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Efficient Certificates of Anti-Concentration Beyond Gaussians
A set of high dimensional points $X=\{x_1, x_2,\ldots, x_n\} \subset R^d$ in isotropic position is said to be $\delta$-anti concentrated if for every direction $v$, the fraction of points in $X$ satisfying $|\langle x_i, v \rangle |\leq \delta$ is at most $O(\delta)$.
Structure learning of Hamiltonians from real-time evolution
We study the problem of Hamiltonian structure learning from real-time evolution: given the ability to apply $e^{-\mathrm{i} Ht}$ for an unknown local Hamiltonian $H = \sum_{a = 1}^m \lambda_a E_a$ on $n$ qubits, the goal is to recover $H$.
A quasi-polynomial time algorithm for Multi-Dimensional Scaling via LP hierarchies
Multi-dimensional Scaling (MDS) is a family of methods for embedding an $n$-point metric into low-dimensional Euclidean space.
Learning quantum Hamiltonians at any temperature in polynomial time
Anshu, Arunachalam, Kuwahara, and Soleimanifar (arXiv:2004. 07266) gave an algorithm to learn a Hamiltonian on $n$ qubits to precision $\epsilon$ with only polynomially many copies of the Gibbs state, but which takes exponential time.
Tensor Decompositions Meet Control Theory: Learning General Mixtures of Linear Dynamical Systems
In this work we give a new approach to learning mixtures of linear dynamical systems that is based on tensor decompositions.
A Near-Linear Time Algorithm for the Chamfer Distance
For any two point sets $A, B \subset \mathbb{R}^d$ of size up to $n$, the Chamfer distance from $A$ to $B$ is defined as $\text{CH}(A, B)=\sum_{a \in A} \min_{b \in B} d_X(a, b)$, where $d_X$ is the underlying distance measure (e. g., the Euclidean or Manhattan distance).
Krylov Methods are (nearly) Optimal for Low-Rank Approximation
In particular, for Spectral LRA, we show that any algorithm requires $\Omega\left(\log(n)/\varepsilon^{1/2}\right)$ matrix-vector products, exactly matching the upper bound obtained by Krylov methods [MM15, BCW22].
A New Approach to Learning Linear Dynamical Systems
Linear dynamical systems are the foundational statistical model upon which control theory is built.
Sub-quadratic Algorithms for Kernel Matrices via Kernel Density Estimation
We build on the recent Kernel Density Estimation framework, which (after preprocessing in time subquadratic in $n$) can return estimates of row/column sums of the kernel matrix.
Low-Rank Approximation with $1/ε^{1/3}$ Matrix-Vector Products
For the special cases of $p=2$ (Frobenius norm) and $p = \infty$ (Spectral norm), Musco and Musco (NeurIPS 2015) obtained an algorithm based on Krylov methods that uses $\tilde{O}(k/\sqrt{\epsilon})$ matrix-vector products, improving on the na\"ive $\tilde{O}(k/\epsilon)$ dependence obtainable by the power method, where $\tilde{O}$ suppresses poly$(\log(dk/\epsilon))$ factors.
Learning a Latent Simplex in Input-Sparsity Time
We consider the problem of learning a latent $k$-vertex simplex $K\subset\mathbb{R}^d$, given access to $A\in\mathbb{R}^{d\times n}$, which can be viewed as a data matrix with $n$ points that are obtained by randomly perturbing latent points in the simplex $K$ (potentially beyond $K$).
Learning a Latent Simplex in Input Sparsity Time
Bhattacharyya and Kannan (SODA 2020) give an algorithm for learning such a $k$-vertex latent simplex in time roughly $O(k\cdot\text{nnz}(\mathbf{A}))$, where $\text{nnz}(\mathbf{A})$ is the number of non-zeros in $\mathbf{A}$.
Robustly Learning Mixtures of $k$ Arbitrary Gaussians
We give a polynomial-time algorithm for the problem of robustly estimating a mixture of $k$ arbitrary Gaussians in $\mathbb{R}^d$, for any fixed $k$, in the presence of a constant fraction of arbitrary corruptions.
Robust Linear Regression: Optimal Rates in Polynomial Time
We obtain robust and computationally efficient estimators for learning several linear models that achieve statistically optimal convergence rate under minimal distributional assumptions.
Outlier-Robust Clustering of Non-Spherical Mixtures
Concretely, our algorithm takes input an $\epsilon$-corrupted sample from a $k$-GMM and whp in $d^{\text{poly}(k/\eta)}$ time, outputs an approximate clustering that misclassifies at most $k^{O(k)}(\epsilon+\eta)$ fraction of the points whenever every pair of mixture components are separated by $1-\exp(-\text{poly}(k/\eta)^k)$ in total variation (TV) distance.
List-Decodable Subspace Recovery: Dimension Independent Error in Polynomial Time
As a result, in addition to Gaussians, our algorithm applies to the uniform distribution on the hypercube and $q$-ary cubes and arbitrary product distributions with subgaussian marginals.
Robust and Sample Optimal Algorithms for PSD Low-Rank Approximation
Our main result is to resolve this question by obtaining an optimal algorithm that queries $O(nk/\epsilon)$ entries of $A$ and outputs a relative-error low-rank approximation in $O(n(k/\epsilon)^{\omega-1})$ time.
Learning Two Layer Rectified Neural Networks in Polynomial Time
Given $n$ samples as a matrix $\mathbf{X} \in \mathbb{R}^{d \times n}$ and the (possibly noisy) labels $\mathbf{U}^* f(\mathbf{V}^* \mathbf{X}) + \mathbf{E}$ of the network on these samples, where $\mathbf{E}$ is a noise matrix, our goal is to recover the weight matrices $\mathbf{U}^*$ and $\mathbf{V}^*$.
Robust Communication-Optimal Distributed Clustering Algorithms
In this work, we study the $k$-median and $k$-means clustering problems when the data is distributed across many servers and can contain outliers.
A Novel Feature Selection and Extraction Technique for Classification
This paper presents a versatile technique for the purpose of feature selection and extraction - Class Dependent Features (CDFs).
