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We establish new hardness results for decision tree optimization problems, adding to a line of work that dates back to Hyafil and Rivest in 1976. We prove, under randomized ETH, superpolynomial lower bounds for two basic problems: given an…
We present a classical algorithm that, for any 3D geometrically-local, polylogarithmic-depth quantum circuit $C$ acting on $n$ qubits, and any bit string $x\in\{0,1\}^n$, can compute the quantity $|< x |C|0^{\otimes n}>|^2$ to within any…
We revisit the problem of learning mixtures of spherical Gaussians. Given samples from mixture $\frac{1}{k}\sum_{j=1}^{k}\mathcal{N}(\mu_j, I_d)$, the goal is to estimate the means $\mu_1, \mu_2, \ldots, \mu_k \in \mathbb{R}^d$ up to a…
Discriminative linear models are a popular tool in machine learning. These can be generally divided into two types: The first is linear classifiers, such as support vector machines, which are well studied and provide state-of-the-art…
We provide new results for noise-tolerant and sample-efficient learning algorithms under $s$-concave distributions. The new class of $s$-concave distributions is a broad and natural generalization of log-concavity, and includes many…
The aim of the paper is to propose a bounded-error quantum polynomial time (BQP) algorithm for the max-bisection and the min-bisection problems. The max-bisection and the min-bisection problems are fundamental NP-hard problems. Given a…
We introduce the notion of a reproducible algorithm in the context of learning. A reproducible learning algorithm is resilient to variations in its samples -- with high probability, it returns the exact same output when run on two samples…
The $2 \rightarrow q$ norm of a matrix $X \in \mathbb{R}^{n \times d}$ is defined as $\lVert X \rVert_{2 \rightarrow q} = \sup_{\lVert v \rVert_2 = 1} \lVert Xv \rVert_q$. We give polynomial-time multiplicative approximation algorithms for…
No polynomial-time algorithm is known to test whether a sparse polynomial G divides another sparse polynomial $F$. While computing the quotient Q=F quo G can be done in polynomial time with respect to the sparsities of F, G and Q, this is…
We study the fundamental problems of agnostically learning halfspaces and ReLUs under Gaussian marginals. In the former problem, given labeled examples $(\mathbf{x}, y)$ from an unknown distribution on $\mathbb{R}^d \times \{ \pm 1\}$,…
Matrix Factorization has emerged as a widely adopted framework for modeling data exhibiting low-rank structures. To address challenges in manifold learning, this paper presents a subspace-constrained quadratic matrix factorization model.…
We present several polynomial- and quasipolynomial-time approximation schemes for a large class of generalized operator norms. Special cases include the $2\rightarrow q$ norm of matrices for $q>2$, the support function of the set of…
A roadmap for a semi-algebraic set $S$ is a curve which has a non-empty and connected intersection with all connected components of $S$. Hence, this kind of object, introduced by Canny, can be used to answer connectivity queries (with…
We give two results on PAC learning DNF formulas using membership queries in the challenging "distribution-free" learning framework, where learning algorithms must succeed for an arbitrary and unknown distribution over $\{0,1\}^n$. (1) We…
This work establishes a novel link between the problem of PAC-learning high-dimensional graphical models and the task of (efficient) counting and sampling of graph structures, using an online learning framework. We observe that if we apply…
We study the problem of learning mixtures of linear classifiers under Gaussian covariates. Given sample access to a mixture of $r$ distributions on $\mathbb{R}^n$ of the form $(\mathbf{x},y_{\ell})$, $\ell\in [r]$, where…
How can non-communicating agents learn to share congested resources efficiently? This is a challenging task when the agents can access the same resource simultaneously (in contrast to multi-agent multi-armed bandit problems) and the…
We give sublinear-time approximation algorithms for some optimization problems arising in machine learning, such as training linear classifiers and finding minimum enclosing balls. Our algorithms can be extended to some kernelized versions…
We present a PTAS for agnostically learning halfspaces w.r.t. the uniform distribution on the $d$ dimensional sphere. Namely, we show that for every $\mu>0$ there is an algorithm that runs in time $\mathrm{poly}(d,\frac{1}{\epsilon})$, and…
Efficiently learning mixture of Gaussians is a fundamental problem in statistics and learning theory. Given samples coming from a random one out of k Gaussian distributions in Rn, the learning problem asks to estimate the means and the…