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Found 19 papers, 0 papers with code
Fast algorithm for overcomplete order-3 tensor decomposition
We develop the first fast spectral algorithm to decompose a random third-order tensor over $\mathbb{R}^d$ of rank up to $O(d^{3/2}/\text{polylog}(d))$.
Beyond Parallel Pancakes: Quasi-Polynomial Time Guarantees for Non-Spherical Gaussian Mixtures
The reason is that such outliers can simulate exponentially small mixing weights even for mixtures with polynomially lower bounded mixing weights.
Robust recovery for stochastic block models
We develop an efficient algorithm for weak recovery in a robust version of the stochastic block model.
Consistent Estimation for PCA and Sparse Regression with Oblivious Outliers
For sparse regression, we achieve consistency for optimal sample size $n\gtrsim (k\log d)/\alpha^2$ and optimal error rate $O(\sqrt{(k\log d)/(n\cdot \alpha^2)})$ where $n$ is the number of observations, $d$ is the number of dimensions and $k$ is the sparsity of the parameter vector, allowing the fraction of inliers to be inverse-polynomial in the number of samples.
SoS Degree Reduction with Applications to Clustering and Robust Moment Estimation
We develop a general framework to significantly reduce the degree of sum-of-squares proofs by introducing new variables.
Sparse PCA: Algorithms, Adversarial Perturbations and Certificates
Despite a long history of prior works, including explicit studies of perturbation resilience, the best known algorithmic guarantees for Sparse PCA are fragile and break down under small adversarial perturbations.
Consistent regression when oblivious outliers overwhelm
Concretely, we show that the Huber loss estimator is consistent for every sample size $n= \omega(d/\alpha^2)$ and achieves an error rate of $O(d/\alpha^2n)^{1/2}$.
Estimating Rank-One Spikes from Heavy-Tailed Noise via Self-Avoiding Walks
For the case of Gaussian noise, the top eigenvector of the given matrix is a widely-studied estimator known to achieve optimal statistical guarantees, e. g., in the sense of the celebrated BBP phase transition.
High-dimensional estimation via sum-of-squares proofs
On one hand, there is a growing body of work utilizing sum-of-squares proofs for recovering solutions to polynomial systems when the system is feasible.
Outlier-robust moment-estimation via sum-of-squares
We develop efficient algorithms for estimating low-degree moments of unknown distributions in the presence of adversarial outliers.
Bayesian estimation from few samples: community detection and related problems
in constant average degree graphs---up to what we conjecture to be the computational threshold for this model.
Fast and robust tensor decomposition with applications to dictionary learning
We develop fast spectral algorithms for tensor decomposition that match the robustness guarantees of the best known polynomial-time algorithms for this problem based on the sum-of-squares (SOS) semidefinite programming hierarchy.
Exact tensor completion with sum-of-squares
We obtain the first polynomial-time algorithm for exact tensor completion that improves over the bound implied by reduction to matrix completion.
Polynomial-time Tensor Decompositions with Sum-of-Squares
We give new algorithms based on the sum-of-squares method for tensor decomposition.
Fast spectral algorithms from sum-of-squares proofs: tensor decomposition and planted sparse vectors
For tensor decomposition, we give an algorithm with running time close to linear in the input size (with exponent $\approx 1. 086$) that approximately recovers a component of a random 3-tensor over $\mathbb R^n$ of rank up to $\tilde \Omega(n^{4/3})$.
Tensor principal component analysis via sum-of-squares proofs
We study a statistical model for the tensor principal component analysis problem introduced by Montanari and Richard: Given a order-$3$ tensor $T$ of the form $T = \tau \cdot v_0^{\otimes 3} + A$, where $\tau \geq 0$ is a signal-to-noise ratio, $v_0$ is a unit vector, and $A$ is a random noise tensor, the goal is to recover the planted vector $v_0$.
Dictionary Learning and Tensor Decomposition via the Sum-of-Squares Method
We give a new approach to the dictionary learning (also known as "sparse coding") problem of recovering an unknown $n\times m$ matrix $A$ (for $m \geq n$) from examples of the form \[ y = Ax + e, \] where $x$ is a random vector in $\mathbb R^m$ with at most $\tau m$ nonzero coordinates, and $e$ is a random noise vector in $\mathbb R^n$ with bounded magnitude.
Sum-of-squares proofs and the quest toward optimal algorithms
Two recent developments, the Unique Games Conjecture (UGC) and the Sum-of-Squares (SOS) method, surprisingly suggest that this tailoring is not necessary and that a single efficient algorithm could achieve best possible guarantees for a wide range of different problems.
Rounding Sum-of-Squares Relaxations
Aside from being a natural relaxation, this is also motivated by a connection to the Small Set Expansion problem shown by Barak et al. (STOC 2012) and our results yield a certain improvement for that problem.
