in order to enhance traffic flow and reduce stops and delays.
Optimization and Control Systems and Control Systems and Control
Sharp Restricted Isometry Bounds for the Inexistence of Spurious Local Minima in Nonconvex Matrix Recovery
Using the technique, we prove that in the case of a rank-1 ground truth, an RIP constant of $\delta<1/2$ is both necessary and sufficient for exact recovery from any arbitrary initial point (such as a random point).
How Much Restricted Isometry is Needed In Nonconvex Matrix Recovery?
When the linear measurements of an instance of low-rank matrix recovery satisfy a restricted isometry property (RIP)---i. e. they are approximately norm-preserving---the problem is known to contain no spurious local minima, so exact recovery is guaranteed.
Large-Scale Sparse Inverse Covariance Estimation via Thresholding and Max-Det Matrix Completion
The sparse inverse covariance estimation problem is commonly solved using an $\ell_{1}$-regularized Gaussian maximum likelihood estimator known as "graphical lasso", but its computational cost becomes prohibitive for large data sets.
Sparse Inverse Covariance Estimation for Chordal Structures
We have also derived a closed-form solution that is optimal when the thresholded sample covariance matrix has an acyclic structure.
Sparse Semidefinite Programs with Guaranteed Near-Linear Time Complexity via Dualized Clique Tree Conversion
Clique tree conversion solves large-scale semidefinite programs by splitting an $n\times n$ matrix variable into up to $n$ smaller matrix variables, each representing a principal submatrix of up to $\omega\times\omega$.
Optimization and Control
